Subgroups — #26125278

Order 12032.

298 conjugacy classes of subgroups.

The subgroup lattice has 138 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×#15688844	C₂	1	1	yes	maximal, characteristic
3	#20241308	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	C₂²	1	2	yes	—
8	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
9	#15688844	C₄	1	2	yes	—
10	#15688844	C₂²	1	2	yes	—
11	C₂×M₃₂×C₄₇	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#346	C₂²	1	2	yes	—
14	C₄×C₁₆×C₄₇	C₄	1	2	yes	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	#13093964	—	2	4	no	—
17	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₄×C₄₇⋊C₈	—	2	2	no	—
19	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₄₇	C₈	1	8	yes	complemented
24	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
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	M₁₆	1	2	yes	—
31	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
35	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₄²	1	1	yes	characteristic
38	C₄²×C₄₇	M₁₆	1	2	yes	—
39	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	—	2	4	no	—
41	C₂×C₄₇⋊C₈	—	4	4	no	—
42	C₁₆×C₄₇	C₂×C₈	1	8	yes	complemented
43	C₄₇⋊C₁₆	—	2	4	no	—
44	C₂³×C₄₇	#97	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#97	1	2	yes	—
49	C₂×C₄×C₄₇	#97	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
52	C₈×C₄₇	C₄×C₈	1	2	yes	—
53	C₈×C₄₇	#97	1	2	yes	—
54	C₈×C₄₇	C₂²×C₈	1	2	yes	—
55	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
56	C₄₇⋊C₈	—	4	4	no	—
57	#11404	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	—	2	2	no	—
59	C₂²×C₄₇	#322	1	2	yes	—
60	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	#322	1	2	yes	—
63	C₄×C₄₇	#347	1	4	yes	—
64	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
66	C₄×M₃₂	—	47	1	no	—
67	C₄×M₃₂	D₄₇	1	1	yes	characteristic
68	C₂×#322	—	47	1	no	—
69	#1313	—	47	4	no	—
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
72	C₂×C₄₇	C₂×#347	1	2	yes	—
73	C₂×M₃₂	—	47	2	no	—
74	C₂×M₃₂	Dic₄₇	1	2	yes	—
75	C₂×C₄×C₈	—	47	2	no	—
76	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
77	#346	—	47	2	no	—
78	#346	C₂×D₄₇	1	2	yes	—
79	C₄×C₁₆	—	47	2	no	—
80	C₄×C₁₆	Dic₄₇	1	2	yes	—
81	#322	—	47	2	no	—
82	#322	—	47	2	no	—
83	C₄₇	#11404	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	94	2	no	—
85	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
87	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
88	M₃₂	—	94	4	no	—
89	M₃₂	C₄₇⋊C₈	1	8	yes	complemented
90	C₂×C₁₆	—	47	4	no	—
91	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
92	C₄×C₈	—	47	4	no	—
93	C₄×C₈	—	94	2	no	—
94	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
95	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
98	C₂²×C₄	#10625324	1	2	yes	—
99	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
100	C₂×C₈	—	94	4	no	—
101	C₂×C₈	—	188	4	no	—
102	C₂×C₈	#10625324	1	4	yes	—
103	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
104	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
105	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
106	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
107	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
108	C₄²	#10625324	1	2	yes	—
109	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
110	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
111	C₁₆	—	94	4	no	—
112	C₁₆	C₂×C₄₇⋊C₈	1	8	yes	complemented
113	C₂³	#13093934	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
116	C₂×C₄	#13093934	1	2	yes	—
117	C₂×C₄	#13093934	1	1	yes	characteristic
118	C₂×C₄	#13093934	1	2	yes	—
119	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
120	C₂×C₄	C₂×#10625324	1	2	yes	—
121	C₈	—	188	4	no	—
122	C₈	C₄×C₄₇⋊C₈	1	2	yes	—
123	C₈	#13093934	1	2	yes	—
124	C₈	C₂²×C₄₇⋊C₈	1	2	yes	—
125	C₈	C₂×#10625324	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#15688844	1	2	yes	—
128	C₂²	C₂×#13093934	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	#15688844	1	2	yes	—
131	C₄	#15688900	1	4	yes	—
132	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
133	C₄	C₂×#13093934	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×#15688844	1	1	yes	characteristic
136	C₂	C₂×#15688900	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂³×C₄₇	#97
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#11404
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20241308	C₂	semidirect
23. M₃₂×C₄₇	C₈	semidirect
42. C₁₆×C₄₇	C₂×C₈	semidirect
83. C₄₇	#11404	semidirect
89. M₃₂	C₄₇⋊C₈	semidirect
112. C₁₆	C₂×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
C₂×#15688844	2
#20241308	2
C₄×M₃₂×C₄₇	2
#11404	47