Subgroups — #30827638

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569158	C₂	1	1	yes	maximal, characteristic, complemented
2	#22568953	C₂	1	1	yes	maximal, characteristic
3	#22559112	C₂	1	2	yes	maximal, complemented
4	#22559088	C₂	1	2	yes	maximal, complemented
5	#22559067	C₂	1	2	yes	maximal, complemented
6	SD₁₆⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal, complemented
7	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558612	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558608	C₂	1	1	yes	maximal, characteristic
10	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3167	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	—	2	1	no	—
13	#16863527	—	2	2	no	—
14	#16863527	C₂²	1	2	yes	—
15	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
16	#16863474	C₂²	1	2	yes	—
17	#16863468	—	2	2	no	—
18	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
19	#16863449	C₂²	1	1	yes	characteristic
20	#16863423	—	2	1	no	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
23	#16863363	—	2	1	no	—
24	#16863352	—	2	1	no	—
25	C₄×#11761612	C₂²	1	1	yes	characteristic
26	#16863149	C₂²	1	1	yes	characteristic
27	#16863149	C₂²	1	1	yes	characteristic
28	#16863149	C₂²	1	1	yes	characteristic
29	#16863149	C₂²	1	1	yes	characteristic
30	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
31	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
32	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
33	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
34	#16863118	C₂²	1	2	yes	—
35	#16863117	C₂²	1	2	yes	—
36	#533×C₆₇	C₂²	1	1	yes	characteristic
37	#532×C₆₇	C₂²	1	1	yes	characteristic
38	#498×C₆₇	C₄	1	1	yes	characteristic
39	#492×C₆₇	C₄	1	1	yes	characteristic
40	#487×C₆₇	C₄	1	1	yes	characteristic
41	#486×C₆₇	C₄	1	1	yes	characteristic
42	#465×C₆₇	C₄	1	2	yes	—
43	#465×C₆₇	C₄	1	2	yes	—
44	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
45	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#14524050	—	4	4	no	—
48	C₂×#11761612	—	2	2	no	—
49	C₂×#11761612	D₄	1	1	yes	characteristic
50	C₂×#11761612	D₄	1	1	yes	characteristic
51	C₂²×C₆₇⋊C₈	—	2	2	no	—
52	C₂²×C₆₇⋊C₈	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	—	2	1	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
55	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
57	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
58	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
59	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
61	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
62	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
63	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
65	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
66	#14523954	—	2	2	no	—
67	#14523954	D₄	1	2	yes	—
68	C₄×C₆₇⋊C₈	—	2	2	no	—
69	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
70	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
71	C₂×#41×C₆₇	—	2	1	no	—
72	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
80	#125×C₆₇	—	2	1	no	—
81	#124×C₆₇	—	2	1	no	—
82	C₂²⋊Q₈×C₆₇	—	2	1	no	—
83	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
84	#121×C₆₇	—	2	1	no	—
85	#121×C₆₇	C₂×C₄	1	2	yes	—
86	C₄×Q₈×C₆₇	—	2	1	no	—
87	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
88	C₄×D₄×C₆₇	—	2	1	no	—
89	C₄×D₄×C₆₇	C₂³	1	2	yes	—
90	#117×C₆₇	—	2	1	no	—
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	#106×C₆₇	C₂×C₄	1	2	yes	—
93	#106×C₆₇	C₂×C₄	1	2	yes	—
94	#103×C₆₇	C₂×C₄	1	2	yes	—
95	#103×C₆₇	C₂×C₄	1	2	yes	—
96	#102×C₆₇	C₂×C₄	1	2	yes	—
97	#102×C₆₇	C₂×C₄	1	2	yes	—
98	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	#11761612	—	2	2	no	—
103	#11761612	—	2	2	no	—
104	#11761612	—	4	4	no	—
105	C₂×C₆₇⋊C₈	—	2	1	no	—
106	C₂×C₆₇⋊C₈	—	2	1	no	—
107	C₂×C₆₇⋊C₈	—	2	1	no	—
108	C₂×C₆₇⋊C₈	—	2	1	no	—
109	C₂×C₆₇⋊C₈	—	2	2	no	—
110	C₂×C₆₇⋊C₈	—	4	4	no	—
111	C₂×C₆₇⋊C₈	—	4	2	no	—
112	C₂×C₆₇⋊C₈	—	4	2	no	—
113	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
114	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
