Subgroups — M₁₆×D₈×C₆₇ #30697285

Order 17152.

794 conjugacy classes of subgroups.

The subgroup lattice has 378 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	D₄×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1589	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1581	C₂	1	4	yes	maximal, complemented
5	C₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1486	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	—	2	2	no	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
12	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
13	C₄×D₈×C₆₇	C₄	1	2	yes	—
14	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
15	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₆₇	C₂²	1	2	yes	—
17	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
18	#436×C₆₇	C₂²	1	4	yes	complemented
19	#435×C₆₇	—	2	4	no	—
20	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
21	#431×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
23	#423×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₄	1	2	yes	—
25	#407×C₆₇	—	2	2	no	—
26	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	#335×C₆₇	C₂²	1	2	yes	complemented
29	#335×C₆₇	C₂²	1	2	yes	complemented
30	D₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
31	D₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
32	#322×C₆₇	C₂²	1	2	yes	complemented
33	C₈²×C₆₇	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂×D₈×C₆₇	—	2	4	no	—
37	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	—	4	8	no	—
43	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₆₇	—	2	2	no	—
49	C₄×D₄×C₆₇	—	2	2	no	—
50	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₄×D₄×C₆₇	C₂³	1	2	yes	—
53	C₄×D₄×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#107×C₆₇	C₂×C₄	1	2	yes	—
58	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
59	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	2	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
63	#102×C₆₇	C₂×C₄	1	2	yes	—
64	#102×C₆₇	C₂×C₄	1	2	yes	—
65	#102×C₆₇	C₂×C₄	1	2	yes	—
66	#102×C₆₇	C₂×C₄	1	2	yes	—
67	C₂²⋊C₈×C₆₇	—	2	1	no	—
68	C₂²⋊C₈×C₆₇	—	2	4	no	—
69	C₂²⋊C₈×C₆₇	—	2	4	no	—
70	#97×C₆₇	D₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	2	yes	—
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂⁴×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	2	4	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	4	2	no	—
81	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	4	4	no	—
89	C₂²×C₄×C₆₇	—	4	4	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
93	D₈×C₆₇	—	2	4	no	—
94	D₈×C₆₇	—	2	2	no	—
95	D₈×C₆₇	M₁₆	1	4	yes	direct product factor
96	M₁₆×C₆₇	—	2	2	no	—
97	M₁₆×C₆₇	—	4	4	no	—
98	M₁₆×C₆₇	—	4	8	no	—
99	M₁₆×C₆₇	D₈	1	4	yes	direct product factor
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	4	1	no	—
104	C₂×C₈×C₆₇	—	4	8	no	—
105	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂²⋊C₄×C₆₇	—	2	2	no	—
116	C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₆₇	#41	1	1	yes	characteristic
121	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₂³×C₆₇	—	2	2	no	—
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	4	4	no	—
125	C₂³×C₆₇	—	4	4	no	—
126	C₂³×C₆₇	—	8	2	no	—
127	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
128	D₄×C₆₇	—	2	2	no	—
129	D₄×C₆₇	—	2	2	no	—
130	D₄×C₆₇	—	2	2	no	—
131	D₄×C₆₇	—	4	4	no	—
132	D₄×C₆₇	C₂×M₁₆	1	4	yes	complemented
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	4	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	4	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	C₄×D₄	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	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₆₇	#135	1	2	yes	—
150	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₇	—	2	1	no	—
154	C₈×C₆₇	—	2	1	no	—
155	C₈×C₆₇	—	2	1	no	—
156	C₈×C₆₇	—	2	2	no	—
157	C₈×C₆₇	—	4	2	no	—
