Subgroups — M₁₆×#11761612 #30821541

Order 17152.

900 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	#22558512	C₂	1	4	yes	maximal, complemented
4	#22558498	C₂	1	2	yes	maximal, complemented
5	#22558496	C₂	1	2	yes	maximal, complemented
6	C₈×#11761612	C₂	1	2	yes	maximal, complemented
7	M₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11761612	C₄	1	2	yes	—
10	#16863453	C₄	1	2	yes	—
11	#16863453	C₄	1	2	yes	—
12	#16863449	C₂²	1	2	yes	—
13	#16863449	C₂²	1	2	yes	—
14	#16863420	C₄	1	2	yes	—
15	#16863360	C₄	1	2	yes	—
16	C₄×#11761612	C₄	1	2	yes	—
17	C₄×#11761612	C₄	1	2	yes	—
18	C₄×#11761612	C₂²	1	1	yes	characteristic
19	C₄×#11761612	C₂²	1	1	yes	characteristic
20	C₄×#11761612	C₂²	1	2	yes	—
21	C₄×#11761612	C₂²	1	2	yes	—
22	C₂×#14523954	C₄	1	2	yes	—
23	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
24	#16863118	C₂²	1	4	yes	complemented
25	#16863118	C₂²	1	4	yes	complemented
26	#16863117	C₂²	1	4	yes	complemented
27	C₈×C₆₇⋊C₈	C₂²	1	4	yes	complemented
28	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
29	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
31	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
32	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
35	C₂×#97×C₆₇	C₄	1	2	yes	—
36	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
37	C₂×#11761612	—	2	8	no	—
38	C₂×#11761612	C₂×C₄	1	4	yes	—
39	C₂×#11761612	C₂×C₄	1	2	yes	—
40	C₂×#11761612	C₂×C₄	1	4	yes	—
41	C₂×#11761612	C₂×C₄	1	2	yes	—
42	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
44	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
45	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
46	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
47	#14523954	C₂×C₄	1	2	yes	—
48	#14523954	C₂×C₄	1	4	yes	—
49	#14523954	C₂×C₄	1	2	yes	—
50	#14523954	C₂×C₄	1	4	yes	—
51	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
52	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
54	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
55	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×M₁₆×C₆₇	—	2	8	no	—
59	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
61	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	#97×C₆₇	C₂×C₄	1	2	yes	—
76	#97×C₆₇	C₂×C₄	1	4	yes	—
77	#97×C₆₇	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
79	C₄×C₈×C₆₇	C₂³	1	2	yes	—
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	#11761612	—	2	4	no	—
82	#11761612	—	2	8	no	—
83	#11761612	—	4	4	no	—
84	#11761612	M₁₆	1	8	yes	direct product factor
85	C₂×C₆₇⋊C₈	—	2	8	no	—
86	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
88	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	M₁₆×C₆₇	—	2	4	no	—
112	M₁₆×C₆₇	—	2	8	no	—
113	M₁₆×C₆₇	M₁₆	1	8	yes	complemented
114	C₂×C₈×C₆₇	—	2	8	no	—
115	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
116	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
117	C₄²×C₆₇	C₄²	1	1	yes	characteristic
118	C₄²×C₆₇	C₄²	1	1	yes	characteristic
119	C₄²×C₆₇	C₄²	1	1	yes	characteristic
120	C₄²×C₆₇	C₄²	1	1	yes	characteristic
121	C₄²×C₆₇	C₄²	1	1	yes	characteristic
122	C₄²×C₆₇	C₄²	1	1	yes	characteristic
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	4	1	no	—
136	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
137	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
138	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	1	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
165	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
169	C₈×C₆₇	—	2	4	no	—
170	C₈×C₆₇	C₂×M₁₆	1	8	yes	complemented
171	C₆₇⋊C₈	—	2	4	no	—
172	C₆₇⋊C₈	—	2	8	no	—
173	C₆₇⋊C₈	C₂×M₁₆	1	8	yes	complemented
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	2	1	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	4	4	no	—
182	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
183	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
184	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
193	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
194	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
195	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
196	C₄×C₆₇	#568	1	2	yes	—
197	M₁₆²	—	67	1	no	maximal, Sylow
198	C₂×C₆₇	—	2	2	no	—
199	C₂×C₆₇	—	2	2	no	—
200	C₂×C₆₇	—	4	1	no	—
201	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
202	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
203	C₂×C₆₇	#2873	1	1	yes	characteristic
204	#2920	—	67	1	no	—
205	C₂×C₄×M₁₆	—	67	1	no	—
206	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
207	#1455	—	67	4	no	—
208	#1451	—	67	2	no	—
209	#1451	—	67	2	no	—
210	C₈×M₁₆	—	67	2	no	—
211	C₈×M₁₆	—	67	2	no	—
212	C₆₇	M₁₆²	1	1	yes	characteristic, complemented, Sylow
213	C₂²×M₁₆	—	67	2	no	—
214	C₂²×M₁₆	Dic₆₇	1	2	yes	—
215	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
216	#432	—	67	2	no	—
217	#432	—	67	2	no	—
218	#431	—	67	2	no	—
219	#431	—	67	2	no	—
220	#423	—	67	2	no	—
221	#407	—	67	2	no	—
222	C₄×M₁₆	—	67	2	no	—
223	C₄×M₁₆	—	67	2	no	—
224	C₄×M₁₆	—	67	1	no	—
225	C₄×M₁₆	—	67	1	no	—
226	C₄×M₁₆	—	67	2	no	—
227	C₄×M₁₆	—	67	2	no	—
228	C₄×M₁₆	Dic₆₇	1	2	yes	—
229	C₄×M₁₆	Dic₆₇	1	2	yes	—
230	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
231	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
232	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
233	C₂×#97	—	67	2	no	—
234	C₂×#97	Dic₆₇	1	2	yes	—
235	C₂×C₄×C₈	—	67	2	no	—
236	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
237	#322	—	67	4	no	—
238	#322	—	67	4	no	—
239	#322	—	67	4	no	—
240	C₈²	—	67	4	no	—
