Subgroups — C₆₇×#35153 #30714404

Order 17152.

2946 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#537×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#536×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2854	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2618	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2429	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2426	C₂	1	8	yes	maximal, complemented
8	C₆₇×#2316	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2297	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₆₇	C₂²	1	2	yes	—
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	#537×C₆₇	—	2	8	no	—
13	#536×C₆₇	—	2	16	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
15	C₂×#124×C₆₇	C₂²	1	4	yes	—
16	C₂×#123×C₆₇	C₂²	1	8	yes	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
19	C₂×#121×C₆₇	C₂²	1	4	yes	—
20	C₂×#121×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	—
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
28	#398×C₆₇	C₂²	1	8	yes	—
29	#396×C₆₇	C₂²	1	16	yes	—
30	#394×C₆₇	C₂²	1	8	yes	—
31	#391×C₆₇	C₂²	1	4	yes	—
32	#390×C₆₇	C₂²	1	4	yes	—
33	#390×C₆₇	C₂²	1	8	yes	—
34	#389×C₆₇	C₂²	1	8	yes	—
35	#389×C₆₇	C₂²	1	8	yes	—
36	#388×C₆₇	C₂²	1	16	yes	—
37	#386×C₆₇	C₂²	1	4	yes	—
38	#386×C₆₇	C₂²	1	4	yes	—
39	#380×C₆₇	C₂²	1	4	yes	—
40	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
41	C₂×#41×C₆₇	—	2	8	no	—
42	C₂×#41×C₆₇	—	2	8	no	—
43	C₂×#41×C₆₇	—	2	8	no	—
44	C₂²×Q₈×C₆₇	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	2	4	no	—
46	C₂²×Q₈×C₆₇	D₄	1	8	yes	complemented
47	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
49	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₆₇	—	2	16	no	—
59	#124×C₆₇	—	2	16	no	—
60	#123×C₆₇	—	2	32	no	—
61	C₂²⋊Q₈×C₆₇	—	2	32	no	—
62	C₂²⋊Q₈×C₆₇	—	2	16	no	—
63	#121×C₆₇	—	2	16	no	—
64	#121×C₆₇	—	2	16	no	—
65	C₄×Q₈×C₆₇	—	2	32	no	—
66	#117×C₆₇	—	2	8	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
68	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₄²×C₆₇	—	2	8	no	—
81	C₂×C₄²×C₆₇	—	2	4	no	—
82	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
84	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
85	#95×C₆₇	C₂³	1	8	yes	—
86	#95×C₆₇	C₂³	1	8	yes	—
87	#95×C₆₇	C₂³	1	8	yes	—
88	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
89	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
90	#41×C₆₇	—	4	32	no	—
91	C₂×Q₈×C₆₇	—	2	32	no	—
92	C₂×Q₈×C₆₇	—	2	8	no	—
93	C₂×Q₈×C₆₇	—	2	4	no	—
94	C₂×Q₈×C₆₇	—	2	4	no	—
95	C₂×Q₈×C₆₇	—	2	8	no	—
96	C₂×Q₈×C₆₇	—	2	4	no	—
97	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	—
98	C₂×D₄×C₆₇	—	2	8	no	—
99	C₂×D₄×C₆₇	—	2	8	no	—
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	8	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	4	8	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	4	4	no	—
115	C₂²×C₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
120	C₂²×C₄×C₆₇	#41	1	4	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
128	C₄⋊C₄×C₆₇	—	2	16	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	32	no	—
132	C₄⋊C₄×C₆₇	—	4	8	no	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
134	C₂²⋊C₄×C₆₇	—	2	8	no	—
135	C₂²⋊C₄×C₆₇	—	2	16	no	—
136	C₂²⋊C₄×C₆₇	—	2	32	no	—
137	C₂²⋊C₄×C₆₇	—	2	16	no	—
138	C₂²⋊C₄×C₆₇	—	4	8	no	—
139	C₄²×C₆₇	—	2	16	no	—
140	C₄²×C₆₇	—	2	8	no	—
141	C₄²×C₆₇	—	2	8	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	4	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	8	no	—
148	C₂³×C₆₇	—	2	8	no	—
149	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
150	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
151	C₂³×C₆₇	C₂×#41	1	2	yes	—
152	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
153	Q₈×C₆₇	—	2	32	no	—
154	Q₈×C₆₇	—	4	8	no	—
155	D₄×C₆₇	—	4	16	no	—
156	D₄×C₆₇	—	4	8	no	—
157	D₄×C₆₇	—	8	8	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	2	16	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	8	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	8	no	—
173	C₂×C₄×C₆₇	—	4	8	no	—
174	C₂×C₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	4	16	no	—
176	C₂×C₄×C₆₇	#121	1	16	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
181	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
