Subgroups — C₆₇×#35158 #30714409

Order 17152.

2442 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#537×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2850	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2639	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2492	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2491	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2490	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2489	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2311	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2297	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	2	yes	—
13	#544×C₆₇	—	2	8	no	—
14	#537×C₆₇	—	2	8	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
16	C₂×#125×C₆₇	C₂²	1	4	yes	—
17	C₂×#124×C₆₇	C₂²	1	4	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
20	C₂×#121×C₆₇	C₂²	1	4	yes	—
21	C₂×#121×C₆₇	C₂²	1	4	yes	—
22	C₂×#117×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
27	#400×C₆₇	C₂²	1	8	yes	—
28	#397×C₆₇	C₂²	1	8	yes	—
29	#394×C₆₇	C₂²	1	8	yes	—
30	#393×C₆₇	C₂²	1	8	yes	—
31	#391×C₆₇	C₂²	1	2	yes	—
32	#391×C₆₇	C₂²	1	2	yes	—
33	#391×C₆₇	C₂²	1	4	yes	—
34	#391×C₆₇	C₂²	1	4	yes	—
35	#390×C₆₇	C₂²	1	2	yes	—
36	#390×C₆₇	C₂²	1	2	yes	—
37	#390×C₆₇	C₂²	1	4	yes	—
38	#390×C₆₇	C₂²	1	4	yes	—
39	#388×C₆₇	C₂²	1	8	yes	—
40	#388×C₆₇	C₂²	1	8	yes	—
41	#386×C₆₇	C₂²	1	4	yes	—
42	#386×C₆₇	C₂²	1	4	yes	—
43	#386×C₆₇	C₂²	1	8	yes	—
44	#382×C₆₇	C₂²	1	4	yes	—
45	#380×C₆₇	C₂²	1	4	yes	—
46	#380×C₆₇	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
51	C₂×#95×C₆₇	C₂²	1	2	yes	—
52	C₂×#41×C₆₇	—	2	8	no	—
53	C₂×#41×C₆₇	—	2	8	no	—
54	C₂×#41×C₆₇	—	2	8	no	—
55	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₇	—	2	16	no	—
67	#125×C₆₇	—	2	16	no	—
68	#124×C₆₇	—	4	8	no	—
69	C₂²⋊Q₈×C₆₇	—	2	16	no	—
70	C₂²⋊Q₈×C₆₇	—	2	16	no	—
71	#121×C₆₇	—	2	16	no	—
72	#121×C₆₇	—	2	16	no	—
73	#117×C₆₇	—	2	8	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
91	C₂×C₄²×C₆₇	—	2	4	no	—
92	C₂×C₄²×C₆₇	—	4	4	no	—
93	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
94	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
98	#95×C₆₇	C₂³	1	8	yes	—
99	#95×C₆₇	C₂³	1	4	yes	—
100	#95×C₆₇	C₂³	1	4	yes	—
101	#95×C₆₇	C₂³	1	4	yes	—
102	#95×C₆₇	C₂³	1	4	yes	—
103	#95×C₆₇	C₂³	1	8	yes	—
104	#95×C₆₇	C₂³	1	8	yes	—
105	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
106	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
107	#41×C₆₇	—	4	32	no	—
108	C₂×Q₈×C₆₇	—	2	4	no	—
109	C₂×Q₈×C₆₇	—	2	8	no	—
110	C₂×Q₈×C₆₇	—	2	4	no	—
111	C₂×Q₈×C₆₇	—	2	4	no	—
112	C₂×Q₈×C₆₇	—	4	4	no	—
113	C₂×Q₈×C₆₇	—	4	4	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	2	8	no	—
116	C₂×D₄×C₆₇	—	2	8	no	—
117	C₂×D₄×C₆₇	—	2	4	no	—
118	C₂×D₄×C₆₇	—	2	8	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	4	4	no	—
122	C₂×D₄×C₆₇	—	4	4	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	8	no	—
129	C₂²×C₄×C₆₇	—	2	8	no	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	—	4	2	no	—
133	C₂²×C₄×C₆₇	—	4	2	no	—
134	C₂²×C₄×C₆₇	—	4	2	no	—
135	C₂²×C₄×C₆₇	—	4	1	no	—
136	C₂²×C₄×C₆₇	—	4	1	no	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₆₇	#41	1	4	yes	—
140	C₂²×C₄×C₆₇	#41	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	8	no	—
153	C₄⋊C₄×C₆₇	—	2	8	no	—
154	C₄⋊C₄×C₆₇	—	2	16	no	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
157	C₂²⋊C₄×C₆₇	—	2	8	no	—
