Subgroups — #30827664

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569214	C₂	1	2	yes	maximal, complemented
2	#22559112	C₂	1	2	yes	maximal, complemented
3	#22559081	C₂	1	2	yes	maximal, complemented
4	#22559074	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558617	C₂	1	2	yes	maximal, complemented
8	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3168	C₂	1	1	yes	maximal, characteristic
10	#16863527	C₂²	1	2	yes	—
11	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
12	#16863474	—	2	2	no	—
13	#16863474	C₂²	1	2	yes	—
14	#16863468	—	2	2	no	—
15	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
16	#16863457	—	2	2	no	—
17	#16863449	C₂²	1	2	yes	—
18	#16863449	C₂²	1	1	yes	characteristic
19	#16863352	—	2	2	no	—
20	C₄×#11761612	C₂²	1	1	yes	characteristic
21	#16863149	C₂²	1	1	yes	characteristic
22	#16863149	C₂²	1	1	yes	characteristic
23	#16863147	C₂²	1	2	yes	—
24	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
25	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
26	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
27	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
28	#16863118	C₂²	1	2	yes	—
29	#16863117	C₂²	1	2	yes	—
30	#529×C₆₇	C₂²	1	2	yes	—
31	#499×C₆₇	C₄	1	1	yes	characteristic
32	#495×C₆₇	C₄	1	1	yes	characteristic
33	#488×C₆₇	C₄	1	2	yes	—
34	#467×C₆₇	C₄	1	1	yes	characteristic
35	#466×C₆₇	C₄	1	1	yes	characteristic
36	#465×C₆₇	C₄	1	2	yes	—
37	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
39	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#11761612	—	2	2	no	—
42	C₂×#11761612	D₄	1	1	yes	characteristic
43	C₂×#11761612	D₄	1	1	yes	characteristic
44	C₂²×C₆₇⋊C₈	—	2	2	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
49	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
50	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
53	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
54	#14523954	—	2	2	no	—
55	#14523954	D₄	1	2	yes	—
56	C₄×C₆₇⋊C₈	—	2	2	no	—
57	C₄×C₆₇⋊C₈	—	2	2	no	—
58	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
59	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#126×C₆₇	—	2	2	no	—
66	#125×C₆₇	C₂×C₄	1	2	yes	—
67	#124×C₆₇	C₂×C₄	1	2	yes	—
68	#123×C₆₇	—	2	2	no	—
69	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
70	#121×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	C₂³	1	2	yes	—
74	#117×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
77	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
78	#106×C₆₇	C₂×C₄	1	2	yes	—
79	#103×C₆₇	C₂×C₄	1	2	yes	—
80	#103×C₆₇	C₂×C₄	1	2	yes	—
81	#102×C₆₇	C₂×C₄	1	2	yes	—
82	#102×C₆₇	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#11761612	—	4	1	no	—
88	#11761612	—	4	1	no	—
89	#11761612	—	4	4	no	—
90	C₂×C₆₇⋊C₈	—	2	2	no	—
91	C₂×C₆₇⋊C₈	—	2	2	no	—
92	C₂×C₆₇⋊C₈	—	2	2	no	—
93	C₂×C₆₇⋊C₈	—	4	4	no	—
94	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	Q₁₆×C₆₇	—	4	1	no	—
104	SD₁₆×C₆₇	—	2	4	no	—
105	D₈×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	4	1	no	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
117	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₂²⋊C₄×C₆₇	—	2	2	no	—
121	C₂²⋊C₄×C₆₇	—	2	2	no	—
122	C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₆₇	#41	1	1	yes	characteristic
127	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
130	Q₈×C₆₇	—	2	2	no	—
131	Q₈×C₆₇	—	2	2	no	—
132	D₄×C₆₇	—	2	2	no	—
133	D₄×C₆₇	—	2	2	no	—
134	D₄×C₆₇	—	8	2	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
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	—	4	1	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	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	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₈×C₆₇	—	2	1	no	—
152	C₈×C₆₇	—	2	1	no	—
153	C₈×C₆₇	—	2	1	no	—
154	C₈×C₆₇	#131	1	2	yes	—
155	C₆₇⋊C₈	—	2	2	no	—
156	C₆₇⋊C₈	—	2	2	no	—
157	C₆₇⋊C₈	—	4	2	no	—
158	C₆₇⋊C₈	—	4	2	no	—
159	C₆₇⋊C₈	—	4	2	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	4	2	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	2	1	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	—	4	2	no	—
173	C₄×C₆₇	—	4	2	no	—
174	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
175	C₄×C₆₇	#568	1	1	yes	characteristic
176	C₄×C₆₇	#575	1	1	yes	characteristic
177	C₄×C₆₇	#575	1	1	yes	characteristic
178	#18074	—	67	1	no	maximal, Sylow
179	C₂×C₆₇	—	4	1	no	—
180	C₂×C₆₇	—	8	2	no	—
181	C₂×C₆₇	#2935	1	1	yes	characteristic
182	C₂×C₆₇	#2943	1	1	yes	characteristic
183	C₂×C₆₇	#2957	1	1	yes	characteristic
184	#3168	D₆₇	1	1	yes	characteristic
185	#2969	—	67	2	no	—
186	#1592	—	67	2	no	—
187	#1587	—	67	2	no	—
188	#1586	—	67	2	no	—
189	SD₁₆⋊C₈	—	67	2	no	—
