Subgroups — C₆₇×#21901 #30701152

Order 17152.

1058 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3455	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1986	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1980	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1897	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1896	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1806	C₂	1	4	yes	maximal, complemented
9	#542×C₆₇	C₄	1	4	yes	complemented
10	#541×C₆₇	C₄	1	2	yes	complemented
11	#540×C₆₇	C₄	1	2	yes	complemented
12	#539×C₆₇	—	2	2	no	—
13	#538×C₆₇	—	2	2	no	—
14	#533×C₆₇	C₂²	1	1	yes	characteristic
15	#532×C₆₇	C₂²	1	1	yes	characteristic
16	#529×C₆₇	—	2	4	no	—
17	#525×C₆₇	C₂²	1	2	yes	—
18	C₄×#41×C₆₇	C₂²	1	2	yes	—
19	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#433×C₆₇	—	2	4	no	—
23	#423×C₆₇	C₂²	1	2	yes	—
24	C₂×#104×C₆₇	C₂²	1	4	yes	—
25	C₂×#104×C₆₇	C₂²	1	4	yes	—
26	#407×C₆₇	C₂²	1	2	yes	—
27	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
28	#389×C₆₇	—	2	2	no	—
29	#385×C₆₇	—	2	2	no	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	#339×C₆₇	C₂²	1	4	yes	—
33	#339×C₆₇	C₂²	1	4	yes	—
34	C₂×#41×C₆₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
39	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	complemented
40	#127×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
41	#126×C₆₇	—	2	4	no	—
42	#126×C₆₇	—	2	4	no	—
43	#125×C₆₇	—	2	4	no	—
44	#125×C₆₇	C₂×C₄	1	2	yes	complemented
45	#124×C₆₇	—	2	4	no	—
46	#124×C₆₇	C₂×C₄	1	2	yes	complemented
47	#123×C₆₇	—	2	2	no	—
48	#123×C₆₇	—	2	4	no	—
49	#123×C₆₇	—	2	4	no	—
50	C₂²⋊Q₈×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	2	4	no	—
52	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
53	#121×C₆₇	—	2	2	no	—
54	#121×C₆₇	—	2	4	no	—
55	#121×C₆₇	C₂×C₄	1	4	yes	—
56	#120×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
59	C₄×D₄×C₆₇	—	2	4	no	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
63	#117×C₆₇	—	2	2	no	—
64	#117×C₆₇	—	4	2	no	—
65	#117×C₆₇	C₂×C₄	1	2	yes	—
66	#117×C₆₇	C₂×C₄	1	2	yes	—
67	#117×C₆₇	D₄	1	4	yes	—
68	#117×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	4	2	no	—
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₆₇	—	2	4	no	—
83	#104×C₆₇	—	4	4	no	—
84	#104×C₆₇	—	4	4	no	—
85	C₂²⋊C₈×C₆₇	—	2	4	no	—
86	#97×C₆₇	—	2	2	no	—
87	C₄×C₈×C₆₇	—	2	2	no	—
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	2	no	—
90	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
91	#41×C₆₇	—	2	4	no	—
92	#41×C₆₇	—	2	4	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	4	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	4	4	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₆₇	—	2	4	no	—
114	M₁₆×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	4	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	4	4	no	—
122	C₄⋊C₄×C₆₇	—	4	4	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
125	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
126	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
127	C₂²⋊C₄×C₆₇	—	2	4	no	—
128	C₂²⋊C₄×C₆₇	—	2	2	no	—
129	C₂²⋊C₄×C₆₇	—	2	4	no	—
130	C₂²⋊C₄×C₆₇	—	2	4	no	—
131	C₂²⋊C₄×C₆₇	—	2	4	no	—
132	C₂²⋊C₄×C₆₇	—	4	2	no	—
133	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
134	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
135	C₄²×C₆₇	—	2	4	no	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	2	4	no	—
138	C₄²×C₆₇	—	2	4	no	—
139	C₄²×C₆₇	—	4	2	no	—
140	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
141	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₆₇	C₂×D₄	1	2	yes	—
144	C₄²×C₆₇	C₂×D₄	1	2	yes	—
145	C₂³×C₆₇	—	2	1	no	—
146	C₂³×C₆₇	—	2	1	no	—
147	C₂³×C₆₇	—	2	2	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	2	4	no	—
152	Q₈×C₆₇	—	8	2	no	—
153	D₄×C₆₇	—	2	4	no	—
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	4	2	no	—
156	D₄×C₆₇	—	8	2	no	—
