Subgroups — C₆₇×#32903 #30712154

Order 17152.

1576 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#460×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#448×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1880	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1803	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
11	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#121×C₆₇	—	2	1	no	—
13	C₂×#121×C₆₇	—	2	1	no	—
14	C₂×#121×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
16	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₆₇	—	2	1	no	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#460×C₆₇	—	2	8	no	—
25	#448×C₆₇	—	2	8	no	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	4	yes	—
28	C₂×#102×C₆₇	C₂²	1	4	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
30	#390×C₆₇	—	2	1	no	—
31	#390×C₆₇	—	2	1	no	—
32	#390×C₆₇	C₂²	1	2	yes	—
33	#384×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	4	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	C₂²×Q₈×C₆₇	—	2	4	no	—
37	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	2	1	no	—
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	D₄	1	4	yes	complemented
41	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	—	2	1	no	—
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂³×C₄×C₆₇	—	2	1	no	—
48	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×SD₁₆×C₆₇	—	2	8	no	—
52	C₂×SD₁₆×C₆₇	—	2	8	no	—
53	C₂×SD₁₆×C₆₇	—	2	8	no	—
54	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
55	#127×C₆₇	—	2	4	no	—
56	#121×C₆₇	—	2	4	no	—
57	#121×C₆₇	—	2	4	no	—
58	#121×C₆₇	—	4	4	no	—
59	#121×C₆₇	—	4	4	no	—
60	C₄×Q₈×C₆₇	—	2	8	no	—
61	C₄×Q₈×C₆₇	—	2	8	no	—
62	C₄×Q₈×C₆₇	—	2	4	no	—
63	C₄×Q₈×C₆₇	—	2	4	no	—
64	C₄×Q₈×C₆₇	—	2	4	no	—
65	C₄×Q₈×C₆₇	D₄	1	8	yes	complemented
66	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₄²×C₆₇	—	2	8	no	—
79	C₂×C₄²×C₆₇	—	2	2	no	—
80	C₂×C₄²×C₆₇	—	2	2	no	—
81	C₂×C₄²×C₆₇	—	2	4	no	—
82	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	—	2	4	no	—
87	#103×C₆₇	—	2	8	no	—
88	#102×C₆₇	—	2	8	no	—
89	C₂²⋊C₈×C₆₇	—	2	4	no	—
90	C₂⁴×C₆₇	—	2	2	no	—
91	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₆₇	—	2	4	no	—
93	C₂×Q₈×C₆₇	—	2	16	no	—
94	C₂×Q₈×C₆₇	—	2	4	no	—
95	C₂×Q₈×C₆₇	—	2	16	no	—
96	C₂×Q₈×C₆₇	—	2	4	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
102	C₂×Q₈×C₆₇	#41	1	4	yes	—
103	C₂×D₄×C₆₇	—	2	4	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	4	no	—
107	C₂×D₄×C₆₇	—	4	2	no	—
108	C₂×D₄×C₆₇	—	4	2	no	—
109	C₂×D₄×C₆₇	—	4	4	no	—
110	C₂×D₄×C₆₇	—	4	4	no	—
111	C₂×D₄×C₆₇	—	4	4	no	—
112	C₂×D₄×C₆₇	—	4	2	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	4	2	no	—
115	C₂×D₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	4	1	no	—
130	C₂²×C₄×C₆₇	—	4	2	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	complemented
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	#41	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	SD₁₆×C₆₇	—	4	16	no	—
142	C₂×C₈×C₆₇	—	2	4	no	—
143	C₂×C₈×C₆₇	—	2	4	no	—
144	C₂×C₈×C₆₇	—	2	4	no	—
145	C₄⋊C₄×C₆₇	—	2	8	no	—
146	C₄⋊C₄×C₆₇	—	2	4	no	—
147	C₄⋊C₄×C₆₇	—	2	8	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	4	2	no	—
152	C₄⋊C₄×C₆₇	—	4	2	no	—
153	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
154	C₂²⋊C₄×C₆₇	—	4	4	no	—
155	C₂²⋊C₄×C₆₇	—	4	4	no	—
156	C₂²⋊C₄×C₆₇	—	4	4	no	—
157	C₄²×C₆₇	—	2	8	no	—
158	C₄²×C₆₇	—	2	8	no	—
159	C₄²×C₆₇	—	2	2	no	—
160	C₄²×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	—	2	2	no	—
162	C₄²×C₆₇	C₂×D₄	1	4	yes	—
163	C₂³×C₆₇	—	2	1	no	—
164	C₂³×C₆₇	—	2	1	no	—
165	C₂³×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	4	2	no	—
168	C₂³×C₆₇	—	4	2	no	—
169	C₂³×C₆₇	—	4	2	no	—
170	C₂³×C₆₇	—	4	2	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	#121	1	2	yes	—
175	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
176	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
177	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₆₇	—	2	16	no	—
179	Q₈×C₆₇	—	2	4	no	—
180	Q₈×C₆₇	—	2	4	no	—
181	Q₈×C₆₇	—	2	4	no	—
182	Q₈×C₆₇	#121	1	8	yes	complemented
183	D₄×C₆₇	—	4	4	no	—
184	D₄×C₆₇	—	4	4	no	—
185	D₄×C₆₇	—	8	4	no	—
186	D₄×C₆₇	—	8	8	no	—
187	D₄×C₆₇	—	8	4	no	—
188	C₂×C₄×C₆₇	—	2	16	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	8	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	#120	1	2	yes	—
