Subgroups — #32150941

Order 18176.

506 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597269	C₂	1	1	yes	maximal, characteristic
2	#22590249	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590247	C₂	1	2	yes	maximal, complemented
4	#22590245	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₆×Dic₁₄₂	C₂	1	2	yes	maximal, complemented
6	C₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590097	C₂	1	2	yes	maximal
8	#22590094	C₂	1	1	yes	maximal, characteristic
9	#22589935	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986553	C₄	1	1	yes	characteristic
13	C₂×#14556067	C₂²	1	2	yes	—
14	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
16	#17986130	C₄	1	1	yes	characteristic
17	#17986128	C₄	1	2	yes	—
18	#17986126	C₄	1	1	yes	characteristic
19	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₄₂	C₂²	1	2	yes	—
21	#17986024	C₄	1	2	yes	—
22	#17986021	C₄	1	1	yes	characteristic
23	#17985953	C₂²	1	2	yes	—
24	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
25	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
27	Dic₇₁⋊C₁₆	C₂²	1	2	yes	—
28	Dic₇₁⋊C₁₆	C₂²	1	2	yes	—
29	Dic₇₁⋊C₁₆	C₂²	1	2	yes	—
30	#17985775	C₂²	1	2	yes	—
31	#17985774	C₂²	1	2	yes	—
32	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₁	C₂²	1	2	yes	—
35	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
39	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
41	C₂×#11765848	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₇₁	C₂³	1	1	yes	characteristic
44	#14556127	C₈	1	1	yes	characteristic
45	#14556126	C₈	1	1	yes	characteristic
46	#14556118	C₈	1	2	yes	—
47	#14556108	C₈	1	1	yes	characteristic
48	#14556106	C₈	1	2	yes	—
49	#14556104	C₈	1	1	yes	characteristic
50	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
52	#14556085	C₂×C₄	1	2	yes	—
53	C₄²×D₇₁	C₂×C₄	1	1	yes	characteristic
54	#14556067	—	2	4	no	—
55	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₁₆×D₇₁	—	2	2	no	—
59	C₁₆×D₇₁	Q₈	1	4	yes	complemented
60	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
62	Dic₇₁⋊C₈	C₂×C₄	1	2	yes	—
63	Dic₇₁⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₇₁⋊C₈	C₂×C₄	1	2	yes	—
65	#14556039	C₂×C₄	1	2	yes	—
66	#14556038	C₂×C₄	1	2	yes	—
67	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
70	C₈×Dic₇₁	C₂³	1	2	yes	—
71	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
74	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₁	C₂³	1	2	yes	—
79	C₂×Dic₁₄₂	C₂×C₈	1	2	yes	—
80	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₇₁	C₂²×C₄	1	1	yes	characteristic
84	#11765848	—	2	4	no	—
85	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
88	C₈×D₇₁	—	2	2	no	—
89	C₈×D₇₁	C₂×Q₈	1	2	yes	—
90	C₈×D₇₁	#41	1	2	yes	—
91	C₄⋊Dic₇₁	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
93	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
94	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
95	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
96	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
97	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
98	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
99	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
100	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
104	C₇₁⋊C₁₆	—	2	1	no	—