190	#1564	—	67	1	no	—
191	#1482	—	67	2	no	—
192	#1451	—	67	1	no	—
193	C₆₇	#18074	1	1	yes	characteristic, complemented, Sylow
194	#529	C₂×D₆₇	1	2	yes	—
195	#499	Dic₆₇	1	1	yes	characteristic
196	#495	Dic₆₇	1	1	yes	characteristic
197	#488	Dic₆₇	1	2	yes	—
198	#467	Dic₆₇	1	1	yes	characteristic
199	#466	Dic₆₇	1	1	yes	characteristic
200	#465	Dic₆₇	1	2	yes	—
201	#446	—	67	2	no	—
202	C₈×Q₈	—	67	2	no	—
203	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
204	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
205	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
206	#436	—	67	2	no	—
207	#436	—	134	2	no	—
208	#435	—	134	2	no	—
209	C₈×D₄	—	67	2	no	—
210	#433	—	134	2	no	—
211	#431	—	67	1	no	—
212	#431	—	67	2	no	—
213	#405	—	134	2	no	—
214	C₄×M₁₆	—	67	1	no	—
215	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
216	#335	—	67	1	no	—
217	#335	—	67	1	no	—
218	#334	—	67	2	no	—
219	Q₈⋊C₈	—	67	2	no	—
220	Q₈⋊C₈	—	67	2	no	—
221	D₄⋊C₈	—	67	2	no	—
222	D₄⋊C₈	—	67	2	no	—
223	#322	—	67	2	no	—
224	#322	—	67	2	no	—
225	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
226	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
227	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
228	C₂×M₁₆	—	67	1	no	—
229	C₂×M₁₆	—	67	1	no	—
230	C₂×M₁₆	—	134	2	no	—
231	C₂²×C₈	—	134	2	no	—
232	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
234	#126	—	2	2	no	—
235	#125	C₂×Dic₆₇	1	2	yes	—
236	#124	C₂×Dic₆₇	1	2	yes	—
237	#123	—	2	2	no	—
238	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
239	#121	C₂×Dic₆₇	1	2	yes	—
240	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	C₂²×D₆₇	1	2	yes	—
243	#117	—	2	2	no	—
244	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
245	#107	C₂×Dic₆₇	1	1	yes	characteristic
246	#107	C₂×Dic₆₇	1	1	yes	characteristic
247	#106	C₂×Dic₆₇	1	2	yes	—
248	C₄⋊C₈	—	67	2	no	—
249	C₄⋊C₈	—	67	2	no	—
250	C₄⋊C₈	—	134	2	no	—
251	C₄⋊C₈	—	134	2	no	—
252	C₄⋊C₈	—	134	1	no	—
253	#103	C₂×Dic₆₇	1	2	yes	—
254	#103	C₂×Dic₆₇	1	2	yes	—
255	#102	C₂×Dic₆₇	1	2	yes	—
256	#102	C₂×Dic₆₇	1	2	yes	—
257	C₂²⋊C₈	—	134	2	no	—
258	C₂²⋊C₈	—	134	2	no	—
259	C₂²⋊C₈	—	134	2	no	—
260	C₂²⋊C₈	—	134	1	no	—
261	#97	—	67	2	no	—
262	#97	—	134	2	no	—
263	C₄×C₈	—	67	2	no	—
264	C₄×C₈	—	67	2	no	—
265	C₄×C₈	—	134	2	no	—
266	C₄×C₈	—	134	2	no	—
267	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
268	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
271	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
277	C₂²×C₄	#11761627	1	1	yes	characteristic
278	Q₁₆	—	4	1	no	—
279	SD₁₆	—	2	4	no	—
280	D₈	—	4	1	no	—
281	M₁₆	—	268	4	no	—
282	M₁₆	—	268	1	no	—
283	M₁₆	—	268	1	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	268	4	no	—
287	C₂×C₈	—	67	2	no	—
288	C₂×C₈	—	67	2	no	—
289	C₂×C₈	—	134	2	no	—
290	C₂×C₈	—	134	2	no	—
291	C₂×C₈	—	134	2	no	—
292	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
301	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
302	C₂²⋊C₄	—	2	2	no	—
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	—	2	2	no	—
308	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
309	C₄²	D₄×D₆₇	1	1	yes	characteristic
310	C₄²	#11761627	1	1	yes	characteristic
311	C₄²	C₂³×D₆₇	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	8	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	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	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
330	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	268	2	no	—
339	C₈	—	268	2	no	—
340	C₈	—	268	2	no	—
341	C₈	—	134	2	no	—
342	C₈	—	134	2	no	—
343	C₈	#14524050	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	C₂×#14524050	1	1	yes	characteristic
359	C₄	#16864163	1	1	yes	characteristic
360	C₄	#16864193	1	1	yes	characteristic
361	C₄	#16864193	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	8	2	no	—
364	C₂	#22568953	1	1	yes	characteristic
365	C₂	#22569010	1	1	yes	characteristic
366	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×#3168	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#466×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3168	D₆₇
hypocenter	C₆₇	#18074
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #22569214	C₂	semidirect
2. #22559112	C₂	semidirect
3. #22559081	C₂	semidirect
4. #22559074	C₂	semidirect
5. SD₁₆⋊(C₆₇⋊C₈)	C₂	semidirect
6. #22558957	C₂	semidirect
7. #22558617	C₂	semidirect
8. #22558498	C₂	semidirect
193. C₆₇	#18074	semidirect

Maximal subgroups

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

subgroup	index
#22569214	2
#22559112	2
#22559081	2
#22559074	2
SD₁₆⋊(C₆₇⋊C₈)	2
#22558957	2
#22558617	2
#22558498	2
C₆₇×#3168	2
#18074	67