Subgroups — #31884317

Order 18176.

230 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	2	yes	maximal
2	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556068	—	2	2	no	—
4	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
5	C₄⋊(C₇₁⋊C₁₆)	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	#322×C₇₁	—	2	2	no	—
10	#14556068	—	4	2	no	—
11	#14556068	—	4	2	no	—
12	C₂×C₇₁⋊C₁₆	—	2	4	no	—
13	C₂²×C₈×C₇₁	—	2	1	no	—
14	C₂²×C₈×C₇₁	C₈	1	2	yes	—
15	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₇₁	—	2	1	no	—
19	C₄×C₈×C₇₁	—	2	1	no	—
20	C₄×C₈×C₇₁	—	2	1	no	—
21	C₄×C₈×C₇₁	C₈	1	2	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₁	D₄	1	2	yes	—
24	C₇₁⋊C₁₆	—	4	4	no	—
25	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
28	C₂×C₈×C₇₁	—	2	1	no	—
29	C₂×C₈×C₇₁	—	2	1	no	—
30	C₂×C₈×C₇₁	—	2	2	no	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	4	2	no	—
33	C₂×C₈×C₇₁	—	4	2	no	—
34	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
35	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
38	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
39	C₄²×C₇₁	C₄²	1	1	yes	characteristic
40	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
43	C₂³×C₇₁	C₄⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
56	C₈×C₇₁	—	2	2	no	—
57	C₈×C₇₁	—	4	2	no	—
58	C₈×C₇₁	—	4	2	no	—
59	C₈×C₇₁	#102	1	2	yes	—
60	C₈×C₇₁	#103	1	2	yes	—
61	C₂²×C₇₁	—	2	1	no	—
62	C₂²×C₇₁	—	2	1	no	—
63	C₂²×C₇₁	#334	1	1	yes	characteristic
64	C₂²×C₇₁	#335	1	1	yes	characteristic
65	C₂²×C₇₁	#336	1	1	yes	characteristic
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	#334	1	1	yes	characteristic
69	C₄×C₇₁	#335	1	1	yes	characteristic
70	C₄×C₇₁	#340	1	1	yes	characteristic
71	C₄×C₇₁	#341	1	1	yes	characteristic
72	C₄×C₇₁	#347	1	2	yes	—
73	C₄×C₇₁	#349	1	2	yes	—
74	#7152	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	—	2	1	no	—
76	C₂×C₇₁	—	2	1	no	—
77	C₂×C₇₁	#1277	1	1	yes	characteristic
78	C₂×C₇₁	#1375	1	1	yes	characteristic
79	C₂×C₇₁	#1377	1	1	yes	characteristic
80	#2150	—	71	2	no	—
81	C₂×#322	D₇₁	1	1	yes	characteristic
82	C₇₁	#7152	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	142	2	no	—
84	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
85	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
87	C₄⋊C₁₆	—	71	4	no	—
88	C₄⋊C₁₆	—	142	2	no	—
89	#322	—	2	2	no	—
90	C₂²×C₈	—	2	1	no	—
91	C₂²×C₈	C₇₁⋊C₈	1	2	yes	—
92	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
93	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
95	M₃₂	—	284	2	no	—
96	M₃₂	—	284	2	no	—
97	C₂×C₁₆	—	142	4	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
102	C₄×C₈	C₄×D₇₁	1	2	yes	—
103	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
104	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	#11765849	1	1	yes	characteristic
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
114	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
115	C₂×C₈	D₅₆₈	1	1	yes	characteristic
116	C₂×C₈	#11765854	1	2	yes	—
117	C₂×C₈	Dic₂₈₄	1	1	yes	characteristic
118	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
119	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
120	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
121	C₄²	#11765849	1	1	yes	characteristic
122	C₁₆	—	284	4	no	—
123	C₂³	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
127	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
128	C₂×C₄	#14556039	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
131	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
132	C₂×C₄	#14556060	1	1	yes	characteristic
133	C₂×C₄	#14556062	1	1	yes	characteristic
134	C₂×C₄	#14556064	1	1	yes	characteristic
135	C₂×C₄	#14556064	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	4	2	no	—
138	C₈	—	4	2	no	—
139	C₈	#14556050	1	2	yes	—
140	C₈	#14556053	1	2	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#17985750	1	1	yes	characteristic
144	C₂²	#17985752	1	1	yes	characteristic
145	C₂²	#17985754	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	#17985750	1	1	yes	characteristic
149	C₄	#17985752	1	1	yes	characteristic
150	C₄	#17985762	1	1	yes	characteristic
151	C₄	#17985764	1	1	yes	characteristic
152	C₄	#17985777	1	2	yes	—
153	C₄	#17985781	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#22586175	1	1	yes	characteristic
157	C₂	#22586420	1	1	yes	characteristic
158	C₂	#22586424	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₇₁⋊#95
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×#322×C₇₁	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#7152
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
82. C₇₁	#7152	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₂×#322×C₇₁	2
#7152	71