Subgroups — #30827558

Order 17152.

618 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569298	C₂	1	2	yes	maximal
2	#22559118	C₂	1	2	yes	maximal, complemented
3	#22559060	C₂	1	4	yes	maximal, complemented
4	#22559028	C₂	1	2	yes	maximal, complemented
5	#22558972	C₂	1	1	yes	maximal, characteristic
6	#22558656	C₂	1	2	yes	maximal
7	#22558498	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863527	—	2	4	no	—
10	#16863527	C₂²	1	4	yes	—
11	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
12	#16863453	—	2	2	no	—
13	#16863453	C₂²	1	2	yes	—
14	#16863449	—	2	2	no	—
15	#16863449	C₂²	1	1	yes	characteristic
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863149	C₂²	1	2	yes	—
18	#16863147	C₂²	1	2	yes	—
19	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	#16863118	C₂²	1	2	yes	—
22	#16863117	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
27	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
28	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
41	#14523954	—	2	4	no	—
42	#14523954	—	2	2	no	—
43	#14523954	—	2	1	no	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₆₇	—	2	4	no	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
59	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	C₂×C₄	1	2	yes	—
65	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#103×C₆₇	C₂×C₄	1	2	yes	—
68	#103×C₆₇	C₂×C₄	1	2	yes	—
69	#103×C₆₇	C₂×C₄	1	2	yes	—
70	#103×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#11761612	—	4	2	no	—
76	C₂×C₆₇⋊C₈	—	2	4	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	4	no	—
81	C₂×Q₈×C₆₇	—	4	2	no	—
82	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
89	Q₁₆×C₆₇	—	2	4	no	—
90	Q₁₆×C₆₇	—	2	2	no	—
91	Q₁₆×C₆₇	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	4	1	no	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	#41	1	1	yes	characteristic
112	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
114	Q₈×C₆₇	—	2	2	no	—
115	Q₈×C₆₇	—	2	2	no	—
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	—	4	4	no	—
118	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	4	4	no	—
129	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
139	C₆₇⋊C₈	—	2	4	no	—
140	C₆₇⋊C₈	—	4	2	no	—
141	C₆₇⋊C₈	—	4	4	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	#444	1	2	yes	—
145	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	4	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
155	C₄×C₆₇	#568	1	1	yes	characteristic
156	C₄×C₆₇	#578	1	2	yes	—
157	#18062	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
160	C₂×C₆₇	#2950	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
162	#2984	—	67	2	no	—
163	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
164	#1593	—	67	2	no	—
165	#1584	—	67	4	no	—
166	#1579	—	67	2	no	—
167	#1568	—	67	1	no	—
168	#1491	—	67	2	no	—
169	#1451	—	67	1	no	—
170	C₆₇	#18062	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
173	#446	—	67	4	no	—
174	#446	—	134	4	no	—
175	C₈×Q₈	—	67	4	no	—
176	C₄×Q₁₆	Dic₆₇	1	2	yes	—
177	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
178	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
179	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
180	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
181	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
182	#432	—	67	2	no	—
183	#432	—	134	2	no	—
184	#431	—	67	1	no	—
185	#431	—	134	2	no	—
186	C₂×#107	Dic₆₇	1	1	yes	characteristic
187	C₂×#103	Dic₆₇	1	2	yes	—
188	C₄×M₁₆	—	67	1	no	—
189	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
190	#335	—	67	2	no	—
191	#334	—	67	2	no	—
192	Q₈⋊C₈	—	67	4	no	—
193	Q₈⋊C₈	—	67	4	no	—
194	#322	—	67	2	no	—
195	#322	—	67	2	no	—
196	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
199	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
200	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂×M₁₆	—	134	1	no	—
202	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
207	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
208	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
209	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
214	#107	C₂×Dic₆₇	1	2	yes	—
215	#107	C₂×Dic₆₇	1	1	yes	characteristic
216	#107	C₂×Dic₆₇	1	1	yes	characteristic
217	C₄⋊C₈	—	67	2	no	—
218	C₄⋊C₈	—	67	4	no	—
219	C₄⋊C₈	—	134	4	no	—
220	C₄⋊C₈	—	134	4	no	—
221	#103	C₂×Dic₆₇	1	2	yes	—
222	#103	C₂×Dic₆₇	1	2	yes	—
223	#103	C₂×Dic₆₇	1	2	yes	—
224	#103	C₂×Dic₆₇	1	2	yes	—
225	C₂²⋊C₈	—	67	2	no	—
226	C₂²⋊C₈	—	134	2	no	—
227	#97	—	134	4	no	—
228	#97	—	134	2	no	—
229	#97	—	134	1	no	—
230	C₄×C₈	—	67	2	no	—
231	C₄×C₈	—	67	2	no	—
232	C₄×C₈	—	134	4	no	—
233	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
234	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
241	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
245	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
246	C₂²×C₄	#11761627	1	1	yes	characteristic
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	#11761612	1	4	yes	complemented
250	M₁₆	—	268	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	67	4	no	—
254	C₂×C₈	—	134	4	no	—
255	C₂×C₈	—	134	2	no	—
256	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
266	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
272	C₄²	D₄×D₆₇	1	1	yes	characteristic
273	C₄²	#11761627	1	1	yes	characteristic
274	C₄²	C₂³×D₆₇	1	1	yes	characteristic
275	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	4	4	no	—
280	Q₈	C₂×#11761612	1	4	yes	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
292	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
293	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	268	2	no	—
301	C₈	—	268	4	no	—
302	C₈	—	134	4	no	—
303	C₈	C₂×#11761612	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	#16863518	1	2	yes	—
307	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	C₂²×#11761612	1	1	yes	characteristic
317	C₄	#16864163	1	1	yes	characteristic
318	C₄	#16864209	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	D₄×#11761612	1	1	yes	characteristic
321	C₂	#22569077	1	1	yes	characteristic
322	C₂	C₂×#16863518	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#18062
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22559118	C₂	semidirect
3. #22559060	C₂	semidirect
4. #22559028	C₂	semidirect
91. Q₁₆×C₆₇	M₁₆	semidirect
170. C₆₇	#18062	semidirect
249. Q₁₆	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22569298	2
#22559118	2
#22559060	2
#22559028	2
#22558972	2
#22558656	2
#22558498	2
C₂×C₄×Q₁₆×C₆₇	2
#18062	67