Subgroups — C₆₇×#32904 #30712155

Order 17152.

1608 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#449×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1875	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#121×C₆₇	—	2	1	no	—
17	C₂×#121×C₆₇	—	2	1	no	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₄×#41×C₆₇	—	2	4	no	—
20	C₄×#41×C₆₇	—	2	4	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	—	2	1	no	—
24	C₂×C₄×D₄×C₆₇	—	2	1	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#117×C₆₇	—	2	1	no	—
28	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₆₇	—	2	1	no	—
30	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	#460×C₆₇	—	2	4	no	—
32	#459×C₆₇	—	2	4	no	—
33	#449×C₆₇	—	2	8	no	—
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#103×C₆₇	C₂²	1	2	yes	—
37	C₂×#102×C₆₇	C₂²	1	2	yes	—
38	C₂×#102×C₆₇	C₂²	1	2	yes	—
39	C₂×#102×C₆₇	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
41	#390×C₆₇	—	2	1	no	—
42	#390×C₆₇	—	2	1	no	—
43	#390×C₆₇	C₂²	1	2	yes	—
44	#384×C₆₇	C₂²	1	1	yes	characteristic
45	#340×C₆₇	C₂²	1	2	yes	—
46	#340×C₆₇	C₂²	1	2	yes	—
47	#336×C₆₇	C₂²	1	2	yes	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	D₄	1	4	yes	complemented
54	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	—	2	1	no	—
56	C₂²×D₄×C₆₇	—	2	2	no	—
57	C₂²×D₄×C₆₇	—	2	1	no	—
58	C₂²×D₄×C₆₇	—	2	1	no	—
59	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	—	2	1	no	—
62	C₂³×C₄×C₆₇	—	2	1	no	—
63	C₂³×C₄×C₆₇	—	2	1	no	—
64	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×SD₁₆×C₆₇	—	2	4	no	—
68	C₂×SD₁₆×C₆₇	—	2	4	no	—
69	C₂×SD₁₆×C₆₇	—	2	4	no	—
70	C₂×D₈×C₆₇	—	2	4	no	—
71	C₂×D₈×C₆₇	—	2	4	no	—
72	C₂×D₈×C₆₇	—	2	4	no	—
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
75	#127×C₆₇	—	2	2	no	—
76	#127×C₆₇	—	2	2	no	—
77	#121×C₆₇	—	2	4	no	—
78	#121×C₆₇	—	2	4	no	—
79	#121×C₆₇	—	4	4	no	—
80	#121×C₆₇	—	4	4	no	—
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	—	2	2	no	—
83	C₄×Q₈×C₆₇	—	2	2	no	—
84	C₄×Q₈×C₆₇	D₄	1	4	yes	complemented
85	C₄×D₄×C₆₇	—	2	8	no	—
86	C₄×D₄×C₆₇	—	2	8	no	—
87	C₄×D₄×C₆₇	—	2	2	no	—
88	C₄×D₄×C₆₇	—	2	2	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	D₄	1	4	yes	complemented
91	#117×C₆₇	—	2	8	no	—
92	#117×C₆₇	—	2	2	no	—
93	#117×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₄²×C₆₇	—	2	8	no	—
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	—	2	1	no	—
110	C₂×C₄²×C₆₇	—	2	1	no	—
111	C₂×C₄²×C₆₇	—	2	2	no	—
112	C₂×C₄²×C₆₇	—	2	2	no	—
113	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₄⋊C₈×C₆₇	—	2	2	no	—
118	C₄⋊C₈×C₆₇	—	2	2	no	—
119	#103×C₆₇	—	2	4	no	—
120	#102×C₆₇	—	2	4	no	—
121	#102×C₆₇	—	2	4	no	—
122	#102×C₆₇	—	2	4	no	—
123	C₂²⋊C₈×C₆₇	—	2	4	no	—
124	C₂⁴×C₆₇	—	2	1	no	—
125	C₂⁴×C₆₇	—	2	2	no	—
126	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
127	#41×C₆₇	—	2	8	no	—
128	#41×C₆₇	—	2	8	no	—
129	#41×C₆₇	—	2	8	no	—
130	#41×C₆₇	—	2	8	no	—
131	C₂×Q₈×C₆₇	—	2	1	no	—