158	C₈×C₆₇	—	4	4	no	—
159	C₈×C₆₇	C₂×M₁₆	1	2	yes	complemented
160	C₈×C₆₇	C₂×D₈	1	4	yes	complemented
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	2	4	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	4	2	no	—
169	C₂²×C₆₇	—	8	8	no	—
170	C₂²×C₆₇	C₄×D₈	1	2	yes	—
171	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	4	2	no	—
176	C₄×C₆₇	—	4	2	no	—
177	C₄×C₆₇	C₄×D₈	1	2	yes	—
178	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
179	C₄×C₆₇	#568	1	1	yes	characteristic
180	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
181	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
182	M₁₆×D₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₆₇	—	2	2	no	—
184	C₂×C₆₇	—	4	4	no	—
185	C₂×C₆₇	—	8	2	no	—
186	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
187	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
188	C₂×C₆₇	#2957	1	1	yes	characteristic
189	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
190	D₄×M₁₆	C₂×C₆₇	1	2	yes	complemented
191	#1589	C₂×C₆₇	1	2	yes	complemented
192	#1581	C₂×C₆₇	1	4	yes	complemented
193	C₈×D₈	C₂×C₆₇	1	2	yes	complemented
194	#1567	C₂×C₆₇	1	1	yes	characteristic, complemented
195	#1486	C₂×C₆₇	1	2	yes	complemented
196	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
197	C₆₇	M₁₆×D₈	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
199	C₂²×M₁₆	—	2	2	no	—
200	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
201	C₄×D₈	C₄×C₆₇	1	2	yes	—
202	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic
203	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic
204	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
205	C₄×D₈	C₂²×C₆₇	1	2	yes	—
206	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
207	#436	C₂²×C₆₇	1	4	yes	complemented
208	#435	—	2	4	no	—
209	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
210	#431	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×#107	C₄×C₆₇	1	1	yes	characteristic
212	#423	C₂²×C₆₇	1	2	yes	—
213	C₂×#102	C₄×C₆₇	1	2	yes	—
214	#407	—	2	2	no	—
215	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
217	#335	C₂²×C₆₇	1	2	yes	complemented
218	#335	C₂²×C₆₇	1	2	yes	complemented
219	D₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
220	D₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
221	#322	C₂²×C₆₇	1	2	yes	complemented
222	C₈²	C₂²×C₆₇	1	2	yes	complemented
223	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
224	C₂³×C₄	—	2	2	no	—
225	C₂×D₈	—	2	4	no	—
226	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
227	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
228	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
229	C₂×M₁₆	—	2	2	no	—
230	C₂×M₁₆	—	2	4	no	—
231	C₂×M₁₆	—	4	8	no	—
232	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
233	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₈	—	2	4	no	—
235	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
236	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
237	C₄×D₄	—	2	2	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
240	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
241	C₄×D₄	C₂³×C₆₇	1	2	yes	—
242	C₄×D₄	C₂³×C₆₇	1	2	yes	—
243	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
244	C₂×C₂²⋊C₄	—	2	2	no	—
245	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
246	#107	C₂×C₄×C₆₇	1	2	yes	—
247	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
248	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
252	#102	C₂×C₄×C₆₇	1	2	yes	—
253	#102	C₂×C₄×C₆₇	1	2	yes	—
254	#102	C₂×C₄×C₆₇	1	2	yes	—
255	#102	C₂×C₄×C₆₇	1	2	yes	—
256	C₂²⋊C₈	—	2	1	no	—
257	C₂²⋊C₈	—	2	4	no	—
258	C₂²⋊C₈	—	2	4	no	—
259	#97	D₄×C₆₇	1	2	yes	—
260	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
261	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₆₇	1	2	yes	—
263	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
