Subgroups — #26125259

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	#20241671	C₂	1	1	yes	maximal, characteristic
3	#20241304	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₄	1	2	yes	—
7	#15689176	C₄	1	2	yes	—
8	#15688899	C₄	1	2	yes	—
9	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15688844	C₄	1	2	yes	—
11	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₄₇	C₄	1	2	yes	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	#13093964	C₈	1	8	yes	complemented
18	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₄⋊(C₄₇⋊C₈)	C₈	1	4	yes	—
20	C₂²⋊(C₄₇⋊C₈)	C₈	1	4	yes	—
21	C₄×C₄₇⋊C₈	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	C₂×C₁₆×C₄₇	C₈	1	8	yes	complemented
26	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
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	4	yes	—
37	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₄²	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×C₈	1	2	yes	—
42	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄₇⋊C₈	C₂×C₈	1	8	yes	—
44	C₁₆×C₄₇	C₂×C₈	1	4	yes	—
45	C₁₆×C₄₇	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	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₄₇	C₄×C₈	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	C₄×C₈	1	2	yes	—
56	C₈×C₄₇	C₄×C₈	1	2	yes	—
57	C₈×C₄₇	C₂²×C₈	1	2	yes	—
58	C₈×C₄₇	C₂²×C₈	1	2	yes	—
59	C₄₇⋊C₈	—	2	8	no	—
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	D₄₇	1	1	yes	characteristic
71	#1450	—	47	1	no	—
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	Dic₄₇	1	2	yes	—
79	#431	—	47	2	no	—
80	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
81	#346	—	47	2	no	—
82	C₄×C₁₆	—	47	2	no	—
83	C₄×C₁₆	Dic₄₇	1	2	yes	—
84	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
85	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
86	#322	—	47	2	no	—
87	C₈²	—	47	2	no	—
88	C₄₇	#11399	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
91	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
92	M₃₂	—	47	8	no	—
93	C₂×C₁₆	—	47	4	no	—
94	C₂×C₁₆	C₄₇⋊C₈	1	8	yes	complemented
95	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
97	C₄⋊C₈	—	47	4	no	—
98	C₂²⋊C₈	—	47	4	no	—
99	C₄×C₈	—	47	4	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	8	no	—
107	C₂×C₈	C₈×D₄₇	1	4	yes	—
108	C₂×C₈	C₂×C₄₇⋊C₈	1	4	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	C₂²×Dic₄₇	1	1	yes	characteristic
113	C₄²	C₈×D₄₇	1	2	yes	—
114	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
115	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
116	C₁₆	—	47	8	no	—
117	C₁₆	C₂×C₄₇⋊C₈	1	4	yes	—
118	C₁₆	C₂×C₄₇⋊C₈	1	4	yes	—
119	C₂³	C₈×Dic₄₇	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	C₈×Dic₄₇	1	1	yes	characteristic
124	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
125	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
126	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
127	C₈	—	94	8	no	—
128	C₈	C₈×Dic₄₇	1	2	yes	—
129	C₈	C₄×C₄₇⋊C₈	1	2	yes	—
130	C₈	C₂×C₈×D₄₇	1	2	yes	—
131	C₈	C₂²×C₄₇⋊C₈	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈×C₄₇⋊C₈	1	2	yes	—
134	C₂²	C₂×C₈×Dic₄₇	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×C₄×C₄₇⋊C₈	1	1	yes	characteristic
139	C₄	#15689539	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×C₄₇⋊C₈	1	1	yes	characteristic
142	C₂	#20244984	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₈²
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×C₈
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#11399
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20241304	C₂	semidirect
4. C₁₆×C₄₇⋊C₈	C₂	semidirect
17. #13093964	C₈	semidirect
25. C₂×C₁₆×C₄₇	C₈	semidirect
88. C₄₇	#11399	semidirect
94. C₂×C₁₆	C₄₇⋊C₈	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
#20241671	2
#20241304	2
C₁₆×C₄₇⋊C₈	2
C₂×C₄×C₁₆×C₄₇	2
#11399	47