Subgroups — #32150101

Order 18176.

818 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986417	C₂	1	1	yes	maximal, characteristic
2	C₂×#17986415	C₂	1	1	yes	maximal, characteristic
3	#22596762	C₂	1	1	yes	maximal, characteristic
4	#22589938	C₂	1	2	yes	maximal, complemented
5	#22589937	C₂	1	1	yes	maximal, characteristic, complemented
6	#22589935	C₂	1	2	yes	maximal, complemented
7	M₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14556067	C₂	1	2	yes	maximal, complemented
9	C₂×#17985774	C₂	1	1	yes	maximal, characteristic, complemented
10	#346×D₇₁	C₂	1	2	yes	maximal, complemented
11	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556136	C₄	1	1	yes	characteristic
13	C₂×#14556134	C₂²	1	1	yes	characteristic
14	#17986467	C₄	1	1	yes	characteristic
15	#17986417	C₄	1	8	yes	complemented
16	#17986415	C₄	1	8	yes	complemented
17	C₂×#14556068	C₂²	1	1	yes	characteristic
18	C₂×#14556067	C₂²	1	2	yes	—
19	C₂×#14556067	C₂²	1	2	yes	—
20	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₇₁	C₂²	1	2	yes	—
22	C₂×C₁₆×D₇₁	C₂²	1	2	yes	—
23	#17985956	C₄	1	2	yes	—
24	#17985955	C₄	1	1	yes	characteristic
25	#17985953	C₂²	1	2	yes	—
26	#17985951	C₄	1	2	yes	—
27	M₁₆×Dic₇₁	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₇₁	C₂²	1	2	yes	—
30	#17985775	C₂²	1	2	yes	—
31	#17985774	C₂²	1	1	yes	characteristic
32	#17985774	C₂²	1	2	yes	—
33	#17985774	C₂²	1	1	yes	characteristic
34	#17985774	C₂²	1	1	yes	characteristic
35	#17985774	C₂²	1	1	yes	characteristic
36	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
37	C₁₆×Dic₇₁	C₂²	1	2	yes	—
38	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
39	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	#405×C₇₁	C₂²	1	1	yes	characteristic
41	#346×C₇₁	C₂²	1	2	yes	—
42	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
43	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
44	#14556136	C₈	1	8	yes	—
45	#14556134	C₂×C₄	1	4	yes	—
46	#14556134	C₂×C₄	1	4	yes	—
47	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
48	C₂×#11765848	C₂×C₄	1	2	yes	—
49	C₂×#11765848	C₂×C₄	1	2	yes	—
50	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₇₁	C₂³	1	2	yes	—
53	#14556093	C₈	1	2	yes	—
54	#14556090	C₈	1	2	yes	—
55	(C₂²⋊D₇₁)⋊C₄	C₈	1	4	yes	—
56	#14556085	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₇₁	C₂×C₄	1	2	yes	—
59	#14556068	C₂×C₄	1	4	yes	complemented
60	#14556067	C₂×C₄	1	8	yes	complemented
61	#14556067	C₂×C₄	1	8	yes	complemented
62	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	complemented
63	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
65	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
66	C₁₆×D₇₁	C₂×C₄	1	8	yes	complemented
67	C₁₆×D₇₁	C₂×C₄	1	8	yes	complemented
68	#14556039	C₂×C₄	1	2	yes	—
69	#14556038	C₂×C₄	1	2	yes	—
70	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
71	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
72	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
