Subgroups — C₆₇×#10870 #30690121

Order 17152.

774 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#343×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×#1736	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
11	C₂×#108×C₆₇	—	2	2	no	—
12	C₂×#108×C₆₇	—	2	2	no	—
13	C₂×#108×C₆₇	C₄	1	4	yes	complemented
14	C₂×#107×C₆₇	—	2	1	no	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	—	2	2	no	—
17	C₂×#107×C₆₇	C₄	1	4	yes	complemented
18	#405×C₆₇	—	2	2	no	—
19	#405×C₆₇	C₄	1	4	yes	complemented
20	C₂×#97×C₆₇	—	2	1	no	—
21	C₂×C₄×C₈×C₆₇	—	2	1	no	—
22	#343×C₆₇	—	2	4	no	—
23	#340×C₆₇	—	2	4	no	—
24	C₂³×C₄×C₆₇	—	2	2	no	—
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₆₇	—	2	1	no	—
27	C₂×M₁₆×C₆₇	—	2	1	no	—
28	C₂×M₁₆×C₆₇	—	2	1	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	2	2	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
37	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
38	C₂²×C₈×C₆₇	—	2	1	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	2	yes	—
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
48	#117×C₆₇	—	2	1	no	—
49	#117×C₆₇	—	2	1	no	—
50	#117×C₆₇	C₂×C₄	1	2	yes	—
51	#117×C₆₇	Q₈	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄²×C₆₇	—	2	1	no	—
61	#108×C₆₇	—	2	8	no	—
62	#107×C₆₇	—	2	4	no	—
63	#107×C₆₇	—	2	4	no	—
64	#97×C₆₇	—	2	4	no	—
65	C₄×C₈×C₆₇	—	2	4	no	—
66	C₂⁴×C₆₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	—	8	4	no	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	M₁₆×C₆₇	—	2	2	no	—
87	M₁₆×C₆₇	—	2	2	no	—
88	M₁₆×C₆₇	—	2	2	no	—
89	M₁₆×C₆₇	—	2	4	no	—
90	M₁₆×C₆₇	—	2	4	no	—
91	M₁₆×C₆₇	—	4	2	no	—
92	M₁₆×C₆₇	—	4	2	no	—
93	M₁₆×C₆₇	C₄⋊C₄	1	4	yes	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	—	4	2	no	—
99	C₂×C₈×C₆₇	—	4	2	no	—
100	C₂×C₈×C₆₇	C₄²	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
104	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
107	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
108	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	2	4	no	—
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	4	2	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
116	C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	#117	1	1	yes	characteristic
123	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
124	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
125	C₂³×C₆₇	#121	1	1	yes	characteristic
126	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂³×C₆₇	#124	1	1	yes	characteristic
128	C₂³×C₆₇	#125	1	1	yes	characteristic
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	8	4	no	—
134	C₂×C₄×C₆₇	—	8	4	no	—
135	C₂×C₄×C₆₇	—	8	4	no	—
136	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
159	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
160	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
164	C₈×C₆₇	—	2	4	no	—
165	C₈×C₆₇	—	4	4	no	—
166	C₈×C₆₇	#95	1	4	yes	—
167	C₈×C₆₇	#95	1	4	yes	—
168	C₂²×C₆₇	—	2	2	no	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	2	2	no	—
172	C₂²×C₆₇	—	2	1	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	2	1	no	—
176	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₆₇	#380	1	1	yes	characteristic
178	C₂²×C₆₇	#382	1	1	yes	characteristic
179	C₂²×C₆₇	#389	1	1	yes	characteristic
180	C₂²×C₆₇	#389	1	1	yes	characteristic
181	C₂²×C₆₇	#390	1	1	yes	characteristic
182	C₂²×C₆₇	#391	1	1	yes	characteristic
183	C₂²×C₆₇	#472	1	2	yes	—
184	C₂²×C₆₇	#473	1	2	yes	—
185	C₄×C₆₇	—	4	4	no	—
186	C₄×C₆₇	—	8	4	no	—
187	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
188	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
189	C₄×C₆₇	#380	1	1	yes	characteristic
190	C₄×C₆₇	#384	1	1	yes	characteristic
191	C₄×C₆₇	#389	1	1	yes	characteristic
192	C₄×C₆₇	#389	1	1	yes	characteristic
193	C₄×C₆₇	#390	1	1	yes	characteristic
194	C₄×C₆₇	#391	1	1	yes	characteristic
195	#10870	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	2	1	no	—
197	C₂×C₆₇	—	2	1	no	—
198	C₂×C₆₇	—	2	1	no	—
199	C₂×C₆₇	—	2	1	no	—
200	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
201	C₂×C₆₇	#1816	1	2	yes	—
202	C₂×C₆₇	#1952	1	2	yes	—
203	C₂×C₆₇	#1978	1	1	yes	characteristic
204	C₂×C₆₇	#1979	1	1	yes	characteristic
205	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic
206	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic
207	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
208	C₂×#343	C₂×C₆₇	1	2	yes	—
209	#1736	C₂×C₆₇	1	2	yes	—
210	C₆₇	#10870	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
212	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
213	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
214	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
215	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
216	C₂×#108	—	2	2	no	—
217	C₂×#108	—	2	2	no	—
218	C₂×#108	C₄×C₆₇	1	4	yes	complemented
219	C₂×#107	—	2	1	no	—
220	C₂×#107	—	2	1	no	—
221	C₂×#107	—	2	2	no	—
222	C₂×#107	C₄×C₆₇	1	4	yes	complemented
223	#405	—	2	2	no	—
224	#405	C₄×C₆₇	1	4	yes	complemented
225	C₂×#97	—	2	1	no	—
226	C₂×C₄×C₈	—	2	1	no	—
227	#343	—	2	4	no	—
