Subgroups — #30824860

Order 17152.

756 conjugacy classes of subgroups.

The subgroup lattice has 412 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×#16863420	C₂	1	1	yes	maximal, characteristic
3	#22558976	C₂	1	4	yes	maximal, complemented
4	#22558968	C₂	1	2	yes	maximal, complemented
5	#22558965	C₂	1	2	yes	maximal, complemented
6	#22558961	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	—	2	1	no	—
9	#16863453	—	2	1	no	—
10	#16863453	C₂²	1	2	yes	—
11	#16863449	—	2	1	no	—
12	#16863449	C₂²	1	2	yes	—
13	#16863420	—	2	1	no	—
14	#16863420	—	2	1	no	—
15	#16863420	C₂²	1	2	yes	—
16	#16863420	C₂²	1	2	yes	—
17	#16863420	C₂²	1	1	yes	characteristic
18	#16863420	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
21	#16863360	—	2	1	no	—
22	C₄×#11761612	—	2	1	no	—
23	#16863149	C₂²	1	8	yes	complemented
24	#16863147	C₂²	1	8	yes	complemented
25	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
28	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
30	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
31	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
32	C₂×#97×C₆₇	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761612	—	2	1	no	—
35	C₂×#11761612	—	2	1	no	—
36	C₂×#11761612	—	2	2	no	—
37	C₂×#11761612	—	2	2	no	—
38	C₂×#11761612	—	4	4	no	—
39	C₂²×C₆₇⋊C₈	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
46	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
47	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
48	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
49	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
50	#14523954	—	2	2	no	—
51	C₄×C₆₇⋊C₈	—	2	2	no	—
52	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	—	2	8	no	—
56	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	#97×C₆₇	C₂×C₄	1	2	yes	—
73	#97×C₆₇	C₂×C₄	1	4	yes	—
74	#97×C₆₇	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₆₇	C₂³	1	4	yes	—
78	#11761612	—	4	4	no	—
79	#11761612	—	4	2	no	—
80	#11761612	—	4	4	no	—
81	#11761612	—	4	2	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₂×C₆₇⋊C₈	—	2	4	no	—
85	C₂×C₆₇⋊C₈	—	4	4	no	—
86	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
109	M₁₆×C₆₇	—	2	4	no	—
110	M₁₆×C₆₇	—	2	8	no	—
111	M₁₆×C₆₇	M₁₆	1	8	yes	complemented
112	C₂×C₈×C₆₇	—	2	8	no	—
113	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
115	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
118	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
119	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
120	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
121	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
128	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
130	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂³×C₆₇	—	2	2	no	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	4	1	no	—
134	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	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×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
165	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₆₇	#137	1	2	yes	—
167	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
170	C₈×C₆₇	—	2	4	no	—
171	C₈×C₆₇	C₂×M₁₆	1	8	yes	complemented
172	C₆₇⋊C₈	—	4	4	no	—
173	C₆₇⋊C₈	—	4	2	no	—
174	C₆₇⋊C₈	—	4	2	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	—	2	1	no	—
182	C₂²×C₆₇	—	4	4	no	—
183	C₂²×C₆₇	#423	1	2	yes	—
184	C₂²×C₆₇	#428	1	2	yes	—
185	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
193	C₄×C₆₇	#423	1	2	yes	—
194	C₄×C₆₇	#428	1	2	yes	—
195	C₄×C₆₇	#430	1	2	yes	—
196	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
197	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
199	#17796	—	67	1	no	maximal, Sylow
200	C₂×C₆₇	—	2	2	no	—
201	C₂×C₆₇	—	2	2	no	—
202	C₂×C₆₇	—	4	1	no	—
203	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
204	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
205	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
206	#2920	—	67	1	no	—
207	C₂×#423	—	67	1	no	—
208	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
209	#1569	—	67	4	no	—
210	#1567	—	67	2	no	—
211	#1566	—	67	2	no	—
212	#1565	—	67	4	no	—
213	C₆₇	#17796	1	1	yes	characteristic, complemented, Sylow
214	C₂²×M₁₆	—	134	1	no	—
215	C₂²×M₁₆	Dic₆₇	1	2	yes	—
216	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
217	#432	—	67	2	no	—
218	#432	—	134	1	no	—
219	#431	—	67	2	no	—
220	#431	—	134	1	no	—
221	#423	—	67	1	no	—
222	#423	—	67	2	no	—
223	#423	—	67	1	no	—
224	#423	—	67	2	no	—
225	#423	—	134	1	no	—
226	#423	—	134	1	no	—
227	C₂×C₄⋊C₈	—	67	2	no	—
228	C₂×C₄⋊C₈	—	134	1	no	—
229	#407	—	134	1	no	—
230	C₄×M₁₆	—	134	1	no	—
231	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
232	C₄×M₁₆	Dic₆₇	1	2	yes	—
233	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
