Subgroups — #31891118

Order 18176.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588492	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556039	C₂	1	1	yes	maximal, characteristic
3	#22586174	C₂	1	4	yes	maximal
4	C₇₁×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14556039	C₄	1	2	yes	—
8	C₂×#14556039	C₄	1	1	yes	characteristic
9	C₂×#14556039	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
12	C₄³×C₇₁	C₂²	1	1	yes	characteristic
13	#336×C₇₁	—	2	2	no	—
14	C₂²×C₇₁⋊C₈	—	2	2	no	—
15	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
17	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
18	C₄⋊(C₇₁⋊C₈)	—	4	4	no	—
19	#14556039	—	2	4	no	—
20	#14556039	—	2	2	no	—
21	#14556039	C₈	1	4	yes	complemented
22	C₄×C₇₁⋊C₈	—	2	2	no	—
23	C₄×C₇₁⋊C₈	—	2	2	no	—
24	C₂²×C₈×C₇₁	—	2	2	no	—
25	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₁	D₄	1	2	yes	—
28	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
29	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₁	—	2	4	no	—
31	C₂×C₇₁⋊C₈	—	2	2	no	—
32	C₂×C₇₁⋊C₈	—	2	2	no	—
33	C₂×C₇₁⋊C₈	—	2	2	no	—
34	C₂×C₇₁⋊C₈	—	2	2	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₂×C₇₁⋊C₈	—	4	4	no	—
37	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
38	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
39	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₇₁	—	4	2	no	—
45	C₂×C₈×C₇₁	—	4	2	no	—
46	C₂×C₈×C₇₁	—	4	2	no	—
47	C₄²×C₇₁	—	2	2	no	—
48	C₄²×C₇₁	—	2	2	no	—
49	C₄²×C₇₁	—	2	2	no	—
50	C₄²×C₇₁	—	4	2	no	—
51	C₄²×C₇₁	C₂×C₈	1	2	yes	—
52	C₄²×C₇₁	M₁₆	1	2	yes	—
53	C₄²×C₇₁	#41	1	2	yes	—
54	C₄²×C₇₁	#41	1	2	yes	—
55	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	—	2	2	no	—
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₇₁	#117	1	2	yes	—
64	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#125	1	2	yes	—
67	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₇₁	#137	1	2	yes	—
72	C₈×C₇₁	—	8	2	no	—
73	C₇₁⋊C₈	—	2	4	no	—
74	C₇₁⋊C₈	—	2	4	no	—
75	C₇₁⋊C₈	—	8	2	no	—
76	C₂²×C₇₁	#382	1	1	yes	characteristic
77	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₇₁	#423	1	1	yes	characteristic
79	C₂²×C₇₁	#428	1	2	yes	—
80	C₂²×C₇₁	#430	1	2	yes	—
81	C₄×C₇₁	—	2	4	no	—
82	C₄×C₇₁	—	2	2	no	—
83	C₄×C₇₁	—	4	2	no	—
84	C₄×C₇₁	#431	1	2	yes	—
85	C₄×C₇₁	#432	1	2	yes	—
86	C₄×C₇₁	#489	1	2	yes	—
87	C₄×C₇₁	#490	1	1	yes	characteristic
88	C₄×C₇₁	#491	1	1	yes	characteristic
89	#8786	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	#1565	1	2	yes	—
91	C₂×C₇₁	#1569	1	2	yes	—
92	C₂×C₇₁	#1831	1	1	yes	characteristic
93	C₂×C₇₁	#1836	1	1	yes	characteristic
94	C₂×C₇₁	#1838	1	1	yes	characteristic
95	#1842	—	71	1	no	—
96	#1831	D₇₁	1	1	yes	characteristic
97	C₄×#97	—	71	1	no	—
98	#1277	—	71	4	no	—
99	C₇₁	#8786	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊C₈	—	71	2	no	—
101	C₂×C₄⋊C₈	—	142	2	no	—
102	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
103	C₂×#97	—	71	1	no	—
104	C₂×#97	—	71	2	no	—
105	C₂×#97	—	71	1	no	—
106	C₂×C₄×C₈	—	71	2	no	—
107	C₄³	C₂×D₇₁	1	1	yes	characteristic
108	#336	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	71	2	no	—
111	C₂²×C₈	—	71	2	no	—
112	C₂²×C₈	—	142	2	no	—
113	C₂×C₄²	Dic₁₄₂	1	2	yes	—
114	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
117	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	284	4	no	—
120	C₄⋊C₈	—	142	4	no	—
121	#97	—	71	4	no	—
122	#97	—	142	4	no	—
123	#97	—	142	2	no	—
124	C₄×C₈	—	142	2	no	—
125	C₄×C₈	—	142	2	no	—
126	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
127	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
128	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	284	2	no	—
135	C₂×C₈	—	284	4	no	—
136	C₂×C₈	—	71	4	no	—
137	C₂×C₈	—	71	4	no	—
138	C₂×C₈	—	142	2	no	—
139	C₂×C₈	—	142	2	no	—
140	C₂×C₈	—	142	2	no	—
141	C₂×C₈	—	142	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₈×D₇₁	1	2	yes	—
147	C₄²	#11765848	1	2	yes	—
148	C₄²	#11765864	1	2	yes	—
149	C₄²	#11765866	1	2	yes	—
150	C₂³	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	Dic₇₁⋊C₈	1	4	yes	—
158	C₂×C₄	#14556086	1	2	yes	—
159	C₂×C₄	#14556115	1	1	yes	characteristic
160	C₂×C₄	#14556117	1	1	yes	characteristic
161	C₂×C₄	#14556119	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
163	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
164	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
165	C₂×C₄	#14556153	1	2	yes	—
166	C₂×C₄	#14556158	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	568	2	no	—
169	C₈	—	142	4	no	—
170	C₈	—	142	4	no	—
171	C₂²	#17985864	1	1	yes	characteristic
172	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
173	C₂²	#17986022	1	1	yes	characteristic
174	C₂²	#17986042	1	2	yes	—
175	C₂²	#17986047	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	#17986051	1	2	yes	—
180	C₄	#17986055	1	2	yes	—
181	C₄	#17986358	1	2	yes	—
182	C₄	#17986365	1	1	yes	characteristic
183	C₄	#17986369	1	1	yes	characteristic
184	C₂	#22587093	1	2	yes	—
185	C₂	#22587108	1	2	yes	—
186	C₂	#22588456	1	1	yes	characteristic
187	C₂	#22588473	1	1	yes	characteristic
188	C₂	#22588480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1831	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1831	D₇₁
hypocenter	C₇₁	#8786
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
21. #14556039	C₈	semidirect
99. C₇₁	#8786	semidirect

Maximal subgroups

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

subgroup	index
#22588492	2
C₄×#14556039	2
#22586174	2
C₇₁×#1831	2
#8786	71