228	#340	—	2	4	no	—
229	C₂³×C₄	—	2	2	no	—
230	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
231	C₂×M₁₆	—	2	1	no	—
232	C₂×M₁₆	—	2	1	no	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
240	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
241	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
242	C₂×M₁₆	Q₈×C₆₇	1	2	yes	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	2	no	—
245	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
246	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
247	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
248	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
249	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₈	D₄×C₆₇	1	2	yes	—
251	C₂²×C₈	D₄×C₆₇	1	2	yes	—
252	C₂²×C₈	Q₈×C₆₇	1	2	yes	—
253	#117	—	2	1	no	—
254	#117	—	2	1	no	—
255	#117	C₂×C₄×C₆₇	1	2	yes	—
256	#117	Q₈×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
263	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
264	C₂×C₂²⋊C₄	—	2	1	no	—
265	C₂×C₄²	—	2	1	no	—
266	#108	—	2	8	no	—
267	#107	—	2	4	no	—
268	#107	—	2	4	no	—
269	#97	—	2	4	no	—
270	C₄×C₈	—	2	4	no	—
271	C₂⁴	#41×C₆₇	1	1	yes	characteristic
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	8	4	no	—
277	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
289	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
291	M₁₆	—	2	2	no	—
292	M₁₆	—	2	2	no	—
293	M₁₆	—	2	2	no	—
294	M₁₆	—	2	4	no	—
295	M₁₆	—	2	4	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	2	no	—
298	M₁₆	C₄⋊C₄×C₆₇	1	4	yes	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	4	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	C₄²×C₆₇	1	4	yes	—
306	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
307	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
308	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
309	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
310	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
311	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
312	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
313	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
314	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₂³	—	2	2	no	—
326	C₂³	—	2	2	no	—
327	C₂³	#117×C₆₇	1	1	yes	characteristic
328	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
329	C₂³	C₄×Q₈×C₆₇	1	1	yes	characteristic
330	C₂³	#121×C₆₇	1	1	yes	characteristic
331	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
332	C₂³	#124×C₆₇	1	1	yes	characteristic
333	C₂³	#125×C₆₇	1	1	yes	characteristic
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	8	4	no	—
339	C₂×C₄	—	8	4	no	—
340	C₂×C₄	—	8	4	no	—
341	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
342	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₄×Q₈×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₄×Q₈×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₄×Q₈×C₆₇	1	1	yes	characteristic
357	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
358	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
359	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
360	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
361	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
362	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
363	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
364	C₂×C₄	#125×C₆₇	1	1	yes	characteristic
365	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
369	C₈	—	2	4	no	—
370	C₈	—	4	4	no	—
371	C₈	#95×C₆₇	1	4	yes	—
372	C₈	#95×C₆₇	1	4	yes	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
382	C₂²	#380×C₆₇	1	1	yes	characteristic
383	C₂²	#382×C₆₇	1	1	yes	characteristic
384	C₂²	#389×C₆₇	1	1	yes	characteristic
385	C₂²	#389×C₆₇	1	1	yes	characteristic
386	C₂²	#390×C₆₇	1	1	yes	characteristic
387	C₂²	#391×C₆₇	1	1	yes	characteristic
388	C₂²	#472×C₆₇	1	2	yes	—
389	C₂²	#473×C₆₇	1	2	yes	—
390	C₄	—	4	4	no	—
391	C₄	—	8	4	no	—
392	C₄	C₂×#95×C₆₇	1	1	yes	characteristic
393	C₄	C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
394	C₄	#380×C₆₇	1	1	yes	characteristic
395	C₄	#384×C₆₇	1	1	yes	characteristic
396	C₄	#389×C₆₇	1	1	yes	characteristic
397	C₄	#389×C₆₇	1	1	yes	characteristic
398	C₄	#390×C₆₇	1	1	yes	characteristic
399	C₄	#391×C₆₇	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	2	1	no	—
404	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
405	C₂	C₆₇×#1816	1	2	yes	—
406	C₂	C₆₇×#1952	1	2	yes	—
407	C₂	C₆₇×#1978	1	1	yes	characteristic
408	C₂	C₆₇×#1979	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#108×C₆₇	C₄	semidirect
17. C₂×#107×C₆₇	C₄	semidirect
19. #405×C₆₇	C₄	semidirect
195. #10870	C₆₇	direct (N × H)
210. C₆₇	#10870	direct (N × H)
218. C₂×#108	C₄×C₆₇	semidirect
222. C₂×#107	C₄×C₆₇	semidirect
224. #405	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#108×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#405×C₆₇	2
C₂×#343×C₆₇	2
C₆₇×#1736	2
#10870	67

Subgroups — C67×#10870 #30690121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10870 #30690121