75	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
79	M₃₂×C₇₁	C₂×C₄	1	4	yes	complemented
80	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
81	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
83	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
84	#97×C₇₁	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₇₁	C₂³	1	2	yes	—
86	#11765865	C₄²	1	4	yes	—
87	#11765865	C₂×C₈	1	4	yes	—
88	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
89	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
90	C₂×C₂²⋊D₇₁	C₂×C₈	1	2	yes	—
91	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
92	C₂×C₄×D₇₁	C₂×C₈	1	4	yes	—
93	C₂×C₄×D₇₁	C₂²×C₄	1	2	yes	—
94	#11765849	C₄²	1	4	yes	—
95	#11765848	C₄²	1	8	yes	—
96	#11765848	C₂×C₈	1	8	yes	—
97	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
98	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₇₁	C₂×C₈	1	8	yes	—
102	C₈×D₇₁	C₂²×C₄	1	4	yes	—
103	C₈×D₇₁	C₂²×C₄	1	4	yes	—
104	Dic₇₁⋊C₄	C₂×C₈	1	4	yes	—
105	(C₂×D₇₁)⋊C₄	C₂×C₈	1	4	yes	—
106	C₄×Dic₇₁	C₂×C₈	1	4	yes	—
107	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
108	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
110	C₇₁⋊C₁₆	C₂²×C₄	1	2	yes	—
111	C₇₁⋊C₁₆	C₂²×C₄	1	4	yes	—
112	C₇₁⋊C₁₆	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
114	M₁₆×C₇₁	C₂×C₈	1	4	yes	—
115	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
121	C₂²⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
122	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
123	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
124	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
125	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
126	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₇₁	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₇₁	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₇₁	C₄×C₈	1	8	yes	—
134	C₂×Dic₇₁	—	4	2	no	—
135	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
136	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
137	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
138	D₂₈₄	C₄×C₈	1	4	yes	—
139	C₄×D₇₁	—	4	4	no	—
140	C₄×D₇₁	C₂×C₄²	1	4	yes	—
141	C₄×D₇₁	C₂²×C₈	1	4	yes	—
142	Dic₁₄₂	C₄×C₈	1	4	yes	—
143	C₈×C₇₁	C₂×C₄²	1	2	yes	—
144	C₈×C₇₁	C₂²×C₈	1	4	yes	—
145	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
146	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
147	C₇₁⋊C₈	C₂×C₄²	1	4	yes	—
148	C₇₁⋊C₈	C₂²×C₈	1	2	yes	—
149	C₇₁⋊C₈	C₂²×C₈	1	2	yes	—
150	C₂²×C₇₁	—	2	2	no	—
151	C₂²×C₇₁	C₂×C₄×C₈	1	2	yes	—
152	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
153	C₂×D₇₁	—	2	4	no	—
154	C₂×D₇₁	C₂×C₄×C₈	1	4	yes	—
155	C₄×C₇₁	—	2	4	no	—
156	C₄×C₇₁	C₂×C₄×C₈	1	2	yes	—
157	C₄×C₇₁	C₂²×C₄²	1	1	yes	characteristic
158	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
159	Dic₇₁	—	2	2	no	—
160	Dic₇₁	—	2	2	no	—
161	Dic₇₁	C₂×C₄×C₈	1	4	yes	—
162	#33326	—	71	1	no	maximal, Sylow
