Subgroups — #30824866

Order 17152.

712 conjugacy classes of subgroups.

The subgroup lattice has 380 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×#16863453	C₂	1	1	yes	maximal, characteristic
3	#22558976	C₂	1	4	yes	maximal, complemented
4	#22558972	C₂	1	4	yes	maximal, complemented
5	#22558961	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863453	—	2	1	no	—
8	#16863453	—	2	1	no	—
9	#16863453	—	2	1	no	—
10	#16863453	C₂²	1	2	yes	—
11	#16863453	C₂²	1	2	yes	—
12	#16863453	C₂²	1	2	yes	—
13	#16863449	—	2	1	no	—
14	#16863449	C₂²	1	2	yes	—
15	#16863420	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	#16863360	—	2	1	no	—
18	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
19	C₄×#11761612	—	2	1	no	—
20	C₂×#14523954	—	2	1	no	—
21	#16863149	C₂²	1	8	yes	complemented
22	#16863147	C₂²	1	8	yes	complemented
23	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
26	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
27	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
28	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
29	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
30	C₂×#97×C₆₇	C₄	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
32	C₂×#11761612	—	2	2	no	—
33	C₂²×C₆₇⋊C₈	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
37	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
38	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
42	#14523954	—	2	2	no	—
43	#14523954	—	2	2	no	—
44	#14523954	—	2	2	no	—
45	C₄×C₆₇⋊C₈	—	2	2	no	—
46	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	—	2	8	no	—
50	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	#97×C₆₇	C₂×C₄	1	2	yes	—
67	#97×C₆₇	C₂×C₄	1	4	yes	—
68	#97×C₆₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂³	1	4	yes	—
72	#11761612	—	4	4	no	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	4	4	no	—
76	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	—	4	2	no	—
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
98	M₁₆×C₆₇	—	2	4	no	—
99	M₁₆×C₆₇	—	2	8	no	—
100	M₁₆×C₆₇	M₁₆	1	8	yes	complemented
101	C₂×C₈×C₆₇	—	2	8	no	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
104	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
118	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	—	4	1	no	—
123	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	—	4	1	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
153	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
156	C₈×C₆₇	—	2	4	no	—
157	C₈×C₆₇	C₂×M₁₆	1	8	yes	—
158	C₆₇⋊C₈	—	4	4	no	—
159	C₆₇⋊C₈	—	4	4	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	2	1	no	—
164	C₂²×C₆₇	—	2	1	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	#423	1	2	yes	—
169	C₂²×C₆₇	#430	1	2	yes	—
170	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
171	C₄×C₆₇	—	2	1	no	—
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	#423	1	2	yes	—
181	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
182	C₄×C₆₇	#575	1	2	yes	—
183	#17797	—	67	1	no	maximal, Sylow
184	C₂×C₆₇	—	2	2	no	—
185	C₂×C₆₇	—	2	2	no	—
186	C₂×C₆₇	—	4	1	no	—
187	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
188	C₂×C₆₇	#2911	1	1	yes	characteristic
189	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
190	#2920	—	67	1	no	—
191	C₂×#432	—	67	1	no	—
192	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
193	#1569	—	67	4	no	—
194	#1568	—	67	4	no	—
195	#1565	—	67	4	no	—
196	C₆₇	#17797	1	1	yes	characteristic, complemented, Sylow
197	C₂²×M₁₆	Dic₆₇	1	2	yes	—
198	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
199	#432	—	67	2	no	—
200	#432	—	67	2	no	—
201	#432	—	67	2	no	—
202	#432	—	134	1	no	—
203	#432	—	134	1	no	—
204	#432	—	134	1	no	—
205	#431	—	67	2	no	—
206	#431	—	134	1	no	—
207	#423	—	67	2	no	—
208	C₂×C₄⋊C₈	—	67	2	no	—
209	#407	—	134	1	no	—
210	C₂×C₂²⋊C₈	—	134	1	no	—
211	C₄×M₁₆	—	134	1	no	—
212	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
213	C₄×M₁₆	Dic₆₇	1	2	yes	—
214	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
215	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
216	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
217	C₂×#97	—	134	1	no	—
218	C₂×#97	Dic₆₇	1	2	yes	—
219	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
220	#335	—	67	8	no	—
221	#334	—	67	8	no	—
222	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
223	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
224	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
225	C₂×M₁₆	—	2	8	no	—
226	C₂×M₁₆	—	134	2	no	—
227	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
228	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
229	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
230	C₂²×C₈	—	134	2	no	—
231	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
232	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
233	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
234	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
235	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
236	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
237	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
239	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
244	C₄⋊C₈	—	67	4	no	—
245	C₄⋊C₈	—	67	4	no	—
246	C₄⋊C₈	—	134	2	no	—
247	C₄⋊C₈	—	134	2	no	—
248	C₂²⋊C₈	—	134	2	no	—
249	C₂²⋊C₈	—	134	2	no	—
250	C₂²⋊C₈	—	134	2	no	—
251	C₂²⋊C₈	—	134	2	no	—
252	#97	—	134	2	no	—
253	#97	—	134	2	no	—
254	#97	—	134	2	no	—
255	#97	C₂×Dic₆₇	1	2	yes	—
256	#97	C₂×Dic₆₇	1	4	yes	—
257	#97	C₂×Dic₆₇	1	2	yes	—
258	C₄×C₈	—	134	2	no	—
259	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
260	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
261	C₄×C₈	C₂²×D₆₇	1	4	yes	—
262	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
272	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
273	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
281	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
282	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
283	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
284	M₁₆	—	2	4	no	—
285	M₁₆	—	2	8	no	—
286	M₁₆	—	268	4	no	—
287	M₁₆	#11761612	1	8	yes	complemented
288	C₂×C₈	—	2	8	no	—
289	C₂×C₈	—	268	4	no	—
290	C₂×C₈	—	134	4	no	—
291	C₂×C₈	—	134	4	no	—
292	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
293	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
294	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
295	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
296	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
297	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
298	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
299	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
300	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
301	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
302	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
303	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
304	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
305	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
306	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
307	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
308	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
309	C₄²	C₂³×D₆₇	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	4	1	no	—
313	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
314	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
315	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
342	C₂×C₄	C₂×#11761612	1	4	yes	—
343	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
345	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	268	4	no	—
348	C₈	—	268	4	no	—
349	C₈	C₂×#11761612	1	8	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	4	4	no	—
358	C₂²	#16863420	1	2	yes	—
359	C₂²	#16863445	1	2	yes	—
360	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	#16863420	1	2	yes	—
371	C₄	C₂²×#11761612	1	2	yes	—
372	C₄	#16864196	1	2	yes	—
373	C₂	—	2	2	no	—
374	C₂	—	2	2	no	—
375	C₂	—	4	1	no	—
376	C₂	C₂×#16863420	1	1	yes	characteristic
377	C₂	#22568799	1	1	yes	characteristic
378	C₂	C₂×#16863445	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#17797
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22558976	C₂	semidirect
4. #22558972	C₂	semidirect
5. #22558961	C₂	semidirect
21. #16863149	C₂²	semidirect
22. #16863147	C₂²	semidirect
100. M₁₆×C₆₇	M₁₆	semidirect
196. C₆₇	#17797	semidirect
287. M₁₆	#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×#16863453	2
#22558976	2
#22558972	2
#22558961	2
C₂×C₄×M₁₆×C₆₇	2
#17797	67