157	D₄×C₆₇	—	8	4	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	4	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	4	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	8	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	8	4	no	—
175	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
176	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
182	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
183	C₈×C₆₇	—	4	4	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
192	C₂²×C₆₇	#535	1	1	yes	characteristic
193	C₂²×C₆₇	#536	1	1	yes	characteristic
194	C₄×C₆₇	—	2	2	no	—
195	C₄×C₆₇	—	2	4	no	—
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	2	2	no	—
198	C₄×C₆₇	—	4	4	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	4	4	no	—
201	C₄×C₆₇	—	8	2	no	—
202	C₄×C₆₇	#520	1	2	yes	—
203	C₄×C₆₇	#537	1	2	yes	—
204	#21901	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	2	1	no	—
207	C₂×C₆₇	—	4	2	no	—
208	C₂×C₆₇	—	8	2	no	—
209	C₂×C₆₇	#2311	1	1	yes	characteristic
210	C₂×C₆₇	#2901	1	2	yes	—
211	#3455	C₂×C₆₇	1	1	yes	characteristic
212	#2921	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#2296	C₂×C₆₇	1	1	yes	characteristic, complemented
214	#1986	C₂×C₆₇	1	2	yes	complemented
215	#1980	C₂×C₆₇	1	2	yes	complemented
216	#1897	C₂×C₆₇	1	2	yes	complemented
217	#1896	C₂×C₆₇	1	2	yes	complemented
218	#1806	C₂×C₆₇	1	4	yes	complemented
219	C₆₇	#21901	1	1	yes	characteristic, direct product factor, Sylow
220	#542	C₄×C₆₇	1	4	yes	complemented
221	#541	C₄×C₆₇	1	2	yes	complemented
222	#540	C₄×C₆₇	1	2	yes	complemented
223	#539	—	2	2	no	—
224	#538	—	2	2	no	—
225	#533	C₂²×C₆₇	1	1	yes	characteristic
226	#532	C₂²×C₆₇	1	1	yes	characteristic
227	#529	—	2	4	no	—
228	#525	C₂²×C₆₇	1	2	yes	—
229	C₄×#41	C₂²×C₆₇	1	2	yes	—
230	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
232	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
233	#433	—	2	4	no	—
234	#423	C₂²×C₆₇	1	2	yes	—
235	C₂×#104	C₂²×C₆₇	1	4	yes	—
236	C₂×#104	C₂²×C₆₇	1	4	yes	—
237	#407	C₂²×C₆₇	1	2	yes	—
238	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
239	#389	—	2	2	no	—
240	#385	—	2	2	no	—
241	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
242	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
243	#339	C₂²×C₆₇	1	4	yes	—
244	#339	C₂²×C₆₇	1	4	yes	—
245	C₂×#41	C₂³×C₆₇	1	2	yes	—
246	C₂³×C₄	—	2	1	no	—
247	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
248	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
249	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
250	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	complemented
251	#127	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
252	#126	—	2	4	no	—
253	#126	—	2	4	no	—
254	#125	—	2	4	no	—
255	#125	C₂×C₄×C₆₇	1	2	yes	complemented
256	#124	—	2	4	no	—
257	#124	C₂×C₄×C₆₇	1	2	yes	complemented
258	#123	—	2	2	no	—
259	#123	—	2	4	no	—
260	#123	—	2	4	no	—
261	C₂²⋊Q₈	—	2	2	no	—
262	C₂²⋊Q₈	—	2	4	no	—
263	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
264	#121	—	2	2	no	—
265	#121	—	2	4	no	—
266	#121	C₂×C₄×C₆₇	1	4	yes	—
267	#120	—	2	2	no	—
268	C₄×Q₈	—	2	2	no	—
269	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
270	C₄×D₄	—	2	4	no	—
271	C₄×D₄	—	2	4	no	—
272	C₄×D₄	—	2	2	no	—
273	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
274	#117	—	2	2	no	—
275	#117	—	4	2	no	—
276	#117	C₂×C₄×C₆₇	1	2	yes	—
277	#117	C₂×C₄×C₆₇	1	2	yes	—
278	#117	D₄×C₆₇	1	4	yes	—
279	#117	C₂³×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
284	C₂×C₂²⋊C₄	—	2	1	no	—
285	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	—	4	2	no	—
289	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
290	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
291	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	C₄⋊C₈	—	2	4	no	—
294	#104	—	4	4	no	—
295	#104	—	4	4	no	—
296	C₂²⋊C₈	—	2	4	no	—
297	#97	—	2	2	no	—
