Subgroups — #31916029

Order 18176.

398 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
3	#22586431	C₂	1	4	yes	maximal
4	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₇₁	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
9	#17985762	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
11	#500×C₇₁	—	2	2	no	—
12	#500×C₇₁	C₄	1	4	yes	complemented
13	C₂×#107×C₇₁	—	2	1	no	—
14	C₂×#107×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₁	—	2	2	no	—
17	C₂×C₄×Dic₇₁	—	4	1	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	4	no	—
20	#14556062	—	2	2	no	—
21	#14556062	Q₈	1	4	yes	complemented
22	C₇₁⋊#95	—	4	1	no	—
23	C₂²×Q₈×C₇₁	—	4	1	no	—
24	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₄⋊Q₈×C₇₁	—	2	4	no	—
27	C₄⋊Q₈×C₇₁	—	4	2	no	—
28	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
30	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
31	#107×C₇₁	—	2	4	no	—
32	#107×C₇₁	—	2	2	no	—
33	#107×C₇₁	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₂²×Dic₇₁	—	4	2	no	—
37	C₄⋊Dic₇₁	—	2	4	no	—
38	C₄⋊Dic₇₁	—	4	2	no	—
39	C₄×Dic₇₁	—	8	2	no	—
40	C₇₁⋊C₁₆	—	4	2	no	—
41	C₂×Q₈×C₇₁	—	4	4	no	—
42	C₂×Q₈×C₇₁	—	8	4	no	—
43	C₂²×C₄×C₇₁	—	4	2	no	—
44	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	2	2	no	—
48	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₇₁	#41	1	2	yes	—
53	C₄⋊C₄×C₇₁	—	2	4	no	—
54	C₄⋊C₄×C₇₁	—	4	4	no	—
55	C₄⋊C₄×C₇₁	—	4	2	no	—
56	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₇₁	SD₁₆	1	2	yes	—
58	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
59	Q₈×C₇₁	—	8	8	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	4	4	no	—
62	C₂×C₄×C₇₁	—	8	4	no	—
63	C₂×C₄×C₇₁	#102	1	2	yes	—
64	C₂×C₄×C₇₁	#102	1	2	yes	—
65	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
66	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
67	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#137	1	2	yes	—
71	C₂×Dic₇₁	—	4	4	no	—
72	C₂×Dic₇₁	—	8	2	no	—
73	C₂×Dic₇₁	—	8	2	no	—
74	C₈×C₇₁	—	2	2	no	—
75	C₈×C₇₁	C₄×Q₈	1	2	yes	—
76	C₈×C₇₁	#124	1	1	yes	characteristic
77	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
79	C₂²×C₇₁	#474	1	2	yes	—
80	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
81	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
82	C₂²×C₇₁	#509	1	2	yes	—
83	C₄×C₇₁	—	2	2	no	—
84	C₄×C₇₁	—	8	4	no	—
85	C₄×C₇₁	#358	1	4	yes	—
86	C₄×C₇₁	#391	1	1	yes	characteristic
87	C₄×C₇₁	#419	1	1	yes	characteristic
88	C₄×C₇₁	#475	1	2	yes	—
89	Dic₇₁	—	8	4	no	—
90	#13276	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	#1988	1	1	yes	characteristic
92	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
93	C₂×C₇₁	#2140	1	1	yes	characteristic
94	C₂×C₇₁	#2226	1	2	yes	—
95	C₂×C₇₁	#2227	1	2	yes	—
96	C₂×#500	D₇₁	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	—	71	1	no	—
98	C₄×#107	—	71	1	no	—
99	#1380	—	71	4	no	—
100	C₇₁	#13276	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	2	1	no	—
102	C₂²×C₁₆	—	71	2	no	—
103	#500	—	2	2	no	—
104	#500	Dic₇₁	1	4	yes	complemented
105	C₂×#107	—	2	1	no	—
106	C₂×#107	—	71	2	no	—
107	C₂×#107	C₂×D₇₁	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	142	1	no	—
110	C₄⋊C₁₆	—	142	2	no	—
111	#340	—	142	1	no	—
112	C₂²×Q₈	—	4	1	no	—
113	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
115	C₄⋊Q₈	—	2	4	no	—
116	C₄⋊Q₈	—	4	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	142	2	no	—
120	C₂×C₄²	—	284	1	no	—
121	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
122	C₂×C₁₆	—	142	2	no	—
123	C₂×C₁₆	—	142	4	no	—
124	#107	—	2	4	no	—
125	#107	—	2	2	no	—
126	#107	—	71	4	no	—
127	#107	—	142	2	no	—
128	#107	C₂×Dic₇₁	1	4	yes	—
129	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
130	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
131	#95	—	284	1	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	8	4	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	284	2	no	—
136	C₂²×C₄	#11765850	1	1	yes	characteristic
137	C₂²×C₄	#11765850	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
141	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₇₁	1	2	yes	—
144	C₂×C₈	#11765866	1	2	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	284	2	no	—
149	C₄⋊C₄	—	142	4	no	—
150	C₄²	—	568	2	no	—
151	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
152	C₄²	#11765853	1	2	yes	—
153	C₁₆	—	284	2	no	—
154	C₂³	C₂×#11765850	1	1	yes	characteristic
155	Q₈	—	8	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	—	284	4	no	—
160	C₂×C₄	—	568	2	no	—
161	C₂×C₄	—	568	2	no	—
162	C₂×C₄	#14556049	1	2	yes	—
163	C₂×C₄	#14556049	1	2	yes	—
164	C₂×C₄	#14556071	1	2	yes	—
165	C₂×C₄	#14556074	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
167	C₂×C₄	#14556104	1	2	yes	—
168	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
169	C₂×C₄	#14556159	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	Q₈×Dic₇₁	1	2	yes	—
172	C₈	#14556117	1	1	yes	characteristic
173	C₈	#14556126	1	1	yes	characteristic
174	C₂²	C₂×#14556049	1	1	yes	characteristic
175	C₂²	#17986276	1	2	yes	—
176	C₂²	C₂×#14556071	1	1	yes	characteristic
177	C₂²	C₂×#14556074	1	1	yes	characteristic
178	C₂²	#17986435	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	8	4	no	—
181	C₄	—	568	4	no	—
182	C₄	#17985803	1	4	yes	—
183	C₄	#17985906	1	1	yes	characteristic
184	C₄	#17986008	1	1	yes	characteristic
185	C₄	#17986281	1	2	yes	—
186	C₂	#22589293	1	1	yes	characteristic
187	C₂	C₂×#17985803	1	1	yes	characteristic
188	C₂	#22590056	1	1	yes	characteristic
189	C₂	#22590484	1	2	yes	—
190	C₂	#22590489	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	#13276
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. #500×C₇₁	C₄	semidirect
21. #14556062	Q₈	semidirect
100. C₇₁	#13276	semidirect
104. #500	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₄×#14556062	2
#22586431	2
C₂×#500×C₇₁	2
#13276	71