Subgroups — #26125260

Order 12032.

330 conjugacy classes of subgroups.

The subgroup lattice has 144 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×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20241305	C₂	1	2	yes	maximal, complemented
4	C₈×C₄₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₄	1	2	yes	—
7	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
8	#15688899	C₄	1	2	yes	—
9	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄₇×#431	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#322	C₄	1	2	yes	—
16	C₈²×C₄₇	C₂²	1	2	yes	—
17	#13093964	C₈	1	8	yes	complemented
18	C₂×C₄₇⋊C₁₆	C₈	1	8	yes	complemented
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	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	C₄⋊C₈×C₄₇	C₈	1	4	yes	—
26	C₂²⋊C₈×C₄₇	C₈	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	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×C₈	1	8	yes	—
37	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₄²	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×C₈	1	2	yes	—
43	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₄₇⋊C₁₆	C₂×C₈	1	4	yes	—
45	C₄₇⋊C₁₆	C₂×C₈	1	4	yes	—
46	C₄₇⋊C₁₆	C₂×C₈	1	8	yes	—
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	2	yes	—
51	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
55	C₈×C₄₇	—	2	8	no	—
56	C₈×C₄₇	C₄×C₈	1	2	yes	—
57	C₈×C₄₇	C₄×C₈	1	2	yes	—
58	C₈×C₄₇	C₂²×C₈	1	2	yes	—
59	C₈×C₄₇	C₂²×C₈	1	2	yes	—
60	#11399	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	C₈²	1	2	yes	—
63	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	C₈²	1	2	yes	—
66	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₄₇	#504	1	4	yes	—
69	C₄×M₃₂	—	47	1	no	—
70	C₂×C₄×C₁₆	—	47	1	no	—
71	#1450	D₄₇	1	1	yes	characteristic
72	#1311	—	47	2	no	—
73	C₈×C₁₆	—	47	2	no	—
74	C₂×C₄₇	—	2	2	no	—
75	C₂×C₄₇	C₂×C₈²	1	1	yes	characteristic
76	C₂×C₄₇	#2108	1	2	yes	—
77	C₂×M₃₂	—	47	2	no	—
78	C₂²×C₁₆	—	47	2	no	—
79	#431	Dic₄₇	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
81	#346	—	47	2	no	—
82	C₄×C₁₆	—	47	2	no	—
83	C₄×C₁₆	—	47	1	no	—
84	C₄×C₁₆	—	47	2	no	—
85	C₄×C₁₆	—	47	1	no	—
86	#322	Dic₄₇	1	2	yes	—
87	C₈²	C₂×D₄₇	1	2	yes	—
88	C₄₇	#11399	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
90	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
91	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
92	M₃₂	—	47	8	no	—
93	C₂×C₁₆	—	47	8	no	—
94	C₂×C₁₆	—	47	2	no	—
95	C₂×C₁₆	—	47	2	no	—
96	C₂×C₁₆	—	47	4	no	—
97	C₄⋊C₈	C₄₇⋊C₈	1	4	yes	—
98	C₂²⋊C₈	C₄₇⋊C₈	1	4	yes	—
99	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
100	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
102	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
105	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
106	C₂×C₈	C₈×D₄₇	1	4	yes	—
107	C₂×C₈	C₈×D₄₇	1	4	yes	—
108	C₂×C₈	C₂×C₄₇⋊C₈	1	8	yes	—
109	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
110	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
111	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
113	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
114	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
115	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
116	C₁₆	—	47	4	no	—
117	C₁₆	—	47	4	no	—
118	C₁₆	—	47	8	no	—
119	C₂³	C₄×C₄₇⋊C₈	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
122	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
123	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
124	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
126	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
127	C₈	—	2	8	no	—
128	C₈	C₈×Dic₄₇	1	2	yes	—
129	C₈	C₈×Dic₄₇	1	2	yes	—
130	C₈	C₂×C₈×D₄₇	1	2	yes	—
131	C₈	C₂×C₈×D₄₇	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈×C₄₇⋊C₈	1	2	yes	—
134	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈×C₄₇⋊C₈	1	2	yes	—
137	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
138	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
139	C₄	#15689540	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×C₄₇⋊C₈	1	1	yes	characteristic
142	C₂	#20244985	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator	C₂×C₄₇	C₂×C₈²
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	C₂×C₄×C₈
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#11399
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20241305	C₂	semidirect
4. C₈×C₄₇⋊C₁₆	C₂	semidirect
17. #13093964	C₈	semidirect
18. C₂×C₄₇⋊C₁₆	C₈	semidirect
88. C₄₇	#11399	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×C₄×C₄₇⋊C₁₆	2
#20241305	2
C₈×C₄₇⋊C₁₆	2
C₄₇×#1450	2
#11399	47