Subgroups — C₆₇×#21059 #30700310

Order 17152.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2297	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1928	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1927	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1926	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1925	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	—	2	1	no	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#428×C₆₇	—	2	4	no	—
21	C₂×#107×C₆₇	C₂²	1	2	yes	—
22	C₂×#106×C₆₇	C₂²	1	2	yes	—
23	#417×C₆₇	—	2	4	no	—
24	#417×C₆₇	C₄	1	8	yes	complemented
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	#391×C₆₇	C₂²	1	1	yes	characteristic
30	#391×C₆₇	C₂²	1	1	yes	characteristic
31	#390×C₆₇	C₂²	1	1	yes	characteristic
32	#390×C₆₇	C₂²	1	1	yes	characteristic
33	#386×C₆₇	—	2	1	no	—
34	#386×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
39	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
40	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
41	C₂×#95×C₆₇	—	2	1	no	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	#336×C₆₇	C₂²	1	2	yes	—
46	C₂×#41×C₆₇	—	2	2	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	—	2	2	no	—
64	C₂×M₁₆×C₆₇	—	2	2	no	—
65	C₂×M₁₆×C₆₇	—	2	2	no	—
66	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	#117×C₆₇	—	2	2	no	—
70	#117×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄²×C₆₇	—	2	2	no	—
87	C₂×C₄²×C₆₇	—	2	2	no	—
88	C₂×C₄²×C₆₇	—	2	2	no	—
89	C₂×C₄²×C₆₇	—	2	2	no	—
90	C₂×C₄²×C₆₇	—	2	1	no	—
91	C₂×C₄²×C₆₇	—	4	4	no	—
92	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	#107×C₆₇	—	2	4	no	—
97	#106×C₆₇	—	2	4	no	—
98	#103×C₆₇	—	2	4	no	—
99	#103×C₆₇	C₂×C₄	1	8	yes	complemented
100	#102×C₆₇	—	2	4	no	—
101	#102×C₆₇	C₂×C₄	1	8	yes	complemented
102	#95×C₆₇	—	2	2	no	—
103	#95×C₆₇	—	2	2	no	—
104	#95×C₆₇	—	2	1	no	—
105	#95×C₆₇	—	2	1	no	—
106	#95×C₆₇	—	2	1	no	—
107	#95×C₆₇	—	2	1	no	—
108	C₂⁴×C₆₇	—	2	1	no	—
109	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
110	#41×C₆₇	—	2	8	no	—
111	#41×C₆₇	—	2	8	no	—
112	#41×C₆₇	—	4	4	no	—
113	#41×C₆₇	—	4	4	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	2	1	no	—
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	—	2	1	no	—
119	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	1	no	—
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	1	no	—
126	C₂×D₄×C₆₇	—	4	2	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	4	1	no	—
142	C₂²×C₄×C₆₇	—	4	1	no	—
143	C₂²×C₄×C₆₇	—	4	2	no	—
144	C₂²×C₄×C₆₇	—	4	2	no	—
145	C₂²×C₄×C₆₇	—	4	2	no	—
146	C₂²×C₄×C₆₇	—	4	2	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	4	2	no	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₆₇	#41	1	2	yes	—
157	C₂²×C₄×C₆₇	#41	1	2	yes	—
158	C₂²×C₄×C₆₇	#41	1	2	yes	—
159	C₂²×C₄×C₆₇	#41	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
161	M₁₆×C₆₇	—	2	8	no	—
162	M₁₆×C₆₇	—	4	4	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	—	2	2	no	—
168	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	4	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	1	no	—
175	C₄⋊C₄×C₆₇	—	2	1	no	—
176	C₄⋊C₄×C₆₇	—	2	1	no	—
177	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
179	C₂²⋊C₄×C₆₇	—	4	2	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	C₄²×C₆₇	—	2	2	no	—
185	C₄²×C₆₇	—	2	2	no	—
186	C₄²×C₆₇	—	2	1	no	—
187	C₄²×C₆₇	—	2	1	no	—
188	C₄²×C₆₇	—	2	1	no	—
189	C₄²×C₆₇	—	2	1	no	—
190	C₄²×C₆₇	—	4	4	no	—
191	C₄²×C₆₇	—	4	2	no	—
192	C₄²×C₆₇	—	4	4	no	—
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	1	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	—	4	1	no	—
203	C₂³×C₆₇	—	4	1	no	—
204	C₂³×C₆₇	C₄×D₄	1	2	yes	—
205	C₂³×C₆₇	#121	1	2	yes	—
206	C₂³×C₆₇	#123	1	2	yes	—
207	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₆₇	—	2	2	no	—
209	Q₈×C₆₇	—	2	2	no	—
210	Q₈×C₆₇	—	4	1	no	—
211	Q₈×C₆₇	—	4	1	no	—
212	D₄×C₆₇	—	2	2	no	—
213	D₄×C₆₇	—	2	2	no	—
214	D₄×C₆₇	—	4	8	no	—
215	D₄×C₆₇	—	4	1	no	—
216	D₄×C₆₇	—	4	1	no	—
217	C₂×C₄×C₆₇	—	2	8	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	4	4	no	—
232	C₂×C₄×C₆₇	—	4	8	no	—
233	C₂×C₄×C₆₇	—	4	8	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	1	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	2	no	—
238	C₂×C₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	4	8	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	—	4	1	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	—	4	1	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	#117	1	4	yes	—
248	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
249	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
