Subgroups — #32151174

Order 18176.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597315	C₂	1	1	yes	maximal, characteristic
2	#22590248	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590173	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590162	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₇₁)⋊C₁₆	C₂	1	2	yes	maximal, complemented
8	#22590146	C₂	1	2	yes	maximal, complemented
9	#22590097	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986533	C₄	1	1	yes	characteristic
13	C₂×#14556067	C₂²	1	2	yes	—
14	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₁₆×D₇₁	C₂²	1	2	yes	—
16	#17986129	C₄	1	1	yes	characteristic
17	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
18	#17986078	C₄	1	1	yes	characteristic
19	#17986070	C₄	1	2	yes	—
20	C₈×D₂₈₄	C₂²	1	1	yes	characteristic
21	C₈×C₂²⋊D₇₁	C₂²	1	2	yes	—
22	#17986058	C₄	1	2	yes	—
23	#17986056	C₄	1	1	yes	characteristic
24	#17985953	C₂²	1	2	yes	—
25	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
28	Dic₇₁⋊C₁₆	C₂²	1	2	yes	—
29	Dic₇₁⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₇₁)⋊C₁₆	C₂²	1	2	yes	—
31	(C₂×D₇₁)⋊C₁₆	C₂²	1	2	yes	—
32	#17985774	C₂²	1	2	yes	—
33	C₁₆×Dic₇₁	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
38	C₂²⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₂×#11765848	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇₁	C₂³	1	2	yes	—
43	#14556122	C₈	1	2	yes	—
44	#14556120	C₈	1	1	yes	characteristic
45	#14556116	C₈	1	1	yes	characteristic
46	#14556111	C₈	1	2	yes	—
47	#14556107	C₈	1	1	yes	characteristic
48	#14556102	C₈	1	1	yes	characteristic
49	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₇₁	C₂×C₄	1	2	yes	—
52	#14556085	C₂×C₄	1	2	yes	—
53	#14556067	—	4	2	no	—
54	C₂×C₇₁⋊C₁₆	—	4	2	no	—
55	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
57	C₁₆×D₇₁	—	4	2	no	—
58	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
59	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
60	Dic₇₁⋊C₈	C₂×C₄	1	2	yes	—
61	Dic₇₁⋊C₈	C₂×C₄	1	2	yes	—
62	C₂²⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
63	(C₂×D₇₁)⋊C₈	C₂×C₄	1	2	yes	—
64	(C₂×D₇₁)⋊C₈	C₂×C₄	1	2	yes	—
65	#14556039	C₂×C₄	1	2	yes	—
66	#14556038	C₂×C₄	1	2	yes	—
67	C₈×Dic₇₁	C₂³	1	2	yes	—
68	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
77	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
80	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
81	C₂×C₂²⋊D₇₁	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₇₁	C₂²×C₄	1	2	yes	—
83	#11765848	—	4	2	no	—
84	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
86	C₈×D₇₁	—	2	2	no	—
87	C₈×D₇₁	—	2	2	no	—
88	C₄⋊Dic₇₁	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₇₁	C₂×C₈	1	1	yes	characteristic
90	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
91	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
92	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
93	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
94	C₂²⋊Dic₇₁	C₂×C₈	1	2	yes	—
95	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
96	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
97	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
98	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
99	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
100	C₇₁⋊C₁₆	—	2	2	no	—
101	C₇₁⋊C₁₆	—	2	2	no	—
102	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₁	—	2	1	no	—