132	C₂×Q₈×C₆₇	—	2	1	no	—
133	C₂×Q₈×C₆₇	—	2	1	no	—
134	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
135	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
136	C₂×Q₈×C₆₇	#41	1	2	yes	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	4	no	—
139	C₂×D₄×C₆₇	—	2	4	no	—
140	C₂×D₄×C₆₇	—	2	1	no	—
141	C₂×D₄×C₆₇	—	2	2	no	—
142	C₂×D₄×C₆₇	—	2	2	no	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	2	1	no	—
145	C₂×D₄×C₆₇	—	2	1	no	—
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	2	2	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	4	2	no	—
150	C₂×D₄×C₆₇	—	4	2	no	—
151	C₂×D₄×C₆₇	—	4	4	no	—
152	C₂×D₄×C₆₇	—	4	4	no	—
153	C₂×D₄×C₆₇	—	4	4	no	—
154	C₂×D₄×C₆₇	—	4	2	no	—
155	C₂×D₄×C₆₇	—	4	2	no	—
156	C₂×D₄×C₆₇	—	4	2	no	—
157	C₂×D₄×C₆₇	—	4	2	no	—
158	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₆₇	#41	1	2	yes	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	1	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	1	no	—
166	C₂²×C₄×C₆₇	—	2	1	no	—
167	C₂²×C₄×C₆₇	—	2	1	no	—
168	C₂²×C₄×C₆₇	—	2	1	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	2	2	no	—
173	C₂²×C₄×C₆₇	—	2	2	no	—
174	C₂²×C₄×C₆₇	—	2	4	no	—
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	2	2	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	4	1	no	—
180	C₂²×C₄×C₆₇	—	4	2	no	—
181	C₂²×C₄×C₆₇	—	4	2	no	—
182	C₂²×C₄×C₆₇	—	4	2	no	—
183	C₂²×C₄×C₆₇	—	4	2	no	—
184	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	#41	1	2	yes	—
191	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
192	SD₁₆×C₆₇	—	4	8	no	—
193	D₈×C₆₇	—	4	8	no	—
194	C₂×C₈×C₆₇	—	2	2	no	—
195	C₂×C₈×C₆₇	—	2	2	no	—
196	C₂×C₈×C₆₇	—	2	2	no	—
197	C₂×C₈×C₆₇	—	2	2	no	—
198	C₂×C₈×C₆₇	—	2	2	no	—
199	C₂×C₈×C₆₇	—	2	2	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	4	no	—
202	C₄⋊C₄×C₆₇	—	2	1	no	—
203	C₄⋊C₄×C₆₇	—	2	1	no	—
204	C₄⋊C₄×C₆₇	—	2	2	no	—
205	C₄⋊C₄×C₆₇	—	2	2	no	—
206	C₄⋊C₄×C₆₇	—	4	2	no	—
207	C₄⋊C₄×C₆₇	—	4	2	no	—
208	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
210	C₂²⋊C₄×C₆₇	—	2	4	no	—
211	C₂²⋊C₄×C₆₇	—	2	4	no	—
212	C₂²⋊C₄×C₆₇	—	2	4	no	—
213	C₂²⋊C₄×C₆₇	—	4	4	no	—
214	C₂²⋊C₄×C₆₇	—	4	4	no	—
215	C₂²⋊C₄×C₆₇	—	4	4	no	—
216	C₄²×C₆₇	—	2	4	no	—
217	C₄²×C₆₇	—	2	4	no	—
218	C₄²×C₆₇	—	2	2	no	—
219	C₄²×C₆₇	—	2	4	no	—
220	C₄²×C₆₇	—	2	2	no	—
221	C₄²×C₆₇	—	2	1	no	—
222	C₄²×C₆₇	—	2	1	no	—
223	C₄²×C₆₇	C₂×D₄	1	2	yes	—
224	C₄²×C₆₇	C₂×D₄	1	2	yes	—
225	C₂³×C₆₇	—	2	1	no	—
226	C₂³×C₆₇	—	2	2	no	—
227	C₂³×C₆₇	—	2	2	no	—
228	C₂³×C₆₇	—	2	1	no	—
229	C₂³×C₆₇	—	2	2	no	—
230	C₂³×C₆₇	—	2	1	no	—
231	C₂³×C₆₇	—	2	1	no	—
232	C₂³×C₆₇	—	2	1	no	—
233	C₂³×C₆₇	—	2	1	no	—
234	C₂³×C₆₇	—	4	1	no	—
235	C₂³×C₆₇	—	4	1	no	—
236	C₂³×C₆₇	—	4	1	no	—
237	C₂³×C₆₇	—	4	1	no	—
238	C₂³×C₆₇	—	4	2	no	—
239	C₂³×C₆₇	—	4	2	no	—
240	C₂³×C₆₇	—	4	2	no	—
241	C₂³×C₆₇	—	4	2	no	—
242	C₂³×C₆₇	—	4	2	no	—
243	C₂³×C₆₇	—	4	2	no	—
244	C₂³×C₆₇	—	4	2	no	—
245	C₂³×C₆₇	#121	1	2	yes	—