213	C₂×C₄×C₆₇	#120	1	2	yes	—
214	C₂×C₄×C₆₇	#121	1	2	yes	—
215	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
217	C₂×C₄×C₆₇	#137	1	2	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
221	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
222	C₈×C₆₇	—	4	4	no	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	2	2	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	2	2	no	—
228	C₂²×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	8	2	no	—
230	C₂²×C₆₇	—	8	2	no	—
231	C₂²×C₆₇	—	8	2	no	—
232	C₂²×C₆₇	—	8	2	no	—
233	C₂²×C₆₇	—	8	2	no	—
234	C₂²×C₆₇	—	8	2	no	—
235	C₂²×C₆₇	—	8	2	no	—
236	C₂²×C₆₇	#460	1	4	yes	complemented
237	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
238	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
239	C₂²×C₆₇	#525	1	1	yes	characteristic
240	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
241	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
243	C₂²×C₆₇	#575	1	1	yes	characteristic
244	C₄×C₆₇	—	2	8	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	4	1	no	—
249	C₄×C₆₇	—	4	4	no	—
250	C₄×C₆₇	—	8	2	no	—
251	C₄×C₆₇	#448	1	4	yes	complemented
252	C₄×C₆₇	C₂×#121	1	2	yes	—
253	C₄×C₆₇	#525	1	2	yes	—
254	#32903	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₆₇	—	2	4	no	—
256	C₂×C₆₇	—	16	2	no	—
257	C₂×C₆₇	#2847	1	1	yes	characteristic
258	C₂×C₆₇	C₂×#448	1	1	yes	characteristic
259	C₂×C₆₇	#3006	1	1	yes	characteristic
260	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
261	C₂×C₆₇	#3037	1	1	yes	characteristic
262	C₂×C₆₇	#3103	1	1	yes	characteristic
263	C₂×C₆₇	#3112	1	1	yes	characteristic
264	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
265	C₂×#460	C₂×C₆₇	1	2	yes	complemented
266	C₂×#448	C₂×C₆₇	1	2	yes	complemented
267	#2856	C₂×C₆₇	1	1	yes	characteristic
268	#1880	C₂×C₆₇	1	4	yes	complemented
269	#1874	C₂×C₆₇	1	2	yes	complemented
270	#1803	C₂×C₆₇	1	2	yes	complemented
271	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₆₇	#32903	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
275	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#121	—	2	1	no	—
277	C₂×#121	—	2	1	no	—
278	C₂×#121	C₂²×C₆₇	1	2	yes	—
279	C₂×C₄×Q₈	—	2	8	no	—
280	C₂×C₄×Q₈	—	2	1	no	—
281	C₂×C₄×Q₈	—	2	1	no	—
282	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
283	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
284	C₂²×C₄⋊C₄	—	2	1	no	—
285	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
286	C₂²×C₄²	—	2	1	no	—
287	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
288	#460	—	2	8	no	—
289	#448	—	2	8	no	—
290	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
291	C₂×#103	C₂²×C₆₇	1	4	yes	—
292	C₂×#102	C₂²×C₆₇	1	4	yes	—
293	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
294	#390	—	2	1	no	—
295	#390	—	2	1	no	—
296	#390	C₂²×C₆₇	1	2	yes	—
297	#384	C₂²×C₆₇	1	1	yes	characteristic
298	#340	C₂²×C₆₇	1	4	yes	—
299	#336	C₂²×C₆₇	1	2	yes	—
300	C₂²×Q₈	—	2	4	no	—
301	C₂²×Q₈	—	2	4	no	—
302	C₂²×Q₈	—	2	1	no	—
303	C₂²×Q₈	—	2	2	no	—
304	C₂²×Q₈	D₄×C₆₇	1	4	yes	complemented
305	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	—	2	1	no	—
308	C₂²×D₄	—	2	1	no	—
309	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	1	no	—
312	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
313	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
315	C₂×SD₁₆	—	2	8	no	—
316	C₂×SD₁₆	—	2	8	no	—
317	C₂×SD₁₆	—	2	8	no	—
318	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
319	#127	—	2	4	no	—
320	#121	—	2	4	no	—
321	#121	—	2	4	no	—
322	#121	—	4	4	no	—
323	#121	—	4	4	no	—
324	C₄×Q₈	—	2	8	no	—
325	C₄×Q₈	—	2	8	no	—
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	—	2	4	no	—
328	C₄×Q₈	—	2	4	no	—
329	C₄×Q₈	D₄×C₆₇	1	8	yes	complemented
330	C₂×C₄⋊C₄	—	2	8	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	4	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₄²	—	2	8	no	—
343	C₂×C₄²	—	2	2	no	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	—	2	4	no	—
346	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
350	C₄⋊C₈	—	2	4	no	—
351	#103	—	2	8	no	—
352	#102	—	2	8	no	—
353	C₂²⋊C₈	—	2	4	no	—
354	C₂⁴	—	2	2	no	—
355	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
356	C₂×Q₈	—	2	4	no	—
357	C₂×Q₈	—	2	16	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	16	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