105	C₇₁⋊C₁₆	—	2	2	no	—
106	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
107	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
108	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
112	C₄⋊C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
115	C₁₆×C₇₁	—	2	1	no	—
116	C₁₆×C₇₁	—	2	2	no	—
117	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
118	C₂²×D₇₁	C₂²×C₈	1	1	yes	characteristic
119	Q₈×C₇₁	—	2	2	no	—
120	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₇₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
127	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
128	C₂×Dic₇₁	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₇₁	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₇₁	—	2	2	no	—
131	C₄×D₇₁	—	2	2	no	—
132	C₄×D₇₁	C₄×Q₈	1	2	yes	—
133	C₄×D₇₁	#131	1	2	yes	—
134	Dic₁₄₂	—	2	2	no	—
135	Dic₁₄₂	—	2	4	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	—	2	2	no	—
138	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
140	C₇₁⋊C₈	—	2	1	no	—
141	C₇₁⋊C₈	—	2	2	no	—
142	C₇₁⋊C₈	C₄×Q₈	1	2	yes	—
143	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₇₁	—	2	1	no	—
145	C₂×D₇₁	—	2	1	no	—
146	C₂×D₇₁	C₈×Q₈	1	2	yes	—
147	C₄×C₇₁	—	2	2	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	2	2	no	—
150	C₄×C₇₁	#504	1	2	yes	—
151	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
153	Dic₇₁	—	2	2	no	—
154	Dic₇₁	—	2	2	no	—
155	Dic₇₁	—	2	1	no	—
156	Dic₇₁	C₈×Q₈	1	2	yes	—
157	Dic₇₁	#504	1	2	yes	—
158	#33474	—	71	1	no	maximal, Sylow
159	C₂×C₇₁	C₂×C₈×Q₈	1	1	yes	characteristic
160	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
161	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
162	D₇₁	—	2	2	no	—
163	#2960	—	71	1	no	—
164	#2183	—	71	1	no	—
165	#2183	—	71	1	no	—
166	#2183	—	71	2	no	—
167	Q₈×C₁₆	—	71	2	no	—
168	Q₈×C₁₆	—	71	1	no	—
169	Q₈×C₁₆	D₇₁	1	1	yes	characteristic, complemented
170	#2151	—	71	2	no	—
171	#2150	—	71	1	no	—
172	#2108	—	71	2	no	—
173	C₂×C₄×C₁₆	—	71	1	no	—
174	C₇₁	#33474	1	1	yes	characteristic, complemented, Sylow
175	#533	—	71	1	no	—
176	C₂×M₃₂	—	71	2	no	—
177	C₂²×C₁₆	—	71	1	no	—
178	C₂²×C₁₆	—	71	1	no	—
179	#446	—	71	1	no	—
180	#446	—	71	1	no	—
181	#446	—	71	2	no	—
182	C₈×Q₈	—	71	2	no	—
183	C₈×Q₈	—	71	1	no	—
184	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
185	#424	—	71	2	no	—
186	#423	—	71	1	no	—
187	#405	—	71	2	no	—
188	C₂×C₄×C₈	—	71	1	no	—
189	C₄⋊C₁₆	—	71	2	no	—
190	C₄⋊C₁₆	—	71	2	no	—
191	C₄⋊C₁₆	—	71	2	no	—
192	C₄⋊C₁₆	—	71	1	no	—
193	C₄⋊C₁₆	—	71	2	no	—
194	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
196	#346	—	71	2	no	—
197	#346	—	71	2	no	—
198	C₄×C₁₆	—	71	1	no	—
199	C₄×C₁₆	—	71	1	no	—
200	C₄×C₁₆	—	71	2	no	—
201	C₄×C₁₆	—	71	1	no	—
202	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
203	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
204	C₂×M₁₆	—	71	2	no	—
205	C₂²×C₈	—	71	1	no	—
206	C₂²×C₈	—	71	1	no	—
207	C₄⋊Q₈	—	71	1	no	—
208	C₄⋊Q₈	—	71	1	no	—
209	#125	—	71	2	no	—
210	C₂²⋊Q₈	—	71	1	no	—
211	C₂²⋊Q₈	—	71	2	no	—
212	C₂²⋊Q₈	—	71	1	no	—
213	C₄×Q₈	—	71	2	no	—
214	C₄×Q₈	—	71	1	no	—
215	C₄×Q₈	C₄×D₇₁	1	1	yes	characteristic
216	#117	—	71	2	no	—
217	C₂×C₄²	—	71	1	no	—
218	M₃₂	—	142	4	no	—
219	C₂×C₁₆	—	71	1	no	—
220	C₂×C₁₆	—	71	4	no	—
221	C₂×C₁₆	—	71	1	no	—
