Subgroups — #32152963

Order 18176.

474 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598873	C₂	1	2	yes	maximal
2	#22590625	C₂	1	2	yes	maximal, complemented
3	#22590623	C₂	1	2	yes	maximal, complemented
4	#22590621	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590619	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590618	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590127	C₂	1	2	yes	maximal
8	#22590123	C₂	1	1	yes	maximal, characteristic
9	#367×D₇₁	C₂	1	1	yes	maximal, characteristic
10	#22589935	C₂	1	1	yes	maximal, characteristic
11	C₇₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986626	—	2	2	no	—
13	C₂×#14556067	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
15	#17986407	—	2	2	no	—
16	#17986405	—	2	2	no	—
17	#17986403	C₂²	1	2	yes	—
18	#17986401	C₂²	1	2	yes	—
19	#17986400	C₂²	1	2	yes	—
20	#17986040	—	2	2	no	—
21	#17986036	C₂²	1	2	yes	—
22	#107×D₇₁	C₂²	1	2	yes	—
23	#17985953	C₂²	1	1	yes	characteristic
24	#17985827	C₂²	1	1	yes	characteristic
25	#17985827	C₂²	1	1	yes	characteristic
26	#17985826	C₂²	1	2	yes	—
27	#17985826	C₂²	1	2	yes	—
28	#17985825	C₂²	1	2	yes	—
29	#17985824	C₂²	1	2	yes	—
30	#17985824	C₂²	1	2	yes	—
31	#17985775	C₂²	1	1	yes	characteristic
32	#17985774	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
34	#500×C₇₁	C₂²	1	2	yes	—
35	#367×C₇₁	C₂²	1	1	yes	characteristic
36	#367×C₇₁	C₂²	1	1	yes	characteristic
37	#367×C₇₁	C₂²	1	2	yes	—
38	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
39	C₂×#11765848	D₄	1	1	yes	characteristic
40	C₂×C₈×D₇₁	C₂³	1	1	yes	characteristic
41	#14556127	—	2	2	no	—
42	#14556126	—	4	2	no	—
43	#14556120	—	2	2	no	—
44	#14556119	—	2	2	no	—
45	#14556118	—	4	2	no	—
46	#14556106	—	4	2	no	—
47	#14556104	—	4	2	no	—
48	#14556086	—	2	2	no	—
49	#14556085	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₇₁	—	2	2	no	—
51	#14556067	Q₈	1	4	yes	complemented
52	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₁₆×D₇₁	Q₈	1	4	yes	complemented
55	#14556062	C₂³	1	2	yes	—
56	#14556062	C₂³	1	2	yes	—
57	#14556061	—	2	2	no	—
58	#14556061	—	2	2	no	—
59	#14556060	—	2	2	no	—
60	#14556059	—	2	2	no	—
61	#14556059	—	2	2	no	—
62	#14556039	D₄	1	1	yes	characteristic
63	#14556038	D₄	1	1	yes	characteristic
64	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₇₁	—	2	2	no	—
66	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
68	#107×C₇₁	—	2	2	no	—
69	#107×C₇₁	C₂³	1	2	yes	—
70	#107×C₇₁	C₂³	1	2	yes	—
71	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₄₂	—	4	2	no	—
73	C₂×C₄×D₇₁	—	4	2	no	—
74	C₂×C₄×D₇₁	C₂×D₄	1	1	yes	characteristic
75	#11765848	Q₁₆	1	4	yes	—
76	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₈×D₇₁	C₂×Q₈	1	2	yes	—
79	C₈×D₇₁	C₂×Q₈	1	2	yes	—
80	C₄⋊Dic₇₁	—	2	2	no	—
81	C₄⋊Dic₇₁	—	2	2	no	—
82	C₄⋊Dic₇₁	—	4	2	no	—
83	Dic₇₁⋊C₄	—	4	2	no	—
84	Dic₇₁⋊C₄	—	4	2	no	—
85	Dic₇₁⋊C₄	—	4	2	no	—
86	Dic₇₁⋊C₄	—	4	2	no	—