182	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	8	no	—
189	C₂²×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	4	8	no	—
192	C₂²×C₆₇	C₂×#121	1	4	yes	—
193	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
194	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
195	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
196	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
197	C₂²×C₆₇	#585	1	2	yes	—
198	C₄×C₆₇	—	2	8	no	—
199	C₄×C₆₇	—	2	8	no	—
200	C₄×C₆₇	—	2	4	no	—
201	C₄×C₆₇	—	4	4	no	—
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	—	4	8	no	—
204	C₄×C₆₇	C₂×#121	1	8	yes	—
205	#35153	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₆₇	—	2	4	no	—
207	C₂×C₆₇	—	8	4	no	—
208	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
209	C₂×C₆₇	#3461	1	2	yes	—
210	C₂×C₆₇	#3485	1	1	yes	characteristic
211	C₂×C₆₇	#3486	1	1	yes	characteristic
212	C₂×C₆₇	#3497	1	2	yes	—
213	C₂×#537	C₂×C₆₇	1	1	yes	characteristic
214	C₂×#536	C₂×C₆₇	1	2	yes	complemented
215	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#2854	C₂×C₆₇	1	4	yes	complemented
217	#2618	C₂×C₆₇	1	4	yes	complemented
218	#2429	C₂×C₆₇	1	8	yes	complemented
219	#2426	C₂×C₆₇	1	8	yes	complemented
220	#2316	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#2297	C₂×C₆₇	1	2	yes	complemented
222	C₆₇	#35153	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²×#41	C₂²×C₆₇	1	2	yes	—
224	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
225	#537	—	2	8	no	—
226	#536	—	2	16	no	—
227	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
228	C₂×#124	C₂²×C₆₇	1	4	yes	—
229	C₂×#123	C₂²×C₆₇	1	8	yes	—
230	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	—
231	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
232	C₂×#121	C₂²×C₆₇	1	4	yes	—
233	C₂×#121	C₂²×C₆₇	1	4	yes	—
234	C₂×C₄×Q₈	—	2	8	no	—
235	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	—
236	C₂×#117	C₂²×C₆₇	1	2	yes	—
237	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
238	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
239	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
240	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
241	#398	C₂²×C₆₇	1	8	yes	—
242	#396	C₂²×C₆₇	1	16	yes	—
243	#394	C₂²×C₆₇	1	8	yes	—
244	#391	C₂²×C₆₇	1	4	yes	—
245	#390	C₂²×C₆₇	1	8	yes	—
246	#390	C₂²×C₆₇	1	4	yes	—
247	#389	C₂²×C₆₇	1	8	yes	—
248	#389	C₂²×C₆₇	1	8	yes	—
249	#388	C₂²×C₆₇	1	16	yes	—
250	#386	C₂²×C₆₇	1	4	yes	—
251	#386	C₂²×C₆₇	1	4	yes	—
252	#380	C₂²×C₆₇	1	4	yes	—
253	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
254	C₂×#41	—	2	8	no	—
255	C₂×#41	—	2	8	no	—
256	C₂×#41	—	2	8	no	—
257	C₂²×Q₈	—	2	4	no	—
258	C₂²×Q₈	—	2	4	no	—
259	C₂²×Q₈	D₄×C₆₇	1	8	yes	complemented
260	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
261	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
262	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
263	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
264	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
265	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
266	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
267	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
270	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	C₄⋊Q₈	—	2	16	no	—
272	#124	—	2	16	no	—
273	#123	—	2	32	no	—
274	C₂²⋊Q₈	—	2	32	no	—
275	C₂²⋊Q₈	—	2	16	no	—
276	#121	—	2	16	no	—
277	#121	—	2	16	no	—
278	C₄×Q₈	—	2	32	no	—
279	#117	—	2	8	no	—
280	C₂×C₄⋊C₄	—	2	8	no	—
281	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
290	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
293	C₂×C₄²	—	2	8	no	—
294	C₂×C₄²	—	2	4	no	—
295	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
296	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
297	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
298	#95	C₂³×C₆₇	1	8	yes	—
299	#95	C₂³×C₆₇	1	8	yes	—
300	#95	C₂³×C₆₇	1	8	yes	—
301	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
302	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
303	#41	—	4	32	no	—
304	C₂×Q₈	—	2	32	no	—
305	C₂×Q₈	—	2	8	no	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	—	2	4	no	—
308	C₂×Q₈	—	2	8	no	—
309	C₂×Q₈	—	2	4	no	—
