Subgroups — #31896677

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589289	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985748	C₂	1	2	yes	maximal
5	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	—	2	1	no	—
9	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985761	—	2	4	no	—
11	#17985753	—	2	1	no	—
12	#17985752	—	2	2	no	—
13	#17985748	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
15	#500×C₇₁	—	2	2	no	—
16	#500×C₇₁	C₄	1	4	yes	complemented
17	C₂×#107×C₇₁	—	2	1	no	—
18	C₂×#107×C₇₁	—	2	1	no	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	—	2	1	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₄⋊Dic₇₁	D₄	1	2	yes	—
23	C₂×C₄×Dic₇₁	—	2	1	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
27	C₄⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
28	C₈×Dic₇₁	—	4	2	no	—
29	C₇₁⋊#95	—	2	1	no	—
30	C₂²×Q₈×C₇₁	—	2	2	no	—
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₇₁	—	2	2	no	—
34	C₄⋊Q₈×C₇₁	—	2	2	no	—
35	C₄⋊Q₈×C₇₁	—	2	2	no	—
36	C₄⋊Q₈×C₇₁	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
39	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	#107×C₇₁	—	2	4	no	—
41	#107×C₇₁	—	2	4	no	—
42	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₇₁	D₄	1	2	yes	—
44	C₂²×Dic₇₁	—	2	2	no	—
45	C₂×C₇₁⋊C₈	—	2	2	no	—
46	C₂×C₇₁⋊C₈	—	4	1	no	—
47	C₂×C₇₁⋊C₈	—	4	1	no	—
48	C₂×C₇₁⋊C₈	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	4	no	—
50	C₄⋊Dic₇₁	—	2	2	no	—
51	C₄⋊Dic₇₁	Q₁₆	1	4	yes	—
52	C₄×Dic₇₁	—	4	1	no	—
53	C₄×Dic₇₁	—	4	1	no	—
54	C₂×Q₈×C₇₁	—	2	4	no	—
55	C₂×Q₈×C₇₁	—	4	2	no	—
56	C₂×Q₈×C₇₁	—	8	4	no	—
57	C₂²×C₄×C₇₁	—	4	2	no	—
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₂×C₈×C₇₁	—	2	1	no	—
62	C₂×C₈×C₇₁	—	2	1	no	—
63	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
66	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
67	C₄⋊C₄×C₇₁	—	2	4	no	—
68	C₄⋊C₄×C₇₁	—	2	4	no	—
69	C₄⋊C₄×C₇₁	—	4	2	no	—
70	C₄⋊C₄×C₇₁	—	4	2	no	—
71	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇₁	#41	1	1	yes	characteristic
75	C₂³×C₇₁	#121	1	1	yes	characteristic
76	Q₈×C₇₁	—	8	4	no	—
77	Q₈×C₇₁	—	8	4	no	—
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	—	4	4	no	—
80	C₂×C₄×C₇₁	—	8	4	no	—
81	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#135	1	2	yes	—
91	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
93	C₂×Dic₇₁	—	2	4	no	—
94	C₂×Dic₇₁	—	4	2	no	—
95	C₂×Dic₇₁	—	4	2	no	—
96	C₈×C₇₁	—	2	2	no	—
97	C₈×C₇₁	#103	1	4	yes	—
98	C₇₁⋊C₈	—	4	2	no	—
99	C₇₁⋊C₈	—	8	2	no	—
100	C₂²×C₇₁	#390	1	1	yes	characteristic
101	C₂²×C₇₁	#461	1	1	yes	characteristic
102	C₂²×C₇₁	#462	1	1	yes	characteristic
103	C₂²×C₇₁	#467	1	2	yes	—
104	C₂²×C₇₁	#472	1	2	yes	—
105	C₄×C₇₁	—	2	1	no	—
106	C₄×C₇₁	—	2	1	no	—
107	C₄×C₇₁	—	8	4	no	—
108	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
109	C₄×C₇₁	#419	1	1	yes	characteristic
110	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
111	C₄×C₇₁	#461	1	1	yes	characteristic
112	C₄×C₇₁	#462	1	1	yes	characteristic
113	C₄×C₇₁	#494	1	1	yes	characteristic
114	C₄×C₇₁	#495	1	1	yes	characteristic
115	Dic₇₁	—	4	4	no	—
116	#9730	—	71	1	no	maximal, Sylow
117	C₂×C₇₁	#1675	1	2	yes	—
118	C₂×C₇₁	#1676	1	2	yes	—
119	C₂×C₇₁	#1969	1	1	yes	characteristic
120	C₂×C₇₁	#1978	1	1	yes	characteristic
121	C₂×C₇₁	#1981	1	1	yes	characteristic
122	C₂×#500	D₇₁	1	1	yes	characteristic
123	#1987	—	71	2	no	—
124	C₈×C₄⋊C₄	—	71	1	no	—
125	C₂×#335	—	71	1	no	—
126	C₂×#333	—	71	2	no	—
127	C₇₁	#9730	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
129	#500	—	2	2	no	—
130	#500	Dic₇₁	1	4	yes	complemented
131	C₂×#107	—	2	1	no	—
132	C₂×#107	—	2	1	no	—