158	C₂²⋊C₄×C₆₇	—	2	16	no	—
159	C₂²⋊C₄×C₆₇	—	2	8	no	—
160	C₂²⋊C₄×C₆₇	—	2	8	no	—
161	C₂²⋊C₄×C₆₇	—	4	4	no	—
162	C₂²⋊C₄×C₆₇	—	4	8	no	—
163	C₂²⋊C₄×C₆₇	—	4	4	no	—
164	C₂²⋊C₄×C₆₇	—	4	8	no	—
165	C₄²×C₆₇	—	2	8	no	—
166	C₄²×C₆₇	—	2	4	no	—
167	C₄²×C₆₇	—	2	4	no	—
168	C₄²×C₆₇	—	2	8	no	—
169	C₄²×C₆₇	—	4	4	no	—
170	C₂³×C₆₇	—	2	1	no	—
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	—	2	1	no	—
173	C₂³×C₆₇	—	2	1	no	—
174	C₂³×C₆₇	—	2	4	no	—
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
182	C₂³×C₆₇	C₂×#41	1	2	yes	—
183	C₂³×C₆₇	C₂×#41	1	2	yes	—
184	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
185	Q₈×C₆₇	—	4	8	no	—
186	Q₈×C₆₇	—	8	8	no	—
187	D₄×C₆₇	—	4	16	no	—
188	D₄×C₆₇	—	4	8	no	—
189	D₄×C₆₇	—	8	8	no	—
190	C₂×C₄×C₆₇	—	2	8	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	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	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	4	8	no	—
213	C₂×C₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	4	4	no	—
215	C₂×C₄×C₆₇	—	4	16	no	—
216	C₂×C₄×C₆₇	#124	1	16	yes	complemented
217	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
220	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
221	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
222	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
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	—	4	4	no	—
230	C₂²×C₆₇	—	4	4	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	C₂×#124	1	4	yes	—
237	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
238	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
239	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
240	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
242	C₂²×C₆₇	#585	1	1	yes	characteristic
243	C₂²×C₆₇	#585	1	1	yes	characteristic
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	8	no	—
247	C₄×C₆₇	—	4	4	no	—
248	C₄×C₆₇	—	4	4	no	—
249	C₄×C₆₇	—	4	2	no	—
250	C₄×C₆₇	—	4	4	no	—
251	C₄×C₆₇	—	4	4	no	—
252	C₄×C₆₇	C₂×#124	1	8	yes	—
253	#35158	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂×C₆₇	—	2	4	no	—
255	C₂×C₆₇	—	8	4	no	—
256	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
257	C₂×C₆₇	#3460	1	1	yes	characteristic
258	C₂×C₆₇	#3461	1	1	yes	characteristic
259	C₂×C₆₇	#3492	1	1	yes	characteristic
260	C₂×C₆₇	#3493	1	1	yes	characteristic
261	C₂×C₆₇	#3496	1	1	yes	characteristic
262	C₂×C₆₇	#3497	1	1	yes	characteristic
263	C₂×#544	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₂×#537	C₂×C₆₇	1	1	yes	characteristic
265	#2850	C₂×C₆₇	1	4	yes	complemented
266	#2639	C₂×C₆₇	1	4	yes	complemented
267	#2492	C₂×C₆₇	1	4	yes	complemented
268	#2491	C₂×C₆₇	1	4	yes	complemented
269	#2490	C₂×C₆₇	1	4	yes	complemented
270	#2489	C₂×C₆₇	1	4	yes	complemented
271	#2311	C₂×C₆₇	1	2	yes	complemented
272	#2297	C₂×C₆₇	1	2	yes	complemented
273	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
274	C₆₇	#35158	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²×#41	C₂²×C₆₇	1	2	yes	—
276	#544	—	2	8	no	—
277	#537	—	2	8	no	—
278	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
279	C₂×#125	C₂²×C₆₇	1	4	yes	—
280	C₂×#124	C₂²×C₆₇	1	4	yes	—
281	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
282	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
283	C₂×#121	C₂²×C₆₇	1	4	yes	—
284	C₂×#121	C₂²×C₆₇	1	4	yes	—
285	C₂×#117	C₂²×C₆₇	1	2	yes	—
286	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
