Subgroups — C₆₇×#22758 #30702009

Order 17152.

978 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1936	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1921	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1879	C₂	1	2	yes	maximal
7	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
18	C₄×D₈×C₆₇	C₄	1	8	yes	complemented
19	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#390×C₆₇	C₂²	1	2	yes	—
28	#389×C₆₇	C₂²	1	2	yes	—
29	#388×C₆₇	—	2	2	no	—
30	#385×C₆₇	—	2	2	no	—
31	#384×C₆₇	C₂²	1	1	yes	characteristic
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#380×C₆₇	—	2	1	no	—
34	#380×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	—	2	1	no	—
46	C₂³×C₄×C₆₇	—	2	2	no	—
47	C₂×D₈×C₆₇	—	2	2	no	—
48	C₂×D₈×C₆₇	—	2	2	no	—
49	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₆₇	—	2	4	no	—
54	C₄×D₄×C₆₇	—	2	2	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	—	4	2	no	—
57	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
59	#117×C₆₇	—	4	4	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄²×C₆₇	—	2	2	no	—
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	#107×C₆₇	C₂×C₄	1	4	yes	complemented
81	C₄⋊C₈×C₆₇	—	2	2	no	—
82	#102×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	4	2	no	—
84	#102×C₆₇	C₂×C₄	1	4	yes	—
85	#102×C₆₇	C₂×C₄	1	4	yes	complemented
86	#97×C₆₇	—	4	2	no	—
87	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	D₄	1	2	yes	—
91	C₂⁴×C₆₇	—	2	1	no	—
92	C₂⁴×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	2	1	no	—
98	C₂×D₄×C₆₇	—	2	1	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂×D₄×C₆₇	#41	1	2	yes	—
104	C₂×D₄×C₆₇	#41	1	2	yes	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	4	4	no	—
112	C₂²×C₄×C₆₇	—	4	4	no	—
113	C₂²×C₄×C₆₇	—	4	2	no	—
114	C₂²×C₄×C₆₇	—	4	2	no	—
115	C₂²×C₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	—	4	2	no	—
117	C₂²×C₄×C₆₇	—	8	2	no	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	D₈×C₆₇	—	2	8	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	4	4	no	—
142	C₄⋊C₄×C₆₇	—	4	2	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	—	8	1	no	—
145	C₄⋊C₄×C₆₇	—	8	1	no	—
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	4	4	no	—
151	C₂²⋊C₄×C₆₇	—	4	4	no	—
152	C₂²⋊C₄×C₆₇	—	8	2	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	—	2	2	no	—
155	C₄²×C₆₇	—	4	4	no	—
156	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
157	C₄²×C₆₇	#41	1	2	yes	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	2	2	no	—
162	C₂³×C₆₇	—	4	1	no	—
163	C₂³×C₆₇	—	4	1	no	—
164	C₂³×C₆₇	—	4	1	no	—
165	C₂³×C₆₇	—	4	1	no	—
166	C₂³×C₆₇	—	4	1	no	—
167	C₂³×C₆₇	—	4	1	no	—
168	C₂³×C₆₇	—	4	1	no	—
169	C₂³×C₆₇	—	4	1	no	—
170	C₂³×C₆₇	—	4	1	no	—
171	C₂³×C₆₇	—	4	1	no	—
172	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	D₄×C₆₇	—	2	2	no	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	2	2	no	—
176	D₄×C₆₇	—	2	2	no	—
177	D₄×C₆₇	—	2	2	no	—
178	D₄×C₆₇	—	4	1	no	—
179	D₄×C₆₇	—	4	1	no	—
180	D₄×C₆₇	#117	1	4	yes	complemented
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	1	no	—
193	C₂×C₄×C₆₇	—	4	1	no	—
194	C₂×C₄×C₆₇	—	4	1	no	—
195	C₂×C₄×C₆₇	—	4	8	no	—
196	C₂×C₄×C₆₇	—	4	8	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	—	8	8	no	—
204	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
214	C₈×C₆₇	—	2	2	no	—
215	C₈×C₆₇	—	2	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	4	2	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	4	2	no	—
222	C₂²×C₆₇	—	4	2	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	4	2	no	—
226	C₂²×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	8	1	no	—
228	C₂²×C₆₇	—	8	1	no	—
229	C₂²×C₆₇	—	8	1	no	—
230	C₂²×C₆₇	—	8	1	no	—
231	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
232	C₂²×C₆₇	#546	1	1	yes	characteristic
233	C₂²×C₆₇	#548	1	1	yes	characteristic
234	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₂²×C₆₇	#576	1	1	yes	characteristic