106	C₂×C₈×C₇₁	—	2	1	no	—
107	C₂×C₈×C₇₁	—	4	1	no	—
108	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
112	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
114	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
116	C₁₆×C₇₁	—	2	1	no	—
117	C₁₆×C₇₁	—	2	1	no	—
118	C₁₆×C₇₁	—	2	1	no	—
119	C₁₆×C₇₁	—	2	1	no	—
120	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
121	C₂²×D₇₁	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₇₁	—	2	2	no	—
123	C₂×C₄×C₇₁	—	2	1	no	—
124	C₂×C₄×C₇₁	—	2	1	no	—
125	C₂×C₄×C₇₁	—	2	2	no	—
126	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
132	C₂²⋊D₇₁	—	2	4	no	—
133	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
134	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
135	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
136	D₂₈₄	—	2	2	no	—
137	C₄×D₇₁	—	2	2	no	—
138	C₄×D₇₁	—	2	2	no	—
139	Dic₁₄₂	—	2	2	no	—
140	C₈×C₇₁	—	2	1	no	—
141	C₈×C₇₁	—	2	1	no	—
142	C₈×C₇₁	—	2	1	no	—
143	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
145	C₇₁⋊C₈	—	2	2	no	—
146	C₇₁⋊C₈	—	2	2	no	—
147	C₂²×C₇₁	—	2	1	no	—
148	C₂²×C₇₁	—	2	1	no	—
149	C₂²×C₇₁	#504	1	2	yes	—
150	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
151	C₂×D₇₁	—	2	2	no	—
152	C₂×D₇₁	—	2	2	no	—
153	C₂×D₇₁	—	2	2	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	1	no	—
157	C₄×C₇₁	—	2	1	no	—
158	C₄×C₇₁	#504	1	2	yes	—
159	C₄×C₇₁	#520	1	1	yes	characteristic
160	C₄×C₇₁	#568	1	1	yes	characteristic
161	Dic₇₁	—	2	2	no	—
162	Dic₇₁	—	2	2	no	—
163	Dic₇₁	—	2	2	no	—
164	#33508	—	71	1	no	maximal, Sylow
165	C₂×C₇₁	—	2	2	no	—
166	C₂×C₇₁	#2988	1	1	yes	characteristic
167	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
168	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
169	D₇₁	—	4	2	no	—
170	#2966	—	71	1	no	—
171	#2183	—	71	1	no	—
172	Q₈×C₁₆	—	71	1	no	—
173	#2169	—	71	1	no	—
174	#2168	—	71	2	no	—
175	D₄×C₁₆	—	71	2	no	—
176	D₄×C₁₆	—	71	1	no	—
177	#2164	—	71	2	no	—
178	#2162	D₇₁	1	1	yes	characteristic, complemented
179	#2151	—	71	2	no	—
180	C₂×C₄⋊C₁₆	—	71	1	no	—
181	C₇₁	#33508	1	1	yes	characteristic, complemented, Sylow
182	#529	—	71	1	no	—
183	C₂×M₃₂	—	71	2	no	—
184	C₂²×C₁₆	—	71	2	no	—
185	C₂²×C₁₆	—	71	2	no	—
186	#446	—	71	1	no	—
187	C₈×Q₈	—	71	1	no	—
188	#436	—	71	1	no	—
189	#435	—	71	2	no	—
190	C₈×D₄	—	71	2	no	—
191	C₈×D₄	—	71	1	no	—
192	#433	—	71	2	no	—
193	#432	—	71	1	no	—
194	#405	—	71	2	no	—
195	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
196	C₄⋊C₁₆	—	71	2	no	—
197	C₄⋊C₁₆	—	71	2	no	—
198	C₄⋊C₁₆	—	71	2	no	—
199	C₄⋊C₁₆	—	71	1	no	—
200	C₄⋊C₁₆	—	71	1	no	—
201	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
202	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
203	C₂²⋊C₁₆	—	71	2	no	—
204	C₂²⋊C₁₆	—	71	2	no	—
205	C₂²⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
206	C₂²⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
207	#346	—	71	2	no	—
208	C₄×C₁₆	—	71	2	no	—
209	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
210	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
211	C₂×M₁₆	—	71	2	no	—
212	C₂²×C₈	—	71	2	no	—
213	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
215	#126	—	71	2	no	—
216	#125	—	71	1	no	—
217	#124	—	71	1	no	—
218	#123	—	71	2	no	—
219	C₂²⋊Q₈	—	71	1	no	—
220	#121	—	71	1	no	—
221	C₄×Q₈	—	71	1	no	—
222	C₄×D₄	—	71	2	no	—
223	C₄×D₄	—	71	1	no	—
224	#117	—	71	2	no	—