87	Dic₇₁⋊C₄	—	4	2	no	—
88	Dic₇₁⋊C₄	D₈	1	2	yes	—
89	(C₂×D₇₁)⋊C₄	—	4	2	no	—
90	(C₂×D₇₁)⋊C₄	D₈	1	2	yes	—
91	C₄×Dic₇₁	—	4	2	no	—
92	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
93	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
94	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
95	C₂×Q₈×C₇₁	—	4	2	no	—
96	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₇₁	—	2	2	no	—
99	C₄⋊C₄×C₇₁	—	2	2	no	—
100	C₄⋊C₄×C₇₁	—	4	2	no	—
101	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
103	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
104	C₂²×D₇₁	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₇₁	—	16	2	no	—
106	C₂×C₄×C₇₁	—	4	2	no	—
107	C₂×C₄×C₇₁	—	4	2	no	—
108	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₇₁	—	4	2	no	—
111	C₂×Dic₇₁	—	4	2	no	—
112	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
113	C₂×Dic₇₁	C₂×D₈	1	1	yes	characteristic
114	C₄×D₇₁	—	16	2	no	—
115	C₄×D₇₁	C₄⋊Q₈	1	2	yes	—
116	C₄×D₇₁	C₂×Q₁₆	1	2	yes	—
117	Dic₁₄₂	—	16	2	no	—
118	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
119	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
121	C₇₁⋊C₈	C₄⋊Q₈	1	2	yes	—
122	C₇₁⋊C₈	C₂×Q₁₆	1	2	yes	—
123	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
124	C₂×D₇₁	—	2	2	no	—
125	C₂×D₇₁	#500	1	2	yes	—
126	C₄×C₇₁	—	2	2	no	—
127	C₄×C₇₁	—	8	2	no	—
128	C₄×C₇₁	—	8	2	no	—
129	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
131	Dic₇₁	—	2	2	no	—
132	Dic₇₁	—	8	2	no	—
133	Dic₇₁	—	8	2	no	—
134	Dic₇₁	#500	1	2	yes	—
135	#33730	—	71	1	no	maximal, Sylow
136	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
137	C₂×C₇₁	#3416	1	2	yes	—
138	D₇₁	—	2	2	no	—
139	#3164	—	71	2	no	—
140	#2255	—	71	2	no	—
141	#2255	—	71	2	no	—
142	#2254	—	71	1	no	—
143	#2254	D₇₁	1	1	yes	characteristic, complemented
144	#2253	—	71	1	no	—
145	#2253	—	71	1	no	—
146	#2159	—	71	2	no	—
147	#2158	—	71	1	no	—
148	C₂×#367	—	71	1	no	—
149	#2108	—	71	1	no	—
150	C₇₁	#33730	1	1	yes	characteristic, complemented, Sylow
151	#544	—	142	2	no	—
152	C₂×M₃₂	—	71	1	no	—
153	C₂²×C₁₆	—	71	1	no	—
154	#501	—	142	2	no	—
155	#501	—	142	2	no	—
156	#500	—	71	2	no	—
157	#500	C₂×D₇₁	1	2	yes	—
158	#499	—	71	2	no	—
159	#499	—	71	2	no	—
160	#428	—	142	2	no	—
161	#427	—	71	2	no	—
162	C₂×#107	—	71	2	no	—
163	#405	—	71	1	no	—
164	#367	—	71	1	no	—
165	#367	—	71	1	no	—
166	#367	—	71	2	no	—
167	#367	—	71	2	no	—
168	#367	C₂×D₇₁	1	1	yes	characteristic
169	#367	C₂×D₇₁	1	1	yes	characteristic
170	#367	C₂×D₇₁	1	2	yes	—
171	#366	—	71	2	no	—
172	#366	—	71	2	no	—
173	#366	—	71	2	no	—
174	#346	—	71	1	no	—
175	#346	—	71	1	no	—
176	C₄×C₁₆	—	71	1	no	—
177	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
178	C₂×M₁₆	—	71	1	no	—
179	C₂²×C₈	—	71	1	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	284	2	no	—
182	C₄⋊Q₈	—	142	2	no	—
183	#125	—	284	2	no	—
184	#125	—	142	2	no	—
185	#125	—	142	2	no	—
186	C₂²⋊Q₈	—	284	2	no	—
