Subgroups — #26125311

Order 12032.

290 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245187	C₂	1	2	yes	maximal
2	#20241306	C₂	1	2	yes	maximal, complemented
3	#20241305	C₂	1	2	yes	maximal, complemented
4	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	—
6	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	—
7	C₄×C₄₇⋊C₁₆	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₈²×C₄₇	C₄	1	1	yes	characteristic
12	C₈²×C₄₇	C₄	1	1	yes	characteristic
13	C₈²×C₄₇	C₂²	1	1	yes	characteristic
14	C₈²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄₇⋊C₁₆	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₄₇	C₈	1	2	yes	—
17	C₂²×C₈×C₄₇	C₈	1	2	yes	—
18	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₈	1	2	yes	—
21	C₄×C₈×C₄₇	C₈	1	2	yes	—
22	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	4	no	—
33	C₂×C₈×C₄₇	—	2	4	no	—
34	C₂×C₈×C₄₇	—	2	4	no	—
35	C₂×C₈×C₄₇	C₄²	1	2	yes	—
36	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
41	C₄²×C₄₇	C₄²	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₄₇⋊C₁₆	—	2	8	no	—
46	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	M₃₂	1	4	yes	—
53	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₄₇	—	2	2	no	—
57	C₈×C₄₇	—	2	2	no	—
58	C₈×C₄₇	—	2	2	no	—
59	C₈×C₄₇	—	2	2	no	—
60	C₈×C₄₇	M₃₂	1	4	yes	—
61	C₈×C₄₇	#131	1	4	yes	—
62	#11411	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	#346	1	2	yes	—
66	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	#346	1	2	yes	—
70	C₄×C₄₇	#405	1	2	yes	—
71	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
72	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
73	#2162	—	47	2	no	—
74	C₂×C₈²	D₄₇	1	1	yes	characteristic
75	#1312	—	47	2	no	—
76	#1311	—	47	2	no	—
77	C₂×C₄₇	—	2	2	no	—
78	C₂×C₄₇	#1450	1	1	yes	characteristic
79	C₂×C₄₇	C₂×#346	1	1	yes	characteristic
80	C₂×C₄₇	C₄×M₃₂	1	1	yes	characteristic
81	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
82	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
84	C₄⋊C₁₆	—	47	4	no	—
85	C₂²⋊C₁₆	—	47	4	no	—
86	C₄×C₁₆	—	47	4	no	—
87	C₈²	Dic₄₇	1	1	yes	characteristic
88	C₈²	Dic₄₇	1	1	yes	characteristic
89	C₈²	C₂×D₄₇	1	1	yes	characteristic
90	C₈²	C₂×D₄₇	1	1	yes	characteristic
91	C₄₇	#11411	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	C₄₇⋊C₈	1	2	yes	—
93	C₂²×C₈	C₄₇⋊C₈	1	2	yes	—
94	C₂²×C₈	C₄×D₄₇	1	2	yes	—
95	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
96	C₂×C₁₆	—	47	8	no	—
97	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
98	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
99	C₄×C₈	C₄×D₄₇	1	2	yes	—
100	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
113	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
114	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
115	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
116	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
118	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
119	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
120	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
121	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
122	C₁₆	—	94	8	no	—
123	C₂³	C₄×C₄₇⋊C₈	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
127	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
128	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
129	C₂×C₄	#13093964	1	4	yes	—
130	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	#13093964	1	4	yes	—
138	C₈	#13094030	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#15688899	1	2	yes	—
142	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	#15688899	1	2	yes	—
146	C₄	#15689078	1	2	yes	—
147	C₄	C₂×#13093964	1	2	yes	—
148	C₄	C₂×#13093964	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#20241671	1	1	yes	characteristic
151	C₂	C₂×#15688899	1	1	yes	characteristic
152	C₂	C₄×#13093964	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂×C₈²×C₄₇	C₂
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#11411
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #20241306	C₂	semidirect
3. #20241305	C₂	semidirect
91. C₄₇	#11411	semidirect

Maximal subgroups

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

subgroup	index
#20245187	2
#20241306	2
#20241305	2
C₂×C₈²×C₄₇	2
#11411	47