246	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	Q₈×C₆₇	—	2	2	no	—
248	Q₈×C₆₇	—	2	2	no	—
249	Q₈×C₆₇	—	2	2	no	—
250	Q₈×C₆₇	#121	1	4	yes	complemented
251	D₄×C₆₇	—	2	16	no	—
252	D₄×C₆₇	—	2	2	no	—
253	D₄×C₆₇	—	2	2	no	—
254	D₄×C₆₇	—	2	2	no	—
255	D₄×C₆₇	—	4	2	no	—
256	D₄×C₆₇	—	4	2	no	—
257	D₄×C₆₇	—	4	2	no	—
258	D₄×C₆₇	—	4	2	no	—
259	D₄×C₆₇	—	8	4	no	—
260	D₄×C₆₇	—	8	8	no	—
261	D₄×C₆₇	—	8	4	no	—
262	D₄×C₆₇	#121	1	4	yes	complemented
263	C₂×C₄×C₆₇	—	2	16	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	1	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	1	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	1	no	—
271	C₂×C₄×C₆₇	—	2	1	no	—
272	C₂×C₄×C₆₇	—	2	1	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	1	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	1	no	—
281	C₂×C₄×C₆₇	—	2	1	no	—
282	C₂×C₄×C₆₇	—	4	4	no	—
283	C₂×C₄×C₆₇	—	4	8	no	—
284	C₂×C₄×C₆₇	—	4	1	no	—
285	C₂×C₄×C₆₇	—	4	1	no	—
286	C₂×C₄×C₆₇	—	4	1	no	—
287	C₂×C₄×C₆₇	—	4	1	no	—
288	C₂×C₄×C₆₇	—	4	4	no	—
289	C₂×C₄×C₆₇	—	4	8	no	—
290	C₂×C₄×C₆₇	—	4	2	no	—
291	C₂×C₄×C₆₇	—	4	2	no	—
292	C₂×C₄×C₆₇	—	4	2	no	—
293	C₂×C₄×C₆₇	#120	1	2	yes	—
294	C₂×C₄×C₆₇	#120	1	2	yes	—
295	C₂×C₄×C₆₇	#121	1	2	yes	—
296	C₂×C₄×C₆₇	#135	1	2	yes	—
297	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
298	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
299	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
303	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
304	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
305	C₈×C₆₇	—	4	2	no	—
306	C₈×C₆₇	—	4	2	no	—
307	C₂²×C₆₇	—	2	2	no	—
308	C₂²×C₆₇	—	2	4	no	—
309	C₂²×C₆₇	—	2	2	no	—
310	C₂²×C₆₇	—	2	2	no	—
311	C₂²×C₆₇	—	4	1	no	—
312	C₂²×C₆₇	—	4	2	no	—
313	C₂²×C₆₇	—	4	1	no	—
314	C₂²×C₆₇	—	4	2	no	—
315	C₂²×C₆₇	—	4	1	no	—
316	C₂²×C₆₇	—	4	2	no	—
317	C₂²×C₆₇	—	4	1	no	—
318	C₂²×C₆₇	—	4	2	no	—
319	C₂²×C₆₇	—	4	2	no	—
320	C₂²×C₆₇	—	4	2	no	—
321	C₂²×C₆₇	—	8	2	no	—
322	C₂²×C₆₇	—	8	2	no	—
323	C₂²×C₆₇	—	8	2	no	—
324	C₂²×C₆₇	—	8	2	no	—
325	C₂²×C₆₇	—	8	2	no	—
326	C₂²×C₆₇	—	8	2	no	—
327	C₂²×C₆₇	—	8	2	no	—
328	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
329	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
330	C₂²×C₆₇	#525	1	1	yes	characteristic
331	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
332	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂²×C₆₇	#575	1	1	yes	characteristic
335	C₄×C₆₇	—	2	2	no	—
336	C₄×C₆₇	—	2	4	no	—
337	C₄×C₆₇	—	2	2	no	—
338	C₄×C₆₇	—	2	2	no	—
339	C₄×C₆₇	—	4	1	no	—
340	C₄×C₆₇	—	4	2	no	—
341	C₄×C₆₇	—	4	2	no	—
342	C₄×C₆₇	—	8	2	no	—
343	C₄×C₆₇	#449	1	4	yes	complemented
344	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
345	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
346	C₄×C₆₇	#525	1	1	yes	characteristic
347	C₄×C₆₇	#525	1	1	yes	characteristic
348	#32904	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
349	C₂×C₆₇	—	4	2	no	—
350	C₂×C₆₇	—	4	4	no	—
351	C₂×C₆₇	—	16	2	no	—