115	#41×C₆₇	—	4	2	no	—
116	#41×C₆₇	—	4	2	no	—
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	—	2	1	no	—
119	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	1	no	—
123	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
128	SD₁₆×C₆₇	—	2	4	no	—
129	SD₁₆×C₆₇	—	4	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	4	1	no	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄×C₆₇	—	2	2	no	—
145	C₂²⋊C₄×C₆₇	—	2	2	no	—
146	C₂²⋊C₄×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₆₇	#41	1	1	yes	characteristic
151	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	—	2	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	8	1	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	—	2	2	no	—
160	D₄×C₆₇	—	4	4	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	8	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄×C₆₇	#137	1	2	yes	—
176	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₈×C₆₇	—	2	1	no	—
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	—	2	1	no	—
182	C₈×C₆₇	#131	1	2	yes	complemented
183	C₆₇⋊C₈	—	2	2	no	—
184	C₆₇⋊C₈	—	2	2	no	—
185	C₆₇⋊C₈	—	4	2	no	—
186	C₆₇⋊C₈	—	4	1	no	—
187	C₆₇⋊C₈	—	4	1	no	—
188	C₆₇⋊C₈	—	4	1	no	—
189	C₆₇⋊C₈	—	4	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	#440	1	2	yes	—
196	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	2	1	no	—
199	C₄×C₆₇	—	2	1	no	—
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	#440	1	2	yes	—
204	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
205	C₄×C₆₇	#568	1	1	yes	characteristic
206	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
208	#18071	—	67	1	no	maximal, Sylow
209	C₂×C₆₇	—	2	2	no	—
210	C₂×C₆₇	—	8	2	no	—
211	C₂×C₆₇	#2935	1	1	yes	characteristic
212	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
213	C₂×C₆₇	#2957	1	1	yes	characteristic
214	#3167	D₆₇	1	1	yes	characteristic
215	#2962	—	67	1	no	—
216	#2935	—	67	1	no	—
217	#1592	—	67	2	no	—
218	#1588	—	67	2	no	—
219	#1585	—	67	2	no	—
220	SD₁₆⋊C₈	—	67	2	no	—
221	C₂×#335	—	67	1	no	—
222	#1481	—	67	1	no	—
223	#1480	—	67	1	no	—
224	#1451	—	67	1	no	—
225	C₆₇	#18071	1	1	yes	characteristic, complemented, Sylow
226	C₂×#131	—	134	1	no	—
227	#533	C₂×D₆₇	1	1	yes	characteristic
228	#532	C₂×D₆₇	1	1	yes	characteristic
229	#498	Dic₆₇	1	1	yes	characteristic
230	#492	Dic₆₇	1	1	yes	characteristic
231	#487	Dic₆₇	1	1	yes	characteristic
232	#486	Dic₆₇	1	1	yes	characteristic
233	#465	Dic₆₇	1	2	yes	—
234	#465	Dic₆₇	1	2	yes	—
235	#446	—	67	2	no	—
236	#446	—	134	2	no	—
237	C₈×Q₈	—	67	2	no	—
238	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
239	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
240	#436	—	67	2	no	—
241	#435	—	134	2	no	—
242	C₈×D₄	—	67	2	no	—
243	#431	—	67	1	no	—
244	#424	—	134	1	no	—
245	C₂×C₄⋊C₈	—	67	1	no	—
246	C₂×C₄⋊C₈	—	67	1	no	—
247	#408	—	134	1	no	—
248	#405	—	134	1	no	—
249	C₄×M₁₆	—	67	1	no	—
250	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
251	#335	—	67	1	no	—
252	#335	—	67	1	no	—
253	#335	—	67	1	no	—
254	#335	—	67	1	no	—
255	Q₈⋊C₈	—	67	2	no	—
256	Q₈⋊C₈	—	67	2	no	—
257	D₄⋊C₈	—	67	2	no	—
258	D₄⋊C₈	—	67	2	no	—
259	#322	—	67	2	no	—
260	#322	—	67	2	no	—
261	C₂×#41	—	2	1	no	—
262	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
263	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
264	#131	—	268	4	no	—
265	C₂×M₁₆	—	67	1	no	—
266	C₂×M₁₆	—	67	1	no	—
267	C₂×M₁₆	—	134	2	no	—
268	C₂²×C₈	—	134	2	no	—
269	C₂²×C₈	—	134	1	no	—
270	C₂²×C₈	—	134	1	no	—
271	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
272	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
273	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
274	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
275	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
276	#127	C₂×Dic₆₇	1	1	yes	characteristic
277	#125	—	2	1	no	—
278	#124	—	2	1	no	—
279	C₂²⋊Q₈	—	2	1	no	—
280	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
281	#121	—	2	1	no	—
282	#121	C₂×Dic₆₇	1	2	yes	—
283	C₄×Q₈	—	2	1	no	—