225	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
226	M₃₂	—	284	2	no	—
227	C₂×C₁₆	—	284	2	no	—
228	C₂×C₁₆	—	284	2	no	—
229	C₂×C₁₆	—	71	2	no	—
230	C₂×C₁₆	—	71	2	no	—
231	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
235	C₄⋊C₈	—	71	2	no	—
236	C₄⋊C₈	—	71	2	no	—
237	C₄⋊C₈	—	71	1	no	—
238	C₄⋊C₈	—	71	1	no	—
239	C₂²⋊C₈	—	71	2	no	—
240	C₂²⋊C₈	—	71	2	no	—
241	C₂²⋊C₈	—	71	2	no	—
242	#97	—	71	2	no	—
243	#97	—	71	2	no	—
244	C₄×C₈	—	71	2	no	—
245	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
246	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
249	C₂×Q₈	—	71	1	no	—
250	C₂×D₄	—	71	2	no	—
251	C₂×D₄	—	71	1	no	—
252	C₂²×C₄	—	71	2	no	—
253	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
254	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
256	M₁₆	—	284	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	71	2	no	—
261	C₂×C₈	—	71	2	no	—
262	C₂×C₈	—	142	2	no	—
263	C₂×C₈	—	142	2	no	—
264	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
267	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
268	C₄⋊C₄	—	71	2	no	—
269	C₄⋊C₄	—	71	2	no	—
270	C₄⋊C₄	—	71	2	no	—
271	C₄⋊C₄	—	71	2	no	—
272	C₄⋊C₄	—	71	1	no	—
273	C₄⋊C₄	—	71	1	no	—
274	C₂²⋊C₄	—	71	2	no	—
275	C₂²⋊C₄	—	71	2	no	—
276	C₂²⋊C₄	—	71	2	no	—
277	C₂²⋊C₄	—	71	2	no	—
278	C₂²⋊C₄	—	71	2	no	—
279	C₄²	—	71	2	no	—
280	C₄²	C₈×D₇₁	1	1	yes	characteristic
281	C₄²	C₈×D₇₁	1	1	yes	characteristic
282	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	142	2	no	—
289	C₁₆	—	142	2	no	—
290	C₂³	—	71	2	no	—
291	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
292	Q₈	—	142	2	no	—
293	D₄	—	142	4	no	—
294	D₄	—	142	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	71	2	no	—
300	C₂×C₄	—	71	2	no	—
301	C₂×C₄	—	71	2	no	—
302	C₂×C₄	—	142	2	no	—
303	C₂×C₄	—	142	2	no	—
304	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
305	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	142	2	no	—
314	C₈	—	142	2	no	—
315	C₈	#14556174	1	1	yes	characteristic
316	C₈	#14556175	1	1	yes	characteristic
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	142	2	no	—
320	C₂²	—	142	2	no	—
321	C₂²	—	142	2	no	—
322	C₂²	#17986415	1	2	yes	—
323	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	142	2	no	—
329	C₄	—	142	2	no	—
330	C₄	—	142	2	no	—
331	C₄	#17986417	1	2	yes	—
332	C₄	#17986479	1	1	yes	characteristic
333	C₄	#17986765	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	284	2	no	—
336	C₂	#22597448	1	1	yes	characteristic
337	C₂	C₂×#17986415	1	1	yes	characteristic
338	C₂	C₂×#17986417	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2162	C₂
prime radical	#14556102	C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2162	D₇₁
hypocenter	C₇₁	#33508
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #22590248	C₂	semidirect
3. C₁₆×Dic₁₄₂	C₂	semidirect
4. #22590173	C₂	semidirect
5. #22590162	C₂	semidirect
6. C₁₆×D₂₈₄	C₂	semidirect
7. (C₂²⋊D₇₁)⋊C₁₆	C₂	semidirect
8. #22590146	C₂	semidirect
9. #22590097	C₂	semidirect
10. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
11. C₇₁×#2162	C₂	semidirect
178. #2162	D₇₁	semidirect
181. C₇₁	#33508	semidirect

Maximal subgroups

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

subgroup	index
#22597315	2
#22590248	2
C₁₆×Dic₁₄₂	2
#22590173	2
#22590162	2
C₁₆×D₂₈₄	2
(C₂²⋊D₇₁)⋊C₁₆	2
#22590146	2
#22590097	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₇₁×#2162	2
#33508	71