250	C₂×C₄×C₆₇	#121	1	2	yes	—
251	C₂×C₄×C₆₇	#123	1	2	yes	—
252	C₂×C₄×C₆₇	#124	1	4	yes	—
253	C₂×C₄×C₆₇	#126	1	4	yes	—
254	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
255	C₂×C₄×C₆₇	#137	1	2	yes	—
256	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
262	C₈×C₆₇	—	2	4	no	—
263	C₈×C₆₇	—	4	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	2	2	no	—
270	C₂²×C₆₇	—	4	2	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	4	2	no	—
273	C₂²×C₆₇	—	8	1	no	—
274	C₂²×C₆₇	—	8	1	no	—
275	C₂²×C₆₇	—	8	1	no	—
276	C₂²×C₆₇	—	8	1	no	—
277	C₂²×C₆₇	#388	1	4	yes	—
278	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
279	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
281	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
283	C₂²×C₆₇	#575	1	1	yes	characteristic
284	C₂²×C₆₇	#575	1	1	yes	characteristic
285	C₄×C₆₇	—	2	2	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	4	1	no	—
288	C₄×C₆₇	—	4	1	no	—
289	C₄×C₆₇	—	4	4	no	—
290	C₄×C₆₇	—	4	4	no	—
291	C₄×C₆₇	—	4	2	no	—
292	C₄×C₆₇	—	8	1	no	—
293	C₄×C₆₇	—	8	1	no	—
294	C₄×C₆₇	#388	1	4	yes	—
295	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
296	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
297	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
298	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
299	#21059	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
300	C₂×C₆₇	—	2	4	no	—
301	C₂×C₆₇	—	8	2	no	—
302	C₂×C₆₇	C₂×#388	1	1	yes	characteristic
303	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
305	C₂×C₆₇	#3036	1	1	yes	characteristic
306	C₂×C₆₇	#3037	1	1	yes	characteristic
307	C₂×C₆₇	#3092	1	1	yes	characteristic
308	C₂×C₆₇	#3093	1	1	yes	characteristic
309	C₂×#428	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₂×#417	C₂×C₆₇	1	1	yes	characteristic
311	C₂×#417	C₂×C₆₇	1	1	yes	characteristic
312	#2297	C₂×C₆₇	1	1	yes	characteristic
313	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#1928	C₂×C₆₇	1	2	yes	complemented
315	#1927	C₂×C₆₇	1	2	yes	complemented
316	#1926	C₂×C₆₇	1	2	yes	complemented
317	#1925	C₂×C₆₇	1	2	yes	complemented
318	#1736	C₂×C₆₇	1	1	yes	characteristic, complemented
319	#1728	C₂×C₆₇	1	1	yes	characteristic, complemented
320	C₆₇	#21059	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
323	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
325	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
326	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
327	C₂²×C₄²	—	2	1	no	—
328	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
329	#428	—	2	4	no	—
330	C₂×#107	C₂²×C₆₇	1	2	yes	—
331	C₂×#106	C₂²×C₆₇	1	2	yes	—
332	#417	—	2	4	no	—
333	#417	C₄×C₆₇	1	8	yes	complemented
334	C₂×#103	C₂²×C₆₇	1	2	yes	—
335	C₂×#103	C₂²×C₆₇	1	2	yes	—
336	C₂×#102	C₂²×C₆₇	1	2	yes	—
337	C₂×#102	C₂²×C₆₇	1	2	yes	—
338	#391	C₂²×C₆₇	1	1	yes	characteristic
339	#391	C₂²×C₆₇	1	1	yes	characteristic
340	#390	C₂²×C₆₇	1	1	yes	characteristic
341	#390	C₂²×C₆₇	1	1	yes	characteristic
342	#386	—	2	1	no	—
343	#386	—	2	1	no	—
344	C₄×C₄⋊C₄	—	2	1	no	—
345	C₄×C₄⋊C₄	—	2	1	no	—
346	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
347	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
348	C₄×C₂²⋊C₄	—	2	1	no	—
349	C₄×C₂²⋊C₄	—	2	1	no	—
350	C₂×#95	—	2	1	no	—
351	#340	C₂²×C₆₇	1	2	yes	—
352	#340	C₂²×C₆₇	1	2	yes	—
353	#336	C₂²×C₆₇	1	2	yes	—
354	#336	C₂²×C₆₇	1	2	yes	—
355	C₂×#41	—	2	2	no	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	—	2	2	no	—
358	C₂×#41	—	2	2	no	—
359	C₂×#41	—	2	2	no	—
360	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
361	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
364	C₂³×C₄	—	2	1	no	—
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
368	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
369	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
370	C₂×M₁₆	—	2	2	no	—
371	C₂×M₁₆	—	2	2	no	—
372	C₂×M₁₆	—	2	2	no	—
373	C₂×M₁₆	—	2	2	no	—
374	C₂×M₁₆	—	2	2	no	—
375	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
376	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
377	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
378	#117	—	2	2	no	—
379	#117	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
385	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	2	1	no	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₄²	—	2	2	no	—
396	C₂×C₄²	—	2	2	no	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	—	2	2	no	—
399	C₂×C₄²	—	2	2	no	—
400	C₂×C₄²	—	4	4	no	—
401	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
405	#107	—	2	4	no	—