234	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
235	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
236	C₂×#97	Dic₆₇	1	2	yes	—
237	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
238	#335	—	67	8	no	—
239	#334	—	67	8	no	—
240	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
241	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
242	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
243	C₂×M₁₆	—	2	8	no	—
244	C₂×M₁₆	—	268	4	no	—
245	C₂×M₁₆	—	134	2	no	—
246	C₂×M₁₆	—	134	2	no	—
247	C₂×M₁₆	—	134	1	no	—
248	C₂×M₁₆	—	134	1	no	—
249	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
250	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
251	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
252	C₂²×C₈	—	134	2	no	—
253	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
254	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
255	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
256	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
257	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
258	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
259	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
260	C₂×C₄²	C₂×Dic₆₇	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	C₂²×D₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
265	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
266	C₄⋊C₈	—	67	2	no	—
267	C₄⋊C₈	—	67	4	no	—
268	C₄⋊C₈	—	67	2	no	—
269	C₄⋊C₈	—	134	2	no	—
270	C₄⋊C₈	—	134	2	no	—
271	C₄⋊C₈	—	134	2	no	—
272	C₄⋊C₈	—	134	1	no	—
273	C₄⋊C₈	—	134	1	no	—
274	C₂²⋊C₈	—	134	2	no	—
275	C₂²⋊C₈	—	134	2	no	—
276	#97	—	134	2	no	—
277	#97	C₂×Dic₆₇	1	2	yes	—
278	#97	C₂×Dic₆₇	1	4	yes	—
279	#97	C₂×Dic₆₇	1	2	yes	—
280	C₄×C₈	—	134	2	no	—
281	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
282	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
283	C₄×C₈	C₂²×D₆₇	1	4	yes	—
284	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	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	C₂²×Dic₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
300	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
302	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
304	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
305	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
306	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
307	M₁₆	—	2	4	no	—
308	M₁₆	—	2	8	no	—
309	M₁₆	—	268	4	no	—
310	M₁₆	—	268	4	no	—
311	M₁₆	—	268	2	no	—
312	M₁₆	—	268	2	no	—
313	M₁₆	#11761612	1	8	yes	complemented
314	C₂×C₈	—	2	8	no	—
315	C₂×C₈	—	268	4	no	—
316	C₂×C₈	—	134	2	no	—
317	C₂×C₈	—	134	2	no	—
318	C₂×C₈	—	134	4	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	C₂²×Dic₆₇	1	1	yes	characteristic
328	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
329	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
330	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
331	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
332	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
333	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
334	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
335	C₄²	C₂×Dic₁₃₄	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	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
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	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
370	C₂×C₄	C₂×#11761612	1	4	yes	—
371	C₂×C₄	#14524071	1	2	yes	—
372	C₂×C₄	#14524076	1	2	yes	—
373	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
375	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
376	C₈	—	2	4	no	—
377	C₈	—	268	4	no	—
378	C₈	—	268	2	no	—
379	C₈	—	268	2	no	—
380	C₈	C₂×#11761612	1	8	yes	complemented
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	4	4	no	—
389	C₂²	#16863420	1	2	yes	—
390	C₂²	#16863438	1	2	yes	—
391	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	#16863420	1	2	yes	—
400	C₄	#16863438	1	2	yes	—
401	C₄	#16863445	1	2	yes	—
402	C₄	C₂²×#11761612	1	2	yes	—
403	C₄	C₂×#14524071	1	1	yes	characteristic
404	C₄	C₂×#14524076	1	1	yes	characteristic
405	C₂	—	2	2	no	—
406	C₂	—	2	2	no	—
407	C₂	—	4	1	no	—
408	C₂	C₂×#16863420	1	1	yes	characteristic
409	C₂	C₂×#16863438	1	1	yes	characteristic
410	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	#17796
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. #22558968	C₂	semidirect
5. #22558965	C₂	semidirect
6. #22558961	C₂	semidirect
23. #16863149	C₂²	semidirect
24. #16863147	C₂²	semidirect
111. M₁₆×C₆₇	M₁₆	semidirect
171. C₈×C₆₇	C₂×M₁₆	semidirect
213. C₆₇	#17796	semidirect
313. M₁₆	#11761612	semidirect
380. 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×#16863420	2
#22558976	2
#22558968	2
#22558965	2
#22558961	2
C₂×C₄×M₁₆×C₆₇	2
#17796	67