163	C₂×C₇₁	—	2	2	no	—
164	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
165	C₂×C₇₁	#3407	1	2	yes	—
166	D₇₁	—	2	4	no	—
167	C₂×#504	—	71	1	no	—
168	C₂×#504	—	71	1	no	—
169	C₄×#131	—	71	1	no	—
170	#2108	—	71	1	no	—
171	#2108	—	71	2	no	—
172	#2108	—	71	2	no	—
173	#2108	D₇₁	1	1	yes	characteristic, complemented
174	C₄×M₃₂	—	71	1	no	—
175	C₄×M₃₂	—	71	2	no	—
176	C₂×#346	—	71	1	no	—
177	C₂×#346	—	71	2	no	—
178	C₇₁	#33326	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	—	71	1	no	—
180	C₂×#131	—	71	1	no	—
181	C₄×#41	—	71	1	no	—
182	#504	—	71	8	no	—
183	#504	—	71	8	no	—
184	C₂×M₃₂	—	71	1	no	—
185	C₂×M₃₂	—	71	2	no	—
186	C₂×M₃₂	—	71	2	no	—
187	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
188	C₂²×C₁₆	—	71	1	no	—
189	C₂²×C₁₆	—	71	2	no	—
190	C₂²×C₁₆	—	71	2	no	—
191	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
192	#405	—	71	1	no	—
193	#405	—	71	2	no	—
194	#405	—	71	2	no	—
195	#405	C₂×D₇₁	1	1	yes	characteristic
196	C₄×M₁₆	—	71	1	no	—
197	C₄×M₁₆	—	71	2	no	—
198	C₂×C₄×C₈	—	71	1	no	—
199	C₂×C₄×C₈	—	71	2	no	—
200	#346	—	71	1	no	—
201	#346	—	71	2	no	—
202	#346	—	71	1	no	—
203	#346	—	71	1	no	—
204	#346	—	71	1	no	—
205	#346	—	71	2	no	—
206	#346	C₂×D₇₁	1	2	yes	—
207	C₄×C₁₆	—	71	2	no	—
208	C₄×C₁₆	—	71	2	no	—
209	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
210	C₂×#41	—	71	1	no	—
211	#131	—	71	8	no	—
212	#131	—	71	4	no	—
213	#131	—	71	4	no	—
214	C₂×M₁₆	—	71	1	no	—
215	C₂×M₁₆	—	71	2	no	—
216	C₂×M₁₆	—	71	2	no	—
217	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
218	C₂²×C₈	—	71	1	no	—
219	C₂²×C₈	—	71	2	no	—
220	C₂²×C₈	—	71	2	no	—
221	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
222	C₄×Q₈	—	71	2	no	—
223	C₄×D₄	—	71	2	no	—
224	C₄×D₄	—	71	4	no	—
225	#117	—	71	2	no	—
226	#117	C₄×D₇₁	1	1	yes	characteristic
227	C₂×C₄²	—	71	1	no	—
228	C₂×C₄²	—	71	2	no	—
229	M₃₂	—	71	4	no	—
230	M₃₂	—	71	8	no	—
231	M₃₂	—	71	8	no	—
232	M₃₂	C₄×D₇₁	1	4	yes	complemented
233	C₂×C₁₆	—	71	4	no	—
234	C₂×C₁₆	—	71	2	no	—
235	C₂×C₁₆	—	71	1	no	—
236	C₂×C₁₆	—	71	1	no	—
237	C₂×C₁₆	—	71	8	no	—
238	C₂×C₁₆	—	71	8	no	—
239	C₂×C₁₆	C₄×D₇₁	1	4	yes	complemented
240	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
241	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
242	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
243	#97	—	71	2	no	—
244	#97	—	71	2	no	—
245	#97	C₄×D₇₁	1	2	yes	—
246	C₄×C₈	—	71	1	no	—
247	C₄×C₈	—	71	2	no	—
248	C₄×C₈	—	71	1	no	—
249	C₄×C₈	—	71	1	no	—
250	C₄×C₈	—	71	1	no	—
251	C₄×C₈	—	71	2	no	—
252	C₄×C₈	C₂²×D₇₁	1	2	yes	—
253	#41	—	71	4	no	—
254	#41	—	71	4	no	—
255	C₂×Q₈	—	71	1	no	—
256	C₂×D₄	—	71	2	no	—
257	C₂×D₄	—	71	1	no	—
258	C₂²×C₄	—	71	2	no	—
259	C₂²×C₄	—	71	4	no	—
260	C₂²×C₄	—	71	2	no	—
261	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
262	M₁₆	—	71	4	no	—
263	M₁₆	—	71	8	no	—
264	M₁₆	—	71	8	no	—
265	M₁₆	C₈×D₇₁	1	4	yes	—
266	C₂×C₈	—	71	1	no	—
267	C₂×C₈	—	71	2	no	—
268	C₂×C₈	—	71	1	no	—