406	#106	—	2	4	no	—
407	#103	—	2	4	no	—
408	#103	C₂×C₄×C₆₇	1	8	yes	complemented
409	#102	—	2	4	no	—
410	#102	C₂×C₄×C₆₇	1	8	yes	complemented
411	#95	—	2	2	no	—
412	#95	—	2	2	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	C₂⁴	—	2	1	no	—
418	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
419	#41	—	2	8	no	—
420	#41	—	2	8	no	—
421	#41	—	4	4	no	—
422	#41	—	4	4	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	4	2	no	—
436	C₂×D₄	—	4	2	no	—
437	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	—	4	1	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	4	2	no	—
455	C₂²×C₄	—	4	2	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
465	C₂²×C₄	#41×C₆₇	1	2	yes	—
466	C₂²×C₄	#41×C₆₇	1	2	yes	—
467	C₂²×C₄	#41×C₆₇	1	2	yes	—
468	C₂²×C₄	#41×C₆₇	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
470	M₁₆	—	2	8	no	—
471	M₁₆	—	4	4	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
487	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
488	C₂²⋊C₄	—	4	2	no	—
489	C₂²⋊C₄	—	4	2	no	—
490	C₂²⋊C₄	—	4	2	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	4	4	no	—
500	C₄²	—	4	2	no	—
501	C₄²	—	4	4	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	C₄×D₄×C₆₇	1	2	yes	—
514	C₂³	#121×C₆₇	1	2	yes	—
515	C₂³	#123×C₆₇	1	2	yes	—
516	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	1	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	2	no	—
523	D₄	—	4	8	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	8	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	8	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	#117×C₆₇	1	4	yes	—
557	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
558	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
559	C₂×C₄	#121×C₆₇	1	2	yes	—
560	C₂×C₄	#123×C₆₇	1	2	yes	—
561	C₂×C₄	#124×C₆₇	1	4	yes	—
562	C₂×C₄	#126×C₆₇	1	4	yes	—
563	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
564	C₂×C₄	#137×C₆₇	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
567	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
568	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
569	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
571	C₈	—	2	4	no	—
572	C₈	—	4	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	8	1	no	—
583	C₂²	—	8	1	no	—
584	C₂²	—	8	1	no	—
585	C₂²	—	8	1	no	—
586	C₂²	#388×C₆₇	1	4	yes	—
587	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
588	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
589	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
590	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
591	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
592	C₂²	#575×C₆₇	1	1	yes	characteristic
593	C₂²	#575×C₆₇	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	4	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	#388×C₆₇	1	4	yes	—
604	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
605	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
606	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
607	C₄	C₂×#126×C₆₇	1	1	yes	characteristic
608	C₂	—	2	4	no	—
609	C₂	—	8	2	no	—
610	C₂	C₂×#388×C₆₇	1	1	yes	characteristic
611	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
612	C₂	C₄×#137×C₆₇	1	1	yes	characteristic
613	C₂	C₆₇×#3036	1	1	yes	characteristic
614	C₂	C₆₇×#3037	1	1	yes	characteristic
615	C₂	C₆₇×#3092	1	1	yes	characteristic
616	C₂	C₆₇×#3093	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂×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×#428×C₆₇	C₂	semidirect
5. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
6. C₆₇×#1928	C₂	semidirect
7. C₆₇×#1927	C₂	semidirect
8. C₆₇×#1926	C₂	semidirect
9. C₆₇×#1925	C₂	semidirect
10. C₆₇×#1736	C₂	semidirect
11. C₆₇×#1728	C₂	semidirect
24. #417×C₆₇	C₄	semidirect
99. #103×C₆₇	C₂×C₄	semidirect
101. #102×C₆₇	C₂×C₄	semidirect
299. #21059	C₆₇	direct (N × H)
309. C₂×#428	C₂×C₆₇	semidirect
313. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
314. #1928	C₂×C₆₇	semidirect
315. #1927	C₂×C₆₇	semidirect
316. #1926	C₂×C₆₇	semidirect
317. #1925	C₂×C₆₇	semidirect
318. #1736	C₂×C₆₇	semidirect
319. #1728	C₂×C₆₇	semidirect
320. C₆₇	#21059	direct (N × H)
333. #417	C₄×C₆₇	semidirect
408. #103	C₂×C₄×C₆₇	semidirect
410. #102	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#428×C₆₇	2
C₂×#417×C₆₇	2
C₆₇×#2297	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1928	2
C₆₇×#1927	2
C₆₇×#1926	2
C₆₇×#1925	2
C₆₇×#1736	2
C₆₇×#1728	2
#21059	67

Subgroups — C67×#21059 #30700310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21059 #30700310