222	C₂×C₁₆	—	71	2	no	—
223	C₂×C₁₆	—	142	2	no	—
224	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
226	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
227	C₄⋊C₈	—	71	2	no	—
228	C₄⋊C₈	—	71	2	no	—
229	C₄⋊C₈	—	71	2	no	—
230	C₄⋊C₈	—	71	1	no	—
231	C₄⋊C₈	—	71	2	no	—
232	C₄⋊C₈	C₄×D₇₁	1	1	yes	characteristic
233	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
234	#97	—	71	2	no	—
235	#97	—	71	2	no	—
236	C₄×C₈	—	71	1	no	—
237	C₄×C₈	—	71	1	no	—
238	C₄×C₈	—	71	2	no	—
239	C₄×C₈	—	71	1	no	—
240	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₇₁	1	2	yes	—
242	C₂×Q₈	—	71	2	no	—
243	C₂×Q₈	—	71	1	no	—
244	C₂×Q₈	C₈×D₇₁	1	1	yes	characteristic
245	C₂²×C₄	—	71	1	no	—
246	C₂²×C₄	—	71	1	no	—
247	C₂²×C₄	—	71	1	no	—
248	M₁₆	—	142	4	no	—
249	C₂×C₈	—	71	1	no	—
250	C₂×C₈	—	71	2	no	—
251	C₂×C₈	—	71	2	no	—
252	C₂×C₈	—	71	1	no	—
253	C₂×C₈	—	71	2	no	—
254	C₂×C₈	—	142	2	no	—
255	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
257	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
258	C₄⋊C₄	—	71	2	no	—
259	C₄⋊C₄	—	71	2	no	—
260	C₄⋊C₄	—	71	2	no	—
261	C₄⋊C₄	—	71	2	no	—
262	C₄⋊C₄	—	71	2	no	—
263	C₄⋊C₄	—	71	2	no	—
264	C₄⋊C₄	—	71	1	no	—
265	C₄⋊C₄	C₈×D₇₁	1	2	yes	—
266	C₄⋊C₄	C₈×D₇₁	1	1	yes	characteristic
267	C₂²⋊C₄	—	71	2	no	—
268	C₂²⋊C₄	—	71	2	no	—
269	C₄²	—	71	1	no	—
270	C₄²	—	71	2	no	—
271	C₄²	—	71	1	no	—
272	C₄²	—	71	1	no	—
273	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₇₁	1	2	yes	—
275	C₁₆	—	2	1	no	—
276	C₁₆	—	2	2	no	—
277	C₁₆	—	71	2	no	—
278	C₁₆	—	142	1	no	—
279	C₁₆	—	142	2	no	—
280	C₁₆	Q₈×D₇₁	1	2	yes	—
281	C₂³	—	71	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	142	4	no	—
284	Q₈	—	142	2	no	—
285	C₂×C₄	—	71	1	no	—
286	C₂×C₄	—	71	1	no	—
287	C₂×C₄	—	71	2	no	—
288	C₂×C₄	—	71	2	no	—
289	C₂×C₄	—	71	1	no	—
290	C₂×C₄	—	71	2	no	—
291	C₂×C₄	—	71	2	no	—
292	C₂×C₄	—	142	2	no	—
293	C₂×C₄	—	142	2	no	—
294	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	71	2	no	—
302	C₈	—	142	1	no	—
303	C₈	—	142	2	no	—
304	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
305	C₈	#41×D₇₁	1	1	yes	characteristic
306	C₂²	—	71	2	no	—
307	C₂²	—	142	1	no	—
308	C₂²	—	142	1	no	—
309	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	71	2	no	—
314	C₄	—	71	2	no	—
315	C₄	—	142	2	no	—
316	C₄	—	142	1	no	—
317	C₄	—	142	2	no	—
318	C₄	#17986415	1	2	yes	—
319	C₄	C₄×Q₈×D₇₁	1	1	yes	characteristic
320	C₄	#131×D₇₁	1	1	yes	characteristic
321	C₂	—	142	2	no	—
322	C₂	C₈×Q₈×D₇₁	1	1	yes	characteristic
323	C₂	#504×D₇₁	1	1	yes	characteristic
324	C₂	C₂×#17986415	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	Q₈×C₁₆×C₇₁	C₂
prime radical	C₂²×D₇₁	C₂²×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	Q₈×C₁₆	D₇₁
hypocenter	C₇₁	#33474
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22590249	C₂	semidirect
3. #22590247	C₂	semidirect
4. #22590245	C₂	semidirect
5. C₁₆×Dic₁₄₂	C₂	semidirect
6. C₁₆×Dic₁₄₂	C₂	semidirect
11. Q₈×C₁₆×C₇₁	C₂	semidirect
59. C₁₆×D₇₁	Q₈	semidirect
169. Q₈×C₁₆	D₇₁	semidirect
174. C₇₁	#33474	semidirect

Maximal subgroups

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

subgroup	index
#22597269	2
#22590249	2
#22590247	2
#22590245	2
C₁₆×Dic₁₄₂	2
#22590097	2
#22590094	2
#22589935	2
C₄×C₁₆×D₇₁	2
Q₈×C₁₆×C₇₁	2
#33474	71