237	C₂²×C₆₇	#578	1	1	yes	characteristic
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	4	2	no	—
240	C₄×C₆₇	—	4	2	no	—
241	C₄×C₆₇	—	4	2	no	—
242	C₄×C₆₇	—	4	2	no	—
243	C₄×C₆₇	—	8	2	no	—
244	C₄×C₆₇	—	8	2	no	—
245	C₄×C₆₇	#442	1	4	yes	—
246	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
247	C₄×C₆₇	#520	1	1	yes	characteristic
248	C₄×C₆₇	#552	1	1	yes	characteristic
249	C₄×C₆₇	#553	1	1	yes	characteristic
250	#22758	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	4	4	no	—
252	C₂×C₆₇	—	8	2	no	—
253	C₂×C₆₇	#2352	1	1	yes	characteristic
254	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
255	C₂×C₆₇	#2951	1	1	yes	characteristic
256	C₂×C₆₇	#3325	1	1	yes	characteristic
257	C₂×C₆₇	#3330	1	1	yes	characteristic
258	C₂×C₆₇	#3366	1	1	yes	characteristic
259	C₂×C₆₇	#3367	1	1	yes	characteristic
260	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
261	#2354	C₂×C₆₇	1	1	yes	characteristic, complemented
262	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#1936	C₂×C₆₇	1	2	yes	complemented
264	#1921	C₂×C₆₇	1	2	yes	complemented
265	#1879	C₂×C₆₇	1	2	yes	—
266	#1834	C₂×C₆₇	1	1	yes	characteristic, complemented
267	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
269	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
270	#1762	C₂×C₆₇	1	2	yes	complemented
271	C₆₇	#22758	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
274	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#117	—	2	1	no	—
277	C₂²×C₄⋊C₄	—	2	1	no	—
278	C₄×D₈	C₄×C₆₇	1	8	yes	complemented
279	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×#102	C₂²×C₆₇	1	2	yes	—
282	C₂×#102	C₂²×C₆₇	1	2	yes	—
283	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
284	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
285	C₂×#97	C₂²×C₆₇	1	2	yes	—
286	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
287	#390	C₂²×C₆₇	1	2	yes	—
288	#389	C₂²×C₆₇	1	2	yes	—
289	#388	—	2	2	no	—
290	#385	—	2	2	no	—
291	#384	C₂²×C₆₇	1	1	yes	characteristic
292	#384	C₂²×C₆₇	1	1	yes	characteristic
293	#380	—	2	1	no	—
294	#380	—	2	1	no	—
295	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
296	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
297	#340	C₂²×C₆₇	1	2	yes	—
298	#340	C₂²×C₆₇	1	2	yes	—
299	#340	C₂²×C₆₇	1	2	yes	—
300	#340	C₂²×C₆₇	1	2	yes	—
301	#336	C₂²×C₆₇	1	2	yes	—
302	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
303	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	—	2	1	no	—
306	C₂³×C₄	—	2	2	no	—
307	C₂×D₈	—	2	2	no	—
308	C₂×D₈	—	2	2	no	—
309	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
310	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
311	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	—	2	2	no	—
315	C₄×D₄	—	2	2	no	—
316	C₄×D₄	—	4	2	no	—
317	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
318	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
319	#117	—	4	4	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	4	4	no	—
326	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
327	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
339	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
340	#107	C₂×C₄×C₆₇	1	4	yes	complemented
341	C₄⋊C₈	—	2	2	no	—
342	#102	—	2	4	no	—
343	#102	—	4	2	no	—
344	#102	C₂×C₄×C₆₇	1	4	yes	—
345	#102	C₂×C₄×C₆₇	1	4	yes	complemented
346	#97	—	4	2	no	—
347	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
348	#95	—	2	2	no	—
349	#95	—	2	2	no	—
350	#95	D₄×C₆₇	1	2	yes	—
351	C₂⁴	—	2	1	no	—
352	C₂⁴	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
362	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
363	C₂×D₄	#41×C₆₇	1	2	yes	—
364	C₂×D₄	#41×C₆₇	1	2	yes	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	4	4	no	—
372	C₂²×C₄	—	4	4	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	—	4	2	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	—	8	2	no	—