365	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
366	C₂×Q₈	#41×C₆₇	1	4	yes	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	—	4	4	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	4	no	—
376	C₂×D₄	—	4	2	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	4	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	4	1	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	complemented
397	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	#41×C₆₇	1	2	yes	—
404	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
405	SD₁₆	—	4	16	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	—	2	4	no	—
408	C₂×C₈	—	2	4	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	—	4	2	no	—
417	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	—	4	4	no	—
421	C₄²	—	2	8	no	—
422	C₄²	—	2	8	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	2	no	—
425	C₄²	—	2	2	no	—
426	C₄²	C₂×D₄×C₆₇	1	4	yes	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	2	1	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	4	2	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	#121×C₆₇	1	2	yes	—
439	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
440	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
441	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
442	Q₈	—	2	16	no	—
443	Q₈	—	2	4	no	—
444	Q₈	—	2	4	no	—
445	Q₈	—	2	4	no	—
446	Q₈	#121×C₆₇	1	8	yes	complemented
447	D₄	—	4	4	no	—
448	D₄	—	4	4	no	—
449	D₄	—	8	4	no	—
450	D₄	—	8	8	no	—
451	D₄	—	8	4	no	—
452	C₂×C₄	—	2	16	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	8	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	#120×C₆₇	1	2	yes	—
477	C₂×C₄	#120×C₆₇	1	2	yes	—
478	C₂×C₄	#121×C₆₇	1	2	yes	—
479	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
481	C₂×C₄	#137×C₆₇	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
484	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
485	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
486	C₈	—	4	4	no	—
487	C₂²	—	2	2	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	8	2	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	2	no	—
499	C₂²	—	8	2	no	—
500	C₂²	#460×C₆₇	1	4	yes	complemented
501	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
502	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
503	C₂²	#525×C₆₇	1	1	yes	characteristic
504	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
505	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
506	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
507	C₂²	#575×C₆₇	1	1	yes	characteristic
508	C₄	—	2	8	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	4	no	—
514	C₄	—	8	2	no	—
515	C₄	#448×C₆₇	1	4	yes	complemented
516	C₄	C₂×#121×C₆₇	1	2	yes	—
517	C₄	#525×C₆₇	1	2	yes	—
518	C₂	—	2	4	no	—
519	C₂	—	16	2	no	—
520	C₂	C₆₇×#2847	1	1	yes	characteristic
521	C₂	C₂×#448×C₆₇	1	1	yes	characteristic
522	C₂	C₆₇×#3006	1	1	yes	characteristic
523	C₂	C₂×#460×C₆₇	1	1	yes	characteristic
524	C₂	C₆₇×#3037	1	1	yes	characteristic
525	C₂	C₆₇×#3103	1	1	yes	characteristic
526	C₂	C₆₇×#3112	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
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. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
2. C₂×#460×C₆₇	C₂	semidirect
3. C₂×#448×C₆₇	C₂	semidirect
5. C₆₇×#1880	C₂	semidirect
6. C₆₇×#1874	C₂	semidirect
7. C₆₇×#1803	C₂	semidirect
8. C₆₇×#1797	C₂	semidirect
40. C₂²×Q₈×C₆₇	D₄	semidirect
65. C₄×Q₈×C₆₇	D₄	semidirect
132. C₂²×C₄×C₆₇	SD₁₆	semidirect
182. Q₈×C₆₇	#121	semidirect
236. C₂²×C₆₇	#460	semidirect
251. C₄×C₆₇	#448	semidirect
254. #32903	C₆₇	direct (N × H)
264. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
265. C₂×#460	C₂×C₆₇	semidirect
266. C₂×#448	C₂×C₆₇	semidirect
268. #1880	C₂×C₆₇	semidirect
269. #1874	C₂×C₆₇	semidirect
270. #1803	C₂×C₆₇	semidirect
271. #1797	C₂×C₆₇	semidirect
272. C₆₇	#32903	direct (N × H)
304. C₂²×Q₈	D₄×C₆₇	semidirect
329. C₄×Q₈	D₄×C₆₇	semidirect
396. C₂²×C₄	SD₁₆×C₆₇	semidirect
446. Q₈	#121×C₆₇	semidirect
500. C₂²	#460×C₆₇	semidirect
515. C₄	#448×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂×#460×C₆₇	2
C₂×#448×C₆₇	2
C₆₇×#2856	2
C₆₇×#1880	2
C₆₇×#1874	2
C₆₇×#1803	2
C₆₇×#1797	2
#32903	67

Subgroups — C67×#32903 #30712154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32903 #30712154