264	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
265	C₄×C₈	C₂³×C₆₇	1	2	yes	—
266	C₂⁴	—	2	2	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
271	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	4	4	no	—
278	C₂²×C₄	—	4	4	no	—
279	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
282	D₈	—	2	4	no	—
283	D₈	—	2	2	no	—
284	D₈	M₁₆×C₆₇	1	4	yes	direct product factor
285	M₁₆	—	2	2	no	—
286	M₁₆	—	4	4	no	—
287	M₁₆	—	4	8	no	—
288	M₁₆	D₈×C₆₇	1	4	yes	direct product factor
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	4	8	no	—
294	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
295	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
296	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
297	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
298	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
299	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
302	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₄²	#41×C₆₇	1	1	yes	characteristic
310	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
311	C₂³	—	2	2	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	4	4	no	—
314	C₂³	—	4	4	no	—
315	C₂³	—	8	2	no	—
316	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	4	4	no	—
321	D₄	C₂×M₁₆×C₆₇	1	4	yes	complemented
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
335	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
336	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
337	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
338	C₂×C₄	#135×C₆₇	1	2	yes	—
339	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	4	4	no	—
348	C₈	C₂×M₁₆×C₆₇	1	2	yes	complemented
349	C₈	C₂×D₈×C₆₇	1	4	yes	complemented
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	4	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	8	8	no	—
359	C₂²	C₄×D₈×C₆₇	1	2	yes	—
360	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	C₄×D₈×C₆₇	1	2	yes	—
367	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
368	C₄	#568×C₆₇	1	1	yes	characteristic
369	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
370	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	4	4	no	—
373	C₂	—	8	2	no	—
374	C₂	D₄×M₁₆×C₆₇	1	1	yes	characteristic
375	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
376	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₈×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. D₄×M₁₆×C₆₇	C₂	semidirect
3. C₆₇×#1589	C₂	semidirect
4. C₆₇×#1581	C₂	semidirect
5. C₈×D₈×C₆₇	C₂	semidirect
6. C₆₇×#1567	C₂	semidirect
7. C₆₇×#1486	C₂	semidirect
8. C₈×M₁₆×C₆₇	C₂	semidirect
18. #436×C₆₇	C₂²	semidirect
20. C₈×D₄×C₆₇	C₂²	semidirect
28. #335×C₆₇	C₂²	semidirect
29. #335×C₆₇	C₂²	semidirect
30. D₄⋊C₈×C₆₇	C₂²	semidirect
31. D₄⋊C₈×C₆₇	C₂²	semidirect
32. #322×C₆₇	C₂²	semidirect
33. C₈²×C₆₇	C₂²	semidirect
95. D₈×C₆₇	M₁₆	direct (N × H)
99. M₁₆×C₆₇	D₈	direct (N × H)
132. D₄×C₆₇	C₂×M₁₆	semidirect
159. C₈×C₆₇	C₂×M₁₆	semidirect
160. C₈×C₆₇	C₂×D₈	semidirect
182. M₁₆×D₈	C₆₇	direct (N × H)
190. D₄×M₁₆	C₂×C₆₇	semidirect
191. #1589	C₂×C₆₇	semidirect
192. #1581	C₂×C₆₇	semidirect
193. C₈×D₈	C₂×C₆₇	semidirect
194. #1567	C₂×C₆₇	semidirect
195. #1486	C₂×C₆₇	semidirect
196. C₈×M₁₆	C₂×C₆₇	semidirect
197. C₆₇	M₁₆×D₈	direct (N × H)
207. #436	C₂²×C₆₇	semidirect
209. C₈×D₄	C₂²×C₆₇	semidirect
217. #335	C₂²×C₆₇	semidirect
218. #335	C₂²×C₆₇	semidirect
219. D₄⋊C₈	C₂²×C₆₇	semidirect
220. D₄⋊C₈	C₂²×C₆₇	semidirect
221. #322	C₂²×C₆₇	semidirect
222. C₈²	C₂²×C₆₇	semidirect
284. D₈	M₁₆×C₆₇	direct (N × H)
288. M₁₆	D₈×C₆₇	direct (N × H)
321. D₄	C₂×M₁₆×C₆₇	semidirect
348. C₈	C₂×M₁₆×C₆₇	semidirect
349. C₈	C₂×D₈×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
D₄×M₁₆×C₆₇	2
C₆₇×#1589	2
C₆₇×#1581	2
C₈×D₈×C₆₇	2
C₆₇×#1567	2
C₆₇×#1486	2
C₈×M₁₆×C₆₇	2
M₁₆×D₈	67

Subgroups — M16×D8×C67 #30697285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×D₈×C₆₇ #30697285