133	C₂×C₄⋊C₈	—	71	1	no	—
134	C₂×C₄⋊C₈	—	71	2	no	—
135	C₂×C₄×C₈	—	142	1	no	—
136	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
137	C₄×C₄⋊C₄	—	71	1	no	—
138	#340	—	142	4	no	—
139	#336	—	142	1	no	—
140	#335	—	142	2	no	—
141	#333	—	142	4	no	—
142	C₂²×Q₈	—	2	2	no	—
143	C₂²×C₈	—	142	2	no	—
144	C₂²×C₈	—	142	1	no	—
145	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	C₂×Dic₇₁	1	4	yes	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	71	2	no	—
154	C₂×C₄²	—	142	1	no	—
155	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
156	#107	—	2	4	no	—
157	#107	—	2	4	no	—
158	C₄⋊C₈	—	71	2	no	—
159	C₄⋊C₈	—	142	2	no	—
160	C₄⋊C₈	—	142	2	no	—
161	C₄⋊C₈	—	142	1	no	—
162	C₄×C₈	—	284	2	no	—
163	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
164	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
165	#95	—	142	1	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	8	4	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	142	2	no	—
171	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
173	C₂²×C₄	#11765864	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	284	1	no	—
177	C₂×C₈	—	284	1	no	—
178	C₂×C₈	—	284	2	no	—
179	C₂×C₈	—	284	4	no	—
180	C₂×C₈	—	142	2	no	—
181	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
182	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
183	C₂×C₈	#11765852	1	2	yes	—
184	C₂×C₈	#11765855	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	71	4	no	—
190	C₄⋊C₄	—	142	2	no	—
191	C₄²	—	284	1	no	—
192	C₄²	—	284	1	no	—
193	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
194	C₄²	D₄×D₇₁	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
196	C₄²	#11765864	1	1	yes	characteristic
197	C₂³	#14556099	1	1	yes	characteristic
198	Q₈	—	8	4	no	—
199	Q₈	—	8	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	4	no	—
203	C₂×C₄	—	284	2	no	—
204	C₂×C₄	—	284	2	no	—
205	C₂×C₄	—	142	4	no	—
206	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
207	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
208	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
209	C₂×C₄	#14556099	1	1	yes	characteristic
210	C₂×C₄	#14556115	1	1	yes	characteristic
211	C₂×C₄	#14556124	1	1	yes	characteristic
212	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
213	C₂×C₄	Q₁₆×D₇₁	1	2	yes	—
214	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
215	C₂×C₄	#14556151	1	2	yes	—
216	C₂×C₄	#14556159	1	1	yes	characteristic
217	C₂×C₄	#14556159	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	284	2	no	—
220	C₈	—	568	2	no	—
221	C₈	#14556052	1	4	yes	—
222	C₂²	#17985902	1	1	yes	characteristic
223	C₂²	#17986201	1	1	yes	characteristic
224	C₂²	#17986206	1	1	yes	characteristic
225	C₂²	#17986237	1	2	yes	—
226	C₂²	#17986266	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	8	4	no	—
230	C₄	—	284	4	no	—
231	C₄	C₂×#14556052	1	1	yes	characteristic
232	C₄	#17986008	1	1	yes	characteristic
233	C₄	Q₁₆×Dic₇₁	1	2	yes	—
234	C₄	#17986201	1	1	yes	characteristic
235	C₄	#17986206	1	1	yes	characteristic
236	C₄	#17986377	1	1	yes	characteristic
237	C₄	#17986384	1	1	yes	characteristic
238	C₂	#22587736	1	2	yes	—
239	C₂	#22587741	1	2	yes	—
240	C₂	#22589202	1	1	yes	characteristic
241	C₂	#22589251	1	1	yes	characteristic
242	C₂	#22589264	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556099
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	#9730
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
16. #500×C₇₁	C₄	semidirect
127. C₇₁	#9730	semidirect
130. #500	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22589289	2
C₈×C₄⋊Dic₇₁	2
C₂×#17985752	2
C₂×#17985748	2
C₂×#500×C₇₁	2
#9730	71