352	C₂×C₆₇	#2847	1	1	yes	characteristic
353	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
354	C₂×C₆₇	#3006	1	1	yes	characteristic
355	C₂×C₆₇	#3035	1	1	yes	characteristic
356	C₂×C₆₇	#3038	1	1	yes	characteristic
357	C₂×C₆₇	#3106	1	1	yes	characteristic
358	C₂×C₆₇	#3115	1	1	yes	characteristic
359	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
360	C₂×#460	C₂×C₆₇	1	1	yes	characteristic, complemented
361	C₂×#459	C₂×C₆₇	1	1	yes	characteristic, complemented
362	C₂×#449	C₂×C₆₇	1	2	yes	complemented
363	#2856	C₂×C₆₇	1	1	yes	characteristic, complemented
364	#1880	C₂×C₆₇	1	2	yes	complemented
365	#1879	C₂×C₆₇	1	2	yes	complemented
366	#1875	C₂×C₆₇	1	2	yes	complemented
367	#1803	C₂×C₆₇	1	1	yes	characteristic, complemented
368	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
369	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
370	C₆₇	#32904	1	1	yes	characteristic, direct product factor, Sylow
371	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
372	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
373	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
374	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
375	C₂×#121	—	2	1	no	—
376	C₂×#121	—	2	1	no	—
377	C₂×#121	C₂²×C₆₇	1	2	yes	—
378	C₄×#41	—	2	4	no	—
379	C₄×#41	—	2	4	no	—
380	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
381	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
382	C₂×C₄×D₄	—	2	1	no	—
383	C₂×C₄×D₄	—	2	1	no	—
384	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
385	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
386	C₂×#117	—	2	1	no	—
387	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
388	C₂²×C₄²	—	2	1	no	—
389	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
390	#460	—	2	4	no	—
391	#459	—	2	4	no	—
392	#449	—	2	8	no	—
393	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
394	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
395	C₂×#103	C₂²×C₆₇	1	2	yes	—
396	C₂×#102	C₂²×C₆₇	1	2	yes	—
397	C₂×#102	C₂²×C₆₇	1	2	yes	—
398	C₂×#102	C₂²×C₆₇	1	2	yes	—
399	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
400	#390	—	2	1	no	—
401	#390	—	2	1	no	—
402	#390	C₂²×C₆₇	1	2	yes	—
403	#384	C₂²×C₆₇	1	1	yes	characteristic
404	#340	C₂²×C₆₇	1	2	yes	—
405	#340	C₂²×C₆₇	1	2	yes	—
406	#336	C₂²×C₆₇	1	2	yes	—
407	C₂×#41	—	2	2	no	—
408	C₂×#41	—	2	2	no	—
409	C₂×#41	—	2	2	no	—
410	C₂×#41	—	2	2	no	—
411	C₂×#41	—	2	2	no	—
412	C₂×#41	D₄×C₆₇	1	4	yes	complemented
413	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
414	C₂²×D₄	—	2	1	no	—
415	C₂²×D₄	—	2	2	no	—
416	C₂²×D₄	—	2	1	no	—
417	C₂²×D₄	—	2	1	no	—
418	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
419	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	—	2	1	no	—
422	C₂³×C₄	—	2	1	no	—
423	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
424	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
425	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