269	C₂×C₈	—	71	4	no	—
270	C₂×C₈	—	71	4	no	—
271	C₂×C₈	—	71	4	no	—
272	C₂×C₈	—	71	8	no	—
273	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
275	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
276	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
277	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
278	C₄⋊C₄	—	71	4	no	—
279	C₄⋊C₄	C₈×D₇₁	1	2	yes	—
280	C₂²⋊C₄	—	71	4	no	—
281	C₂²⋊C₄	C₈×D₇₁	1	2	yes	—
282	C₄²	—	71	1	no	—
283	C₄²	—	71	1	no	—
284	C₄²	—	71	4	no	—
285	C₄²	—	71	2	no	—
286	C₄²	C₂×C₄×D₇₁	1	2	yes	—
287	C₁₆	—	71	2	no	—
288	C₁₆	—	71	4	no	—
289	C₁₆	—	71	2	no	—
290	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
291	C₁₆	C₂×C₄×D₇₁	1	4	yes	—
292	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
293	C₂³	—	71	2	no	—
294	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
295	Q₈	—	71	4	no	—
296	D₄	—	71	8	no	—
297	D₄	—	71	4	no	—
298	C₂×C₄	—	284	4	no	—
299	C₂×C₄	—	284	2	no	—
300	C₂×C₄	—	71	2	no	—
301	C₂×C₄	—	71	4	no	—
302	C₂×C₄	—	71	4	no	—
303	C₂×C₄	—	71	2	no	—
304	C₂×C₄	—	71	2	no	—
305	C₂×C₄	C₄²×D₇₁	1	2	yes	—
306	C₂×C₄	C₂×C₈×D₇₁	1	4	yes	—
307	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
310	C₈	—	71	4	no	—
311	C₈	—	71	2	no	—
312	C₈	—	71	2	no	—
313	C₈	C₄²×D₇₁	1	2	yes	—
314	C₈	C₂×C₈×D₇₁	1	4	yes	—
315	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
316	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	71	4	no	—
319	C₂²	—	142	4	no	—
320	C₂²	C₄×C₈×D₇₁	1	2	yes	—
321	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	71	4	no	—
324	C₄	—	142	2	no	—
325	C₄	—	142	2	no	—
326	C₄	C₄×C₈×D₇₁	1	2	yes	—
327	C₄	C₂×C₄²×D₇₁	1	1	yes	characteristic
328	C₄	C₂²×C₈×D₇₁	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	142	4	no	—
331	C₂	C₂×C₄×C₈×D₇₁	1	1	yes	characteristic
332	C₂	#22601441	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator	C₂×C₇₁	C₂²×C₄×C₈
Fitting	C₇₁×#2108	C₂
socle	C₂²×C₇₁	C₂³×C₈
prime radical	C₂×#11765865	C₂×C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#33326
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
4. #22589938	C₂	semidirect
5. #22589937	C₂	semidirect
6. #22589935	C₂	semidirect
7. M₃₂×Dic₇₁	C₂	semidirect
8. C₄×#14556067	C₂	semidirect
9. C₂×#17985774	C₂	semidirect
10. #346×D₇₁	C₂	semidirect
11. C₇₁×#2108	C₂	semidirect
15. #17986417	C₄	semidirect
16. #17986415	C₄	semidirect
59. #14556068	C₂×C₄	semidirect
60. #14556067	C₂×C₄	semidirect
61. #14556067	C₂×C₄	semidirect
62. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
66. C₁₆×D₇₁	C₂×C₄	semidirect
67. C₁₆×D₇₁	C₂×C₄	semidirect
79. M₃₂×C₇₁	C₂×C₄	semidirect
80. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
173. #2108	D₇₁	semidirect
178. C₇₁	#33326	semidirect
232. M₃₂	C₄×D₇₁	semidirect
239. C₂×C₁₆	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986417	2
C₂×#17986415	2
#22596762	2
#22589938	2
#22589937	2
#22589935	2
M₃₂×Dic₇₁	2
C₄×#14556067	2
C₂×#17985774	2
#346×D₇₁	2
C₇₁×#2108	2
#33326	71