Subgroups — #26132715

Order 12032.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688951	C₂	1	4	yes	maximal
2	#20243548	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093958	—	2	2	no	—
5	C₂×#13093958	C₂²	1	4	yes	—
6	#15688993	—	2	2	no	—
7	#15688951	—	2	8	no	—
8	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₄₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₄₇	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	—	4	4	no	—
13	#13093958	—	2	8	no	—
14	#13093958	—	4	4	no	—
15	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
20	C₂×C₁₆×C₄₇	Q₈	1	4	yes	—
21	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₄₇	D₈	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	4	4	no	—
26	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₄₇	D₈	1	2	yes	—
28	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
29	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
31	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
32	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
34	C₄²×C₄₇	Q₁₆	1	2	yes	—
35	C₄⋊Dic₄₇	—	4	4	no	—
36	C₄⋊Dic₄₇	—	4	4	no	—
37	C₄⋊Dic₄₇	—	8	8	no	—
38	C₁₆×C₄₇	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₄₇	#107	1	4	yes	—
41	C₂×C₄×C₄₇	SD₃₂	1	8	yes	—
42	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₄₇	—	8	4	no	—
49	C₂×Dic₄₇	—	8	8	no	—
50	C₈×C₄₇	#107	1	4	yes	—
51	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
52	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
53	#12906	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
55	C₂²×C₄₇	#493	1	1	yes	characteristic
56	C₂²×C₄₇	#500	1	1	yes	characteristic
57	C₂²×C₄₇	C₂×SD₃₂	1	4	yes	—
58	C₄×C₄₇	#367	1	8	yes	complemented
59	C₄×C₄₇	#384	1	1	yes	characteristic
60	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
61	C₄×C₄₇	#494	1	1	yes	characteristic
62	C₄×C₄₇	#500	1	1	yes	characteristic
63	Dic₄₇	—	16	4	no	—
64	C₂×#367	—	47	4	no	—
65	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
66	#1845	—	47	2	no	—
67	C₂×C₄₇	#1845	1	1	yes	characteristic
68	C₂×C₄₇	C₂×#367	1	2	yes	—
69	C₂×C₄₇	#2248	1	2	yes	—
70	C₂×C₄₇	#2254	1	2	yes	—
71	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
72	C₂×#107	—	47	4	no	—
73	C₂×#107	—	94	2	no	—
74	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
75	#384	—	94	2	no	—
76	#367	—	94	8	no	—
77	C₄×C₁₆	Dic₄₇	1	4	yes	complemented
78	C₄₇	#12906	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	94	4	no	—
82	C₂×C₄⋊C₄	—	188	4	no	—
83	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
84	C₂×C₁₆	Dic₉₄	1	4	yes	—
85	C₂×C₁₆	D₁₈₈	1	4	yes	—
86	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
87	#107	—	94	8	no	—
88	#107	—	188	4	no	—
89	C₄×C₈	Dic₉₄	1	2	yes	—
90	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
91	C₂²×C₄	—	188	4	no	—
92	C₂²×C₄	D₃₇₆	1	2	yes	—
93	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
94	C₂×C₈	D₃₇₆	1	2	yes	—
95	C₂×C₈	Dic₁₈₈	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
97	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
98	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
100	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
101	C₄⋊C₄	—	376	8	no	—
102	C₄⋊C₄	—	188	4	no	—
103	C₄⋊C₄	—	188	4	no	—
104	C₄²	Dic₁₈₈	1	2	yes	—
105	C₄²	C₄⋊Dic₄₇	1	2	yes	—
106	C₁₆	C₄⋊Dic₄₇	1	8	yes	complemented
107	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
108	C₂×C₄	—	376	4	no	—
109	C₂×C₄	—	376	8	no	—
110	C₂×C₄	#13093958	1	4	yes	—
111	C₂×C₄	#13093969	1	8	yes	—
112	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
113	C₂×C₄	#13094021	1	1	yes	characteristic
114	C₂×C₄	#13094024	1	1	yes	characteristic
115	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
118	C₈	#13093958	1	4	yes	—
119	C₈	C₂×C₄⋊Dic₄₇	1	2	yes	—
120	C₈	#13094024	1	2	yes	—
121	C₂²	C₂×#13093958	1	1	yes	characteristic
122	C₂²	#15689500	1	1	yes	characteristic
123	C₂²	#15689528	1	1	yes	characteristic
124	C₂²	C₂×#13093969	1	4	yes	—
125	C₄	—	752	4	no	—
126	C₄	#15688951	1	8	yes	complemented
127	C₄	#15688993	1	1	yes	characteristic
128	C₄	C₂×#13093958	1	1	yes	characteristic
129	C₄	#15689503	1	1	yes	characteristic
130	C₄	#15689528	1	1	yes	characteristic
131	C₂	#20243548	1	1	yes	characteristic
132	C₂	C₂×#15688951	1	2	yes	—
133	C₂	#20245644	1	2	yes	—
134	C₂	#20245670	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#12906
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₄₇	C₄	semidirect
38. C₁₆×C₄₇	C₄⋊C₄	semidirect
58. C₄×C₄₇	#367	semidirect
77. C₄×C₁₆	Dic₄₇	semidirect
78. C₄₇	#12906	semidirect
106. C₁₆	C₄⋊Dic₄₇	semidirect
126. C₄	#15688951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688951	2
#20243548	2
C₂×C₄×C₁₆×C₄₇	2
#12906	47