426	C₂×SD₁₆	—	2	4	no	—
427	C₂×SD₁₆	—	2	4	no	—
428	C₂×SD₁₆	—	2	4	no	—
429	C₂×D₈	—	2	4	no	—
430	C₂×D₈	—	2	4	no	—
431	C₂×D₈	—	2	4	no	—
432	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
433	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
434	#127	—	2	2	no	—
435	#127	—	2	2	no	—
436	#121	—	2	4	no	—
437	#121	—	2	4	no	—
438	#121	—	4	4	no	—
439	#121	—	4	4	no	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	—	2	2	no	—
442	C₄×Q₈	—	2	2	no	—
443	C₄×Q₈	D₄×C₆₇	1	4	yes	complemented
444	C₄×D₄	—	2	8	no	—
445	C₄×D₄	—	2	8	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	—	2	2	no	—
448	C₄×D₄	—	2	2	no	—
449	C₄×D₄	D₄×C₆₇	1	4	yes	complemented
450	#117	—	2	8	no	—
451	#117	—	2	2	no	—
452	#117	—	2	2	no	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	—	2	1	no	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	—	2	1	no	—
457	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
458	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
459	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
460	C₂×C₂²⋊C₄	—	2	1	no	—
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	—	2	1	no	—
463	C₂×C₂²⋊C₄	—	2	2	no	—
464	C₂×C₂²⋊C₄	—	2	2	no	—
465	C₂×C₂²⋊C₄	—	2	2	no	—
466	C₂×C₄²	—	2	8	no	—
467	C₂×C₄²	—	2	1	no	—
468	C₂×C₄²	—	2	1	no	—
469	C₂×C₄²	—	2	1	no	—
470	C₂×C₄²	—	2	2	no	—
471	C₂×C₄²	—	2	2	no	—
472	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
473	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
474	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
475	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
476	C₄⋊C₈	—	2	2	no	—
477	C₄⋊C₈	—	2	2	no	—
478	#103	—	2	4	no	—
479	#102	—	2	4	no	—
480	#102	—	2	4	no	—
481	#102	—	2	4	no	—
482	C₂²⋊C₈	—	2	4	no	—
483	C₂⁴	—	2	1	no	—
484	C₂⁴	—	2	2	no	—
485	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
486	#41	—	2	8	no	—
487	#41	—	2	8	no	—
488	#41	—	2	8	no	—
489	#41	—	2	8	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
494	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
495	C₂×Q₈	#41×C₆₇	1	2	yes	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	4	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	2	2	no	—
508	C₂×D₄	—	4	2	no	—
509	C₂×D₄	—	4	2	no	—
510	C₂×D₄	—	4	4	no	—
511	C₂×D₄	—	4	4	no	—
512	C₂×D₄	—	4	4	no	—
513	C₂×D₄	—	4	2	no	—
514	C₂×D₄	—	4	2	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	4	2	no	—
517	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
518	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
519	C₂×D₄	#41×C₆₇	1	2	yes	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	4	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	4	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	4	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	2	no	—
536	C₂²×C₄	—	2	2	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	4	1	no	—
539	C₂²×C₄	—	4	2	no	—
540	C₂²×C₄	—	4	2	no	—