310	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	—
311	C₂×D₄	—	2	8	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	2	8	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	2	4	no	—
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	4	8	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	2	8	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	4	4	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
330	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
331	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
333	C₂²×C₄	#41×C₆₇	1	4	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
336	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
337	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
338	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
339	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
341	C₄⋊C₄	—	2	16	no	—
342	C₄⋊C₄	—	2	8	no	—
343	C₄⋊C₄	—	2	8	no	—
344	C₄⋊C₄	—	2	32	no	—
345	C₄⋊C₄	—	4	8	no	—
346	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
347	C₂²⋊C₄	—	2	8	no	—
348	C₂²⋊C₄	—	2	16	no	—
349	C₂²⋊C₄	—	2	32	no	—
350	C₂²⋊C₄	—	2	16	no	—
351	C₂²⋊C₄	—	4	8	no	—
352	C₄²	—	2	16	no	—
353	C₄²	—	2	8	no	—
354	C₄²	—	2	8	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	2	2	no	—
357	C₂³	—	2	4	no	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	8	no	—
361	C₂³	—	2	8	no	—
362	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
363	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
364	C₂³	C₂×#41×C₆₇	1	2	yes	—
365	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
366	Q₈	—	2	32	no	—
367	Q₈	—	4	8	no	—
368	D₄	—	4	16	no	—
369	D₄	—	4	8	no	—
370	D₄	—	8	8	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	16	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	8	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	8	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	8	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	16	no	—
389	C₂×C₄	#121×C₆₇	1	16	yes	—
390	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
391	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
392	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
393	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
394	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
395	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
396	C₂²	—	2	4	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	4	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	4	8	no	—
402	C₂²	—	4	4	no	—
403	C₂²	—	4	4	no	—
404	C₂²	—	4	8	no	—
405	C₂²	C₂×#121×C₆₇	1	4	yes	—
406	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
407	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
408	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
409	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
410	C₂²	#585×C₆₇	1	2	yes	—
411	C₄	—	2	8	no	—
412	C₄	—	2	8	no	—
413	C₄	—	2	4	no	—
414	C₄	—	4	4	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	8	no	—
417	C₄	C₂×#121×C₆₇	1	8	yes	—
418	C₂	—	2	4	no	—
419	C₂	—	8	4	no	—
420	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
421	C₂	C₆₇×#3461	1	2	yes	—
422	C₂	C₆₇×#3485	1	1	yes	characteristic
423	C₂	C₆₇×#3486	1	1	yes	characteristic
424	C₂	C₆₇×#3497	1	2	yes	—

	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
2. C₂×#536×C₆₇	C₂	semidirect
3. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#2854	C₂	semidirect
5. C₆₇×#2618	C₂	semidirect
6. C₆₇×#2429	C₂	semidirect
7. C₆₇×#2426	C₂	semidirect
8. C₆₇×#2316	C₂	semidirect
9. C₆₇×#2297	C₂	semidirect
46. C₂²×Q₈×C₆₇	D₄	semidirect
205. #35153	C₆₇	direct (N × H)
214. C₂×#536	C₂×C₆₇	semidirect
215. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
216. #2854	C₂×C₆₇	semidirect
217. #2618	C₂×C₆₇	semidirect
218. #2429	C₂×C₆₇	semidirect
219. #2426	C₂×C₆₇	semidirect
220. #2316	C₂×C₆₇	semidirect
221. #2297	C₂×C₆₇	semidirect
222. C₆₇	#35153	direct (N × H)
259. C₂²×Q₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#537×C₆₇	2
C₂×#536×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2854	2
C₆₇×#2618	2
C₆₇×#2429	2
C₆₇×#2426	2
C₆₇×#2316	2
C₆₇×#2297	2
#35153	67

Subgroups — C67×#35153 #30714404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35153 #30714404