298	C₄×C₈	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
302	#41	—	2	4	no	—
303	#41	—	2	4	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	4	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	4	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
323	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
324	M₁₆	—	2	4	no	—
325	M₁₆	—	2	4	no	—
326	C₂×C₈	—	2	4	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	4	4	no	—
334	C₄⋊C₄	—	4	4	no	—
335	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
336	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
337	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
338	C₂²⋊C₄	—	2	4	no	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	2	4	no	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₂²⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	4	2	no	—
344	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
345	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
346	C₄²	—	2	4	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	4	no	—
349	C₄²	—	2	4	no	—
350	C₄²	—	4	2	no	—
351	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
352	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
353	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₆₇	1	2	yes	—
355	C₄²	C₂×D₄×C₆₇	1	2	yes	—
356	C₂³	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
361	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	8	2	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	2	no	—
368	D₄	—	8	4	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	8	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	8	4	no	—
386	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
387	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
388	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
389	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
393	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
394	C₈	—	4	4	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	4	no	—
402	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
403	C₂²	#535×C₆₇	1	1	yes	characteristic
404	C₂²	#536×C₆₇	1	1	yes	characteristic
405	C₄	—	2	2	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	4	no	—
412	C₄	—	8	2	no	—
413	C₄	#520×C₆₇	1	2	yes	—
414	C₄	#537×C₆₇	1	2	yes	—
415	C₂	—	2	1	no	—
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	8	2	no	—
419	C₂	C₆₇×#2311	1	1	yes	characteristic
420	C₂	C₆₇×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2921	C₂	semidirect
3. C₆₇×#2296	C₂	semidirect
4. C₆₇×#1986	C₂	semidirect
5. C₆₇×#1980	C₂	semidirect
6. C₆₇×#1897	C₂	semidirect
7. C₆₇×#1896	C₂	semidirect
8. C₆₇×#1806	C₂	semidirect
9. #542×C₆₇	C₄	semidirect
10. #541×C₆₇	C₄	semidirect
11. #540×C₆₇	C₄	semidirect
38. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
39. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
40. #127×C₆₇	C₂×C₄	semidirect
44. #125×C₆₇	C₂×C₄	semidirect
46. #124×C₆₇	C₂×C₄	semidirect
58. C₄×Q₈×C₆₇	C₂×C₄	semidirect
62. C₄×D₄×C₆₇	C₂×C₄	semidirect
204. #21901	C₆₇	direct (N × H)
212. #2921	C₂×C₆₇	semidirect
213. #2296	C₂×C₆₇	semidirect
214. #1986	C₂×C₆₇	semidirect
215. #1980	C₂×C₆₇	semidirect
216. #1897	C₂×C₆₇	semidirect
217. #1896	C₂×C₆₇	semidirect
218. #1806	C₂×C₆₇	semidirect
219. C₆₇	#21901	direct (N × H)
220. #542	C₄×C₆₇	semidirect
221. #541	C₄×C₆₇	semidirect
222. #540	C₄×C₆₇	semidirect
249. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
250. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
251. #127	C₂×C₄×C₆₇	semidirect
255. #125	C₂×C₄×C₆₇	semidirect
257. #124	C₂×C₄×C₆₇	semidirect
269. C₄×Q₈	C₂×C₄×C₆₇	semidirect
273. C₄×D₄	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
C₆₇×#3455	2
C₆₇×#2921	2
C₆₇×#2296	2
C₆₇×#1986	2
C₆₇×#1980	2
C₆₇×#1897	2
C₆₇×#1896	2
C₆₇×#1806	2
#21901	67

Subgroups — C67×#21901 #30701152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21901 #30701152