541	C₂²×C₄	—	4	2	no	—
542	C₂²×C₄	—	4	2	no	—
543	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
544	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
545	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
546	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
547	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
548	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
549	C₂²×C₄	#41×C₆₇	1	2	yes	—
550	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
551	SD₁₆	—	4	8	no	—
552	D₈	—	4	8	no	—
553	C₂×C₈	—	2	2	no	—
554	C₂×C₈	—	2	2	no	—
555	C₂×C₈	—	2	2	no	—
556	C₂×C₈	—	2	2	no	—
557	C₂×C₈	—	2	2	no	—
558	C₂×C₈	—	2	2	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	4	2	no	—
566	C₄⋊C₄	—	4	2	no	—
567	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
568	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
569	C₂²⋊C₄	—	2	4	no	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	4	no	—
572	C₂²⋊C₄	—	4	4	no	—
573	C₂²⋊C₄	—	4	4	no	—
574	C₂²⋊C₄	—	4	4	no	—
575	C₄²	—	2	4	no	—
576	C₄²	—	2	4	no	—
577	C₄²	—	2	2	no	—
578	C₄²	—	2	4	no	—
579	C₄²	—	2	2	no	—
580	C₄²	—	2	1	no	—
581	C₄²	—	2	1	no	—
582	C₄²	C₂×D₄×C₆₇	1	2	yes	—
583	C₄²	C₂×D₄×C₆₇	1	2	yes	—
584	C₂³	—	2	1	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	2	no	—
587	C₂³	—	2	1	no	—
588	C₂³	—	2	2	no	—
589	C₂³	—	2	1	no	—
590	C₂³	—	2	1	no	—
591	C₂³	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	4	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	1	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	4	2	no	—
598	C₂³	—	4	2	no	—
599	C₂³	—	4	2	no	—
600	C₂³	—	4	2	no	—
601	C₂³	—	4	2	no	—
602	C₂³	—	4	2	no	—
603	C₂³	—	4	2	no	—
604	C₂³	#121×C₆₇	1	2	yes	—
605	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
606	Q₈	—	2	2	no	—
607	Q₈	—	2	2	no	—
608	Q₈	—	2	2	no	—
609	Q₈	#121×C₆₇	1	4	yes	complemented
610	D₄	—	2	16	no	—
611	D₄	—	2	2	no	—
612	D₄	—	2	2	no	—
613	D₄	—	2	2	no	—
614	D₄	—	4	2	no	—
615	D₄	—	4	2	no	—
616	D₄	—	4	2	no	—
617	D₄	—	4	2	no	—
618	D₄	—	8	4	no	—
619	D₄	—	8	8	no	—
620	D₄	—	8	4	no	—
621	D₄	#121×C₆₇	1	4	yes	complemented
622	C₂×C₄	—	2	16	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	4	4	no	—
642	C₂×C₄	—	4	8	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	4	no	—
648	C₂×C₄	—	4	8	no	—
649	C₂×C₄	—	4	2	no	—
650	C₂×C₄	—	4	2	no	—
651	C₂×C₄	—	4	2	no	—
652	C₂×C₄	#120×C₆₇	1	2	yes	—
653	C₂×C₄	#120×C₆₇	1	2	yes	—
654	C₂×C₄	#121×C₆₇	1	2	yes	—
655	C₂×C₄	#135×C₆₇	1	2	yes	—
656	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
657	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
658	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
660	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
661	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
662	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
663	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
664	C₈	—	4	2	no	—
665	C₈	—	4	2	no	—
666	C₂²	—	2	2	no	—