241	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
242	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
243	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
244	C₂×M₁₆	—	2	8	no	—
245	C₂×M₁₆	—	67	4	no	—
246	C₂×M₁₆	—	67	2	no	—
247	C₂×M₁₆	—	67	4	no	—
248	C₂×M₁₆	—	67	2	no	—
249	C₂×M₁₆	—	134	8	no	—
250	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
251	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
252	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
253	C₂²×C₈	—	67	4	no	—
254	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
255	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
256	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
257	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
258	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
259	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
260	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
261	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
262	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
263	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
265	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
266	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
267	C₄⋊C₈	—	67	4	no	—
268	C₄⋊C₈	—	67	4	no	—
269	C₂²⋊C₈	—	67	4	no	—
270	C₂²⋊C₈	—	67	4	no	—
271	#97	—	67	2	no	—
272	#97	—	67	4	no	—
273	#97	—	67	2	no	—
274	#97	—	67	4	no	—
275	#97	C₂×Dic₆₇	1	2	yes	—
276	#97	C₂×Dic₆₇	1	4	yes	—
277	#97	C₂×Dic₆₇	1	2	yes	—
278	C₄×C₈	—	67	4	no	—
279	C₄×C₈	—	67	2	no	—
280	C₄×C₈	—	67	4	no	—
281	C₄×C₈	—	67	2	no	—
282	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
283	C₄×C₈	C₂²×D₆₇	1	2	yes	—
284	C₄×C₈	C₂²×D₆₇	1	2	yes	—
285	C₂⁴	C₄×Dic₆₇	1	1	yes	characteristic
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
291	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
300	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
304	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
307	M₁₆	—	2	4	no	—
308	M₁₆	—	2	8	no	—
309	M₁₆	—	268	4	no	—
310	M₁₆	—	67	8	no	—
311	M₁₆	—	134	8	no	—
312	M₁₆	—	134	4	no	—
313	M₁₆	#11761612	1	8	yes	direct product factor
314	C₂×C₈	—	2	8	no	—
315	C₂×C₈	—	67	4	no	—
316	C₂×C₈	—	67	4	no	—
317	C₂×C₈	—	67	8	no	—
318	C₂×C₈	—	134	8	no	—
319	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
320	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
321	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
322	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
323	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
324	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
325	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
326	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
327	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
328	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
329	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
330	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
331	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
332	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
333	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
334	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
335	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
336	C₄²	C₂³×D₆₇	1	1	yes	characteristic
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	4	1	no	—
340	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
341	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
342	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
360	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
361	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
362	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
363	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
364	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
368	C₂×C₄	M₁₆×D₆₇	1	4	yes	—
369	C₂×C₄	C₂×#11761612	1	4	yes	—
370	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
373	C₈	—	2	4	no	—
374	C₈	—	67	8	no	—
375	C₈	—	134	4	no	—
376	C₈	—	134	8	no	—
377	C₈	C₂×#11761612	1	8	yes	complemented
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	4	4	no	—
386	C₂²	M₁₆×Dic₆₇	1	2	yes	—
387	C₂²	C₄×#11761612	1	2	yes	—
388	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	M₁₆×Dic₆₇	1	2	yes	—
397	C₄	C₄×#11761612	1	2	yes	—
398	C₄	C₂×M₁₆×D₆₇	1	2	yes	—
399	C₄	C₂²×#11761612	1	2	yes	—
400	C₄	#16864162	1	2	yes	—
401	C₂	—	2	2	no	—
402	C₂	—	2	2	no	—
403	C₂	—	4	1	no	—
404	C₂	C₂×M₁₆×Dic₆₇	1	1	yes	characteristic
405	C₂	C₂×C₄×#11761612	1	1	yes	characteristic
406	C₂	#22568627	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₄²
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₄²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	M₁₆²
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22558512	C₂	semidirect
4. #22558498	C₂	semidirect
5. #22558496	C₂	semidirect
6. C₈×#11761612	C₂	semidirect
7. M₁₆×C₆₇⋊C₈	C₂	semidirect
24. #16863118	C₂²	semidirect
25. #16863118	C₂²	semidirect
26. #16863117	C₂²	semidirect
27. C₈×C₆₇⋊C₈	C₂²	semidirect
84. #11761612	M₁₆	direct (N × H)
113. M₁₆×C₆₇	M₁₆	semidirect
170. C₈×C₆₇	C₂×M₁₆	semidirect
173. C₆₇⋊C₈	C₂×M₁₆	semidirect
212. C₆₇	M₁₆²	semidirect
313. M₁₆	#11761612	direct (N × H)
377. C₈	C₂×#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
C₂×C₄×#11761612	2
#22558512	2
#22558498	2
#22558496	2
C₈×#11761612	2
M₁₆×C₆₇⋊C₈	2
C₂×C₄×M₁₆×C₆₇	2
M₁₆²	67

Subgroups — M16×#11761612 #30821541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×#11761612 #30821541