287	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
288	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
289	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
290	#400	C₂²×C₆₇	1	8	yes	—
291	#397	C₂²×C₆₇	1	8	yes	—
292	#394	C₂²×C₆₇	1	8	yes	—
293	#393	C₂²×C₆₇	1	8	yes	—
294	#391	C₂²×C₆₇	1	4	yes	—
295	#391	C₂²×C₆₇	1	4	yes	—
296	#391	C₂²×C₆₇	1	2	yes	—
297	#391	C₂²×C₆₇	1	2	yes	—
298	#390	C₂²×C₆₇	1	4	yes	—
299	#390	C₂²×C₆₇	1	4	yes	—
300	#390	C₂²×C₆₇	1	2	yes	—
301	#390	C₂²×C₆₇	1	2	yes	—
302	#388	C₂²×C₆₇	1	8	yes	—
303	#388	C₂²×C₆₇	1	8	yes	—
304	#386	C₂²×C₆₇	1	8	yes	—
305	#386	C₂²×C₆₇	1	4	yes	—
306	#386	C₂²×C₆₇	1	4	yes	—
307	#382	C₂²×C₆₇	1	4	yes	—
308	#380	C₂²×C₆₇	1	4	yes	—
309	#380	C₂²×C₆₇	1	4	yes	—
310	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
311	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
312	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
313	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
314	C₂×#95	C₂²×C₆₇	1	2	yes	—
315	C₂×#41	—	2	8	no	—
316	C₂×#41	—	2	8	no	—
317	C₂×#41	—	2	8	no	—
318	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
319	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
320	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
321	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
322	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
324	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
325	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
326	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
328	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
329	C₄⋊Q₈	—	2	16	no	—
330	#125	—	2	16	no	—
331	#124	—	4	8	no	—
332	C₂²⋊Q₈	—	2	16	no	—
333	C₂²⋊Q₈	—	2	16	no	—
334	#121	—	2	16	no	—
335	#121	—	2	16	no	—
336	#117	—	2	8	no	—
337	C₂×C₄⋊C₄	—	2	4	no	—
338	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	—
339	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
345	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
354	C₂×C₄²	—	2	4	no	—
355	C₂×C₄²	—	4	4	no	—
356	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
357	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
358	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
359	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
360	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
361	#95	C₂³×C₆₇	1	8	yes	—
362	#95	C₂³×C₆₇	1	4	yes	—
363	#95	C₂³×C₆₇	1	4	yes	—
364	#95	C₂³×C₆₇	1	4	yes	—
365	#95	C₂³×C₆₇	1	4	yes	—
366	#95	C₂³×C₆₇	1	8	yes	—
367	#95	C₂³×C₆₇	1	8	yes	—
368	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
369	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
370	#41	—	4	32	no	—
371	C₂×Q₈	—	2	4	no	—
372	C₂×Q₈	—	2	8	no	—
373	C₂×Q₈	—	2	4	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	4	4	no	—
376	C₂×Q₈	—	4	4	no	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	8	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	8	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	4	4	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	8	no	—
392	C₂²×C₄	—	2	8	no	—
393	C₂²×C₄	—	2	8	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	—	4	1	no	—
400	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
401	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
402	C₂²×C₄	#41×C₆₇	1	4	yes	—