667	C₂²	—	2	2	no	—
668	C₂²	—	2	2	no	—
669	C₂²	—	2	4	no	—
670	C₂²	—	4	2	no	—
671	C₂²	—	4	2	no	—
672	C₂²	—	4	1	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	1	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	4	1	no	—
677	C₂²	—	4	2	no	—
678	C₂²	—	4	1	no	—
679	C₂²	—	4	2	no	—
680	C₂²	—	8	2	no	—
681	C₂²	—	8	2	no	—
682	C₂²	—	8	2	no	—
683	C₂²	—	8	2	no	—
684	C₂²	—	8	2	no	—
685	C₂²	—	8	2	no	—
686	C₂²	—	8	2	no	—
687	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
688	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
689	C₂²	#525×C₆₇	1	1	yes	characteristic
690	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
691	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
692	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
693	C₂²	#575×C₆₇	1	1	yes	characteristic
694	C₄	—	2	2	no	—
695	C₄	—	2	4	no	—
696	C₄	—	2	2	no	—
697	C₄	—	2	2	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	8	2	no	—
702	C₄	#449×C₆₇	1	4	yes	complemented
703	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
704	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
705	C₄	#525×C₆₇	1	1	yes	characteristic
706	C₄	#525×C₆₇	1	1	yes	characteristic
707	C₂	—	4	2	no	—
708	C₂	—	4	4	no	—
709	C₂	—	16	2	no	—
710	C₂	C₆₇×#2847	1	1	yes	characteristic
711	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
712	C₂	C₆₇×#3006	1	1	yes	characteristic
713	C₂	C₆₇×#3035	1	1	yes	characteristic
714	C₂	C₆₇×#3038	1	1	yes	characteristic
715	C₂	C₆₇×#3106	1	1	yes	characteristic
716	C₂	C₆₇×#3115	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₄×#41×C₆₇	C₂	semidirect
2. C₂×#460×C₆₇	C₂	semidirect
3. C₂×#459×C₆₇	C₂	semidirect
4. C₂×#449×C₆₇	C₂	semidirect
5. C₆₇×#2856	C₂	semidirect
6. C₆₇×#1880	C₂	semidirect
7. C₆₇×#1879	C₂	semidirect
8. C₆₇×#1875	C₂	semidirect
9. C₆₇×#1803	C₂	semidirect
10. C₆₇×#1802	C₂	semidirect
11. C₆₇×#1797	C₂	semidirect
53. C₂×#41×C₆₇	D₄	semidirect
84. C₄×Q₈×C₆₇	D₄	semidirect
90. C₄×D₄×C₆₇	D₄	semidirect
250. Q₈×C₆₇	#121	semidirect
262. D₄×C₆₇	#121	semidirect
343. C₄×C₆₇	#449	semidirect
348. #32904	C₆₇	direct (N × H)
359. C₂×C₄×#41	C₂×C₆₇	semidirect
360. C₂×#460	C₂×C₆₇	semidirect
361. C₂×#459	C₂×C₆₇	semidirect
362. C₂×#449	C₂×C₆₇	semidirect
363. #2856	C₂×C₆₇	semidirect
364. #1880	C₂×C₆₇	semidirect
365. #1879	C₂×C₆₇	semidirect
366. #1875	C₂×C₆₇	semidirect
367. #1803	C₂×C₆₇	semidirect
368. #1802	C₂×C₆₇	semidirect
369. #1797	C₂×C₆₇	semidirect
370. C₆₇	#32904	direct (N × H)
412. C₂×#41	D₄×C₆₇	semidirect
443. C₄×Q₈	D₄×C₆₇	semidirect
449. C₄×D₄	D₄×C₆₇	semidirect
609. Q₈	#121×C₆₇	semidirect
621. D₄	#121×C₆₇	semidirect
702. C₄	#449×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂×#460×C₆₇	2
C₂×#459×C₆₇	2
C₂×#449×C₆₇	2
C₆₇×#2856	2
C₆₇×#1880	2
C₆₇×#1879	2
C₆₇×#1875	2
C₆₇×#1803	2
C₆₇×#1802	2
C₆₇×#1797	2
#32904	67

Subgroups — C67×#32904 #30712155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32904 #30712155