378	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
379	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
383	D₈	—	2	8	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
392	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	4	4	no	—
402	C₄⋊C₄	—	4	2	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	8	1	no	—
405	C₄⋊C₄	—	8	1	no	—
406	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
407	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
408	C₂²⋊C₄	—	2	4	no	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	—	4	4	no	—
411	C₂²⋊C₄	—	4	4	no	—
412	C₂²⋊C₄	—	8	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	4	4	no	—
416	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
417	C₄²	#41×C₆₇	1	2	yes	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	1	no	—
432	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
433	D₄	—	2	2	no	—
434	D₄	—	2	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	4	1	no	—
439	D₄	—	4	1	no	—
440	D₄	#117×C₆₇	1	4	yes	complemented
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	8	8	no	—
464	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
465	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
466	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
467	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
468	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
472	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
473	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	2	2	no	—
476	C₈	—	4	2	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	1	no	—
491	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
492	C₂²	#546×C₆₇	1	1	yes	characteristic
493	C₂²	#548×C₆₇	1	1	yes	characteristic
494	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
495	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
496	C₂²	#576×C₆₇	1	1	yes	characteristic
497	C₂²	#578×C₆₇	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	8	2	no	—
504	C₄	—	8	2	no	—
505	C₄	#442×C₆₇	1	4	yes	—
506	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
507	C₄	#520×C₆₇	1	1	yes	characteristic
508	C₄	#552×C₆₇	1	1	yes	characteristic
509	C₄	#553×C₆₇	1	1	yes	characteristic
510	C₂	—	4	4	no	—
511	C₂	—	8	2	no	—
512	C₂	C₆₇×#2352	1	1	yes	characteristic
513	C₂	C₂×#442×C₆₇	1	1	yes	characteristic
514	C₂	C₆₇×#2951	1	1	yes	characteristic
515	C₂	C₆₇×#3325	1	1	yes	characteristic
516	C₂	C₆₇×#3330	1	1	yes	characteristic
517	C₂	C₆₇×#3366	1	1	yes	characteristic
518	C₂	C₆₇×#3367	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×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2354	C₂	semidirect
3. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#1936	C₂	semidirect
5. C₆₇×#1921	C₂	semidirect
7. C₆₇×#1834	C₂	semidirect
8. C₆₇×#1802	C₂	semidirect
9. C₆₇×#1802	C₂	semidirect
10. C₆₇×#1770	C₂	semidirect
11. C₆₇×#1762	C₂	semidirect
18. C₄×D₈×C₆₇	C₄	semidirect
58. C₄×D₄×C₆₇	C₂×C₄	semidirect
80. #107×C₆₇	C₂×C₄	semidirect
85. #102×C₆₇	C₂×C₄	semidirect
87. C₄×C₈×C₆₇	C₂×C₄	semidirect
180. D₄×C₆₇	#117	semidirect
250. #22758	C₆₇	direct (N × H)
261. #2354	C₂×C₆₇	semidirect
262. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
263. #1936	C₂×C₆₇	semidirect
264. #1921	C₂×C₆₇	semidirect
266. #1834	C₂×C₆₇	semidirect
267. #1802	C₂×C₆₇	semidirect
268. #1802	C₂×C₆₇	semidirect
269. #1770	C₂×C₆₇	semidirect
270. #1762	C₂×C₆₇	semidirect
271. C₆₇	#22758	direct (N × H)
278. C₄×D₈	C₄×C₆₇	semidirect
318. C₄×D₄	C₂×C₄×C₆₇	semidirect
340. #107	C₂×C₄×C₆₇	semidirect
345. #102	C₂×C₄×C₆₇	semidirect
347. C₄×C₈	C₂×C₄×C₆₇	semidirect
440. D₄	#117×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×D₈×C₆₇	2
C₆₇×#2354	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1936	2
C₆₇×#1921	2
C₆₇×#1879	2
C₆₇×#1834	2
C₆₇×#1802	2
C₆₇×#1770	2
C₆₇×#1762	2
#22758	67

Subgroups — C67×#22758 #30702009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22758 #30702009