Subgroups — #31909010

Order 18176.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22587088	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
3	#22586406	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986052	—	2	1	no	—
6	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	#17985752	—	2	1	no	—
8	#17985750	—	2	1	no	—
9	#17985750	C₂²	1	2	yes	—
10	#17985750	C₂²	1	2	yes	—
11	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
12	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#346×C₇₁	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
16	C₂²×C₇₁⋊C₈	—	4	1	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	4	1	no	—
21	C₄×C₇₁⋊C₈	—	4	1	no	—
22	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
25	M₃₂×C₇₁	—	2	2	no	—
26	M₃₂×C₇₁	C₈	1	4	yes	complemented
27	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₂×C₇₁⋊C₈	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	8	2	no	—
36	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
44	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
48	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
49	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₇₁	—	2	2	no	—
51	C₁₆×C₇₁	C₂×C₈	1	4	yes	complemented
52	C₂³×C₇₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#108	1	2	yes	—
60	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₇₁	—	2	1	no	—
64	C₈×C₇₁	—	2	1	no	—
65	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
66	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
68	C₇₁⋊C₈	—	8	2	no	—
69	C₇₁⋊C₈	—	8	2	no	—
70	C₂²×C₇₁	—	2	1	no	—
71	C₂²×C₇₁	—	2	1	no	—
72	C₂²×C₇₁	#334	1	2	yes	—
73	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	—	2	1	no	—
76	C₄×C₇₁	—	2	1	no	—
77	C₄×C₇₁	#334	1	2	yes	—
78	C₄×C₇₁	#365	1	2	yes	—
79	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
81	#12000	—	71	1	no	maximal, Sylow
82	C₂×C₇₁	—	2	2	no	—
83	C₂×C₇₁	C₂×#334	1	1	yes	characteristic
84	C₂×C₇₁	C₂×#365	1	1	yes	characteristic
85	C₂×C₇₁	#2155	1	1	yes	characteristic
86	C₄×M₃₂	D₇₁	1	1	yes	characteristic
87	#1564	—	71	1	no	—
88	C₂×#334	—	71	1	no	—
89	#1368	—	71	4	no	—
90	C₇₁	#12000	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
92	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
93	#431	—	142	1	no	—
94	C₂×C₄⋊C₈	—	142	1	no	—
95	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
96	#346	C₂×D₇₁	1	2	yes	—
97	C₄×C₁₆	Dic₇₁	1	2	yes	—
98	#335	—	142	1	no	—
99	#334	—	71	2	no	—
100	#334	—	71	2	no	—
101	#334	—	142	1	no	—
102	C₂²×C₈	—	284	1	no	—
103	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
105	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₇₁⋊C₈	1	4	yes	complemented
108	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
110	C₄⋊C₈	—	284	1	no	—
111	C₄⋊C₈	—	142	2	no	—
112	C₄⋊C₈	—	142	2	no	—
113	C₂²⋊C₈	—	284	1	no	—
114	C₄×C₈	—	284	1	no	—
115	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
116	C₄×C₈	D₂₈₄	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
120	C₂²×C₄	#11765854	1	1	yes	characteristic
121	C₂²×C₄	#11765854	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	284	2	no	—
124	C₂×C₈	—	284	2	no	—
125	C₂×C₈	—	568	2	no	—
126	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
128	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
129	C₂×C₈	#11765849	1	2	yes	—
130	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
133	C₄²	#11765854	1	1	yes	characteristic
134	C₄²	#11765854	1	1	yes	characteristic
135	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	complemented
138	C₂³	#14556060	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
142	C₂×C₄	#14556060	1	1	yes	characteristic
143	C₂×C₄	#14556060	1	1	yes	characteristic
144	C₂×C₄	#14556060	1	1	yes	characteristic
145	C₂×C₄	#14556064	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
147	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
148	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	568	2	no	—
152	C₈	—	568	2	no	—
153	C₈	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
154	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
155	C₈	C₂×#11765849	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#17985750	1	2	yes	—
159	C₂²	C₂×#14556060	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#17985750	1	2	yes	—
164	C₄	#17985822	1	2	yes	—
165	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
166	C₄	C₂×#14556064	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#17985750	1	1	yes	characteristic
169	C₂	C₂×#17985822	1	1	yes	characteristic
170	C₂	#22590113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556060
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₄×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#106
prime radical	C₂³×C₇₁	#106
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₄×M₃₂	D₇₁
hypocenter	C₇₁	#12000
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22586406	C₂	semidirect
26. M₃₂×C₇₁	C₈	semidirect
51. C₁₆×C₇₁	C₂×C₈	semidirect
90. C₇₁	#12000	semidirect
107. M₃₂	C₇₁⋊C₈	semidirect
137. C₁₆	C₂×C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22587088	2
C₂×#17985750	2
#22586406	2
C₄×M₃₂×C₇₁	2
#12000	71