Subgroups — #26125272

Order 12032.

322 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241307	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₄	1	2	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
7	#15688899	C₄	1	2	yes	—
8	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
10	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
11	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂×M₃₂×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#346	C₄	1	2	yes	—
15	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₈	C₈	1	4	yes	—
17	#13093964	—	2	4	no	—
18	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₄×C₄₇⋊C₈	C₈	1	4	yes	—
20	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₄₇	—	2	4	no	—
26	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
36	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₄₇	C₄²	1	1	yes	characteristic
40	C₄²×C₄₇	C₂×C₈	1	2	yes	—
41	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄₇⋊C₈	—	2	8	no	—
43	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
44	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
45	C₁₆×C₄₇	—	2	4	no	—
46	C₄₇⋊C₁₆	—	2	4	no	—
47	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#97	1	2	yes	—
52	C₂×C₄×C₄₇	#97	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
55	C₈×C₄₇	#97	1	2	yes	—
56	C₈×C₄₇	#97	1	2	yes	—
57	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
58	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
59	C₄₇⋊C₈	—	2	4	no	—
60	C₄₇⋊C₈	M₃₂	1	8	yes	complemented
61	#11402	—	47	1	no	maximal, Sylow
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	#322	1	2	yes	—
64	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	#322	1	2	yes	—
67	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
68	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
69	C₄×C₄₇	C₂×M₃₂	1	4	yes	—
70	C₄×M₃₂	—	47	1	no	—
71	C₄×M₃₂	D₄₇	1	1	yes	characteristic
72	C₂×C₈²	—	47	1	no	—
73	#1312	—	47	4	no	—
74	C₂×C₄₇	—	2	2	no	—
75	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
76	C₂×C₄₇	C₄×M₃₂	1	2	yes	—
77	C₂×M₃₂	—	47	2	no	—
78	C₂×M₃₂	Dic₄₇	1	2	yes	—
79	C₂×C₄×C₈	—	47	2	no	—
80	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
81	#346	—	47	2	no	—
82	#346	Dic₄₇	1	2	yes	—
83	C₄×C₁₆	—	47	2	no	—
84	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
85	C₈²	—	47	1	no	—
86	C₈²	—	47	1	no	—
87	C₈²	—	47	2	no	—
88	C₄₇	#11402	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	47	4	no	—
90	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	94	4	no	—
95	C₂×C₁₆	—	47	4	no	—
96	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
97	C₄×C₈	—	47	4	no	—
98	C₄×C₈	—	47	2	no	—
99	C₄×C₈	—	47	2	no	—
100	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
101	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	C₂²×C₄	C₈×D₄₇	1	2	yes	—
105	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
106	C₂×C₈	—	47	4	no	—
107	C₂×C₈	—	47	4	no	—
108	C₂×C₈	—	94	8	no	—
109	C₂×C₈	#10625324	1	4	yes	—
110	C₂×C₈	#10625329	1	4	yes	—
111	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
112	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
113	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
114	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
115	C₄²	C₈×D₄₇	1	2	yes	—
116	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
117	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	C₁₆	—	94	4	no	—
120	C₂³	C₈×Dic₄₇	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
123	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
124	C₂×C₄	#13093934	1	2	yes	—
125	C₂×C₄	#13093935	1	2	yes	—
126	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
128	C₈	—	47	8	no	—
129	C₈	—	94	4	no	—
130	C₈	#13093934	1	2	yes	—
131	C₈	#13093935	1	2	yes	—
132	C₈	C₂×#10625324	1	2	yes	—
133	C₈	C₂×#10625329	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#15688845	1	2	yes	—
136	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	#15688845	1	2	yes	—
139	C₄	C₂×#13093934	1	1	yes	characteristic
140	C₄	C₂×#13093935	1	1	yes	characteristic
141	C₄	M₃₂×D₄₇	1	4	yes	—
142	C₂	—	2	2	no	—
143	C₂	C₂×#15688845	1	1	yes	characteristic
144	C₂	M₃₂×Dic₄₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄×M₃₂×C₄₇	C₂
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#11402
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20241307	C₂	semidirect
60. C₄₇⋊C₈	M₃₂	semidirect
88. C₄₇	#11402	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093964	2
C₂×C₈×C₄₇⋊C₈	2
#20241307	2
C₄×M₃₂×C₄₇	2
#11402	47