187	C₂²⋊Q₈	—	284	2	no	—
188	#117	—	71	1	no	—
189	#117	—	142	2	no	—
190	C₂×C₄⋊C₄	—	142	2	no	—
191	M₃₂	—	71	4	no	—
192	C₂×C₁₆	—	71	1	no	—
193	C₂×C₁₆	—	71	1	no	—
194	C₂×C₁₆	—	71	4	no	—
195	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
196	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
197	#107	—	2	2	no	—
198	#107	—	71	2	no	—
199	#107	—	71	2	no	—
200	#107	—	142	2	no	—
201	#107	—	142	2	no	—
202	#107	C₂²×D₇₁	1	2	yes	—
203	#107	C₂²×D₇₁	1	2	yes	—
204	#106	—	142	2	no	—
205	#106	—	142	2	no	—
206	#106	—	142	2	no	—
207	#97	—	71	1	no	—
208	#97	—	71	1	no	—
209	C₄×C₈	—	71	1	no	—
210	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	284	2	no	—
213	C₂²×C₄	—	284	2	no	—
214	C₂²×C₄	—	71	1	no	—
215	M₁₆	—	71	4	no	—
216	C₂×C₈	—	71	1	no	—
217	C₂×C₈	—	71	1	no	—
218	C₂×C₈	—	71	2	no	—
219	C₂×C₈	—	71	2	no	—
220	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	284	2	no	—
226	C₄⋊C₄	—	284	2	no	—
227	C₄⋊C₄	—	284	2	no	—
228	C₄⋊C₄	—	284	2	no	—
229	C₄⋊C₄	—	284	2	no	—
230	C₄⋊C₄	—	284	2	no	—
231	C₄⋊C₄	—	71	2	no	—
232	C₄⋊C₄	—	142	2	no	—
233	C₄⋊C₄	—	142	2	no	—
234	C₂²⋊C₄	—	284	2	no	—
235	C₂²⋊C₄	—	71	2	no	—
236	C₄²	—	284	2	no	—
237	C₄²	—	71	1	no	—
238	C₄²	D₄×D₇₁	1	1	yes	characteristic
239	C₁₆	—	71	2	no	—
240	C₁₆	—	71	2	no	—
241	C₁₆	Q₈×D₇₁	1	2	yes	—
242	C₁₆	Q₈×D₇₁	1	2	yes	—
243	C₂³	—	71	1	no	—
244	Q₈	—	16	2	no	—
245	Q₈	—	1136	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	284	2	no	—
249	C₂×C₄	—	284	2	no	—
250	C₂×C₄	—	71	1	no	—
251	C₂×C₄	—	71	2	no	—
252	C₂×C₄	—	71	2	no	—
253	C₂×C₄	—	71	2	no	—
254	C₂×C₄	—	1136	2	no	—
255	C₂×C₄	D₈×D₇₁	1	2	yes	—
256	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
257	C₈	—	71	2	no	—
258	C₈	—	71	2	no	—
259	C₈	Q₁₆×D₇₁	1	2	yes	—
260	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
262	C₂²	—	71	2	no	—
263	C₂²	—	142	2	no	—
264	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	568	2	no	—
269	C₄	—	568	2	no	—
270	C₄	—	71	2	no	—
271	C₄	—	142	2	no	—
272	C₄	C₄⋊Q₈×D₇₁	1	1	yes	characteristic
273	C₄	C₂×Q₁₆×D₇₁	1	1	yes	characteristic
274	C₂	—	142	2	no	—
275	C₂	#500×D₇₁	1	1	yes	characteristic
276	C₂	#22601482	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₁
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2254	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂²×D₇₁	C₂×D₈
hypercenter	#2254	D₇₁
hypocenter	C₇₁	#33730
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
2. #22590625	C₂	semidirect
3. #22590623	C₂	semidirect
4. #22590621	C₂	semidirect
5. #22590619	C₂	semidirect
6. #22590618	C₂	semidirect
11. C₇₁×#2254	C₂	semidirect
51. #14556067	Q₈	semidirect
54. C₁₆×D₇₁	Q₈	semidirect
143. #2254	D₇₁	semidirect
150. C₇₁	#33730	semidirect

Maximal subgroups

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

subgroup	index
#22598873	2
#22590625	2
#22590623	2
#22590621	2
#22590619	2
#22590618	2
#22590127	2
#22590123	2
#367×D₇₁	2
#22589935	2
C₇₁×#2254	2
#33730	71