284	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	C₂²×D₆₇	1	2	yes	—
287	#117	—	2	1	no	—
288	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
289	#106	C₂×Dic₆₇	1	2	yes	—
290	#106	C₂×Dic₆₇	1	2	yes	—
291	C₄⋊C₈	—	67	1	no	—
292	C₄⋊C₈	—	67	1	no	—
293	C₄⋊C₈	—	67	1	no	—
294	C₄⋊C₈	—	67	1	no	—
295	C₄⋊C₈	—	134	2	no	—
296	C₄⋊C₈	—	134	2	no	—
297	C₄⋊C₈	—	134	1	no	—
298	C₄⋊C₈	—	134	1	no	—
299	C₄⋊C₈	—	134	1	no	—
300	#103	C₂×Dic₆₇	1	2	yes	—
301	#103	C₂×Dic₆₇	1	2	yes	—
302	#102	C₂×Dic₆₇	1	2	yes	—
303	#102	C₂×Dic₆₇	1	2	yes	—
304	C₂²⋊C₈	—	134	2	no	—
305	C₂²⋊C₈	—	134	2	no	—
306	C₂²⋊C₈	—	134	1	no	—
307	#97	—	67	2	no	—
308	#97	—	134	2	no	—
309	C₄×C₈	—	67	2	no	—
310	C₄×C₈	—	67	2	no	—
311	C₄×C₈	—	134	2	no	—
312	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
313	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
314	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
315	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
316	#41	—	4	2	no	—
317	#41	—	4	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
327	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
328	C₂²×C₄	#11761627	1	1	yes	characteristic
329	SD₁₆	—	2	4	no	—
330	SD₁₆	—	4	2	no	—
331	M₁₆	—	268	4	no	—
332	M₁₆	—	134	2	no	—
333	M₁₆	—	134	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	268	2	no	—
337	C₂×C₈	—	268	2	no	—
338	C₂×C₈	—	268	4	no	—
339	C₂×C₈	—	67	2	no	—
340	C₂×C₈	—	67	2	no	—
341	C₂×C₈	—	134	1	no	—
342	C₂×C₈	—	134	1	no	—
343	C₂×C₈	—	134	1	no	—
344	C₂×C₈	—	134	1	no	—
345	C₂×C₈	—	134	2	no	—
346	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
347	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
348	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
349	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
357	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
363	C₄²	D₄×D₆₇	1	1	yes	characteristic
364	C₄²	#11761627	1	1	yes	characteristic
365	C₄²	C₂³×D₆₇	1	1	yes	characteristic
366	C₂³	—	2	2	no	—
367	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
368	Q₈	—	2	2	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	8	1	no	—
372	D₄	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	4	no	—
375	D₄	—	4	2	no	—
376	D₄	—	8	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
386	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
387	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
388	C₂×C₄	#14524070	1	2	yes	—
389	C₂×C₄	#14524072	1	2	yes	—
390	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
391	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	268	2	no	—
397	C₈	—	268	1	no	—
398	C₈	—	268	1	no	—
399	C₈	—	268	1	no	—
400	C₈	—	268	1	no	—
401	C₈	—	134	2	no	—
402	C₈	—	134	2	no	—
403	C₈	#14524050	1	2	yes	complemented
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	#16863497	1	2	yes	—
410	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	2	no	—
417	C₄	#16863497	1	2	yes	—
418	C₄	C₂×#14524050	1	1	yes	characteristic
419	C₄	#16864163	1	1	yes	characteristic
420	C₄	C₂×#14524070	1	1	yes	characteristic
421	C₄	C₂×#14524072	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	8	2	no	—
424	C₂	#22568953	1	1	yes	characteristic
425	C₂	C₂×#16863497	1	1	yes	characteristic
426	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3167	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3167	D₆₇
hypocenter	C₆₇	#18071
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3167

Complements

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

N	H ≅ G/N	type
1. #22569158	C₂	semidirect
3. #22559112	C₂	semidirect
4. #22559088	C₂	semidirect
5. #22559067	C₂	semidirect
6. SD₁₆⋊(C₆₇⋊C₈)	C₂	semidirect
7. C₂×#16863149	C₂	semidirect
8. #22558612	C₂	semidirect
10. #22558498	C₂	semidirect
182. C₈×C₆₇	#131	semidirect
225. C₆₇	#18071	semidirect
403. C₈	#14524050	semidirect

Maximal subgroups

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

subgroup	index
#22569158	2
#22568953	2
#22559112	2
#22559088	2
#22559067	2
SD₁₆⋊(C₆₇⋊C₈)	2
C₂×#16863149	2
#22558612	2
#22558608	2
#22558498	2
C₆₇×#3167	2
#18071	67