403	C₂²×C₄	#41×C₆₇	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	16	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	16	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	4	8	no	—
425	C₂²⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	—	4	8	no	—
427	C₂²⋊C₄	—	4	4	no	—
428	C₄²	—	2	8	no	—
429	C₄²	—	2	4	no	—
430	C₄²	—	2	4	no	—
431	C₄²	—	2	8	no	—
432	C₄²	—	4	4	no	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	4	no	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	4	no	—
444	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
445	C₂³	C₂×#41×C₆₇	1	2	yes	—
446	C₂³	C₂×#41×C₆₇	1	2	yes	—
447	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
448	Q₈	—	4	8	no	—
449	Q₈	—	8	8	no	—
450	D₄	—	4	16	no	—
451	D₄	—	4	8	no	—
452	D₄	—	8	8	no	—
453	C₂×C₄	—	2	8	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	16	no	—
479	C₂×C₄	#124×C₆₇	1	16	yes	complemented
480	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
481	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
482	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
483	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
484	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
485	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
486	C₂²	—	2	2	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	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	4	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	C₂×#124×C₆₇	1	4	yes	—
500	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
501	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
502	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
503	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
504	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
505	C₂²	#585×C₆₇	1	1	yes	characteristic
506	C₂²	#585×C₆₇	1	1	yes	characteristic
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	8	no	—
510	C₄	—	4	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	C₂×#124×C₆₇	1	8	yes	—
516	C₂	—	2	4	no	—
517	C₂	—	8	4	no	—
518	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
519	C₂	C₆₇×#3460	1	1	yes	characteristic
520	C₂	C₆₇×#3461	1	1	yes	characteristic
521	C₂	C₆₇×#3492	1	1	yes	characteristic
522	C₂	C₆₇×#3493	1	1	yes	characteristic
523	C₂	C₆₇×#3496	1	1	yes	characteristic
524	C₂	C₆₇×#3497	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#537×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#544×C₆₇	C₂	semidirect
3. C₆₇×#2850	C₂	semidirect
4. C₆₇×#2639	C₂	semidirect
5. C₆₇×#2492	C₂	semidirect
6. C₆₇×#2491	C₂	semidirect
7. C₆₇×#2490	C₂	semidirect
8. C₆₇×#2489	C₂	semidirect
9. C₆₇×#2311	C₂	semidirect
10. C₆₇×#2297	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
216. C₂×C₄×C₆₇	#124	semidirect
253. #35158	C₆₇	direct (N × H)
263. C₂×#544	C₂×C₆₇	semidirect
265. #2850	C₂×C₆₇	semidirect
266. #2639	C₂×C₆₇	semidirect
267. #2492	C₂×C₆₇	semidirect
268. #2491	C₂×C₆₇	semidirect
269. #2490	C₂×C₆₇	semidirect
270. #2489	C₂×C₆₇	semidirect
271. #2311	C₂×C₆₇	semidirect
272. #2297	C₂×C₆₇	semidirect
273. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
274. C₆₇	#35158	direct (N × H)
479. C₂×C₄	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#544×C₆₇	2
C₂×#537×C₆₇	2
C₆₇×#2850	2
C₆₇×#2639	2
C₆₇×#2492	2
C₆₇×#2491	2
C₆₇×#2490	2
C₆₇×#2489	2
C₆₇×#2311	2
C₆₇×#2297	2
C₂×C₄×C₄⋊C₄×C₆₇	2
#35158	67

Subgroups — C67×#35158 #30714409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35158 #30714409