Subgroups — #26131158

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	2	yes	maximal
2	#20241480	C₂	1	4	yes	maximal, complemented
3	C₂³×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093983	—	2	2	no	—
5	#15689161	—	2	4	no	—
6	#15689161	C₄	1	8	yes	complemented
7	C₂×#13093958	C₂²	1	4	yes	—
8	C₂×#13093956	—	2	2	no	—
9	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₄₇	—	2	4	no	—
11	C₂²×C₁₆×C₄₇	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
13	#13093983	—	2	8	no	—
14	#13093983	—	4	4	no	—
15	C₂×C₄⋊Dic₄₇	—	2	4	no	—
16	C₂×C₂²⋊Dic₄₇	—	4	2	no	—
17	C₂×C₄×Dic₄₇	—	4	2	no	—
18	#13093958	—	2	4	no	—
19	#13093958	C₂×C₄	1	8	yes	—
20	#13093956	—	2	8	no	—
21	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	D₄	1	4	yes	—
26	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	—	2	8	no	—
29	C₂×C₁₆×C₄₇	—	2	4	no	—
30	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
31	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	4	4	no	—
39	C₂×C₈×C₄₇	C₄²	1	4	yes	—
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
44	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²⋊Dic₄₇	—	4	8	no	—
49	C₄⋊Dic₄₇	—	2	8	no	—
50	C₄⋊Dic₄₇	—	4	4	no	—
51	C₄×Dic₄₇	—	4	8	no	—
52	C₁₆×C₄₇	—	2	4	no	—
53	C₂³×C₄₇	—	2	4	no	—
54	C₂³×C₄₇	#102	1	4	yes	—
55	C₂³×C₄₇	#107	1	2	yes	—
56	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#95	1	4	yes	—
58	C₂×C₄×C₄₇	#102	1	4	yes	—
59	C₂×C₄×C₄₇	#103	1	8	yes	—
60	C₂×C₄×C₄₇	#106	1	4	yes	—
61	C₂×C₄×C₄₇	#107	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₄₇	—	4	8	no	—
67	C₂×Dic₄₇	—	8	8	no	—
68	C₈×C₄₇	#95	1	4	yes	—
69	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
72	#12556	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	8	no	—
74	C₂²×C₄₇	—	2	4	no	—
75	C₂²×C₄₇	#340	1	4	yes	—
76	C₂²×C₄₇	C₂×#102	1	2	yes	—
77	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
78	C₂²×C₄₇	#508	1	4	yes	—
79	C₄×C₄₇	#340	1	4	yes	—
80	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
81	C₄×C₄₇	C₂×#103	1	2	yes	—
82	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
83	Dic₄₇	—	8	8	no	—
84	C₂³×C₁₆	D₄₇	1	1	yes	characteristic
85	C₂×#427	—	47	2	no	—
86	#1380	—	47	4	no	—
87	C₂×C₄₇	—	2	4	no	—
88	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
89	C₂×C₄₇	#2138	1	4	yes	—
90	C₂×C₄₇	#2158	1	2	yes	—
91	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
92	C₂×#117	—	94	2	no	—
93	C₂²×C₁₆	—	2	4	no	—
94	C₂²×C₁₆	Dic₄₇	1	4	yes	complemented
95	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
96	#427	—	47	8	no	—
97	#427	—	94	4	no	—
98	C₂×#107	—	47	4	no	—
99	C₂×#106	—	94	2	no	—
100	C₄₇	#12556	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
102	C₂²×C₈	Dic₉₄	1	2	yes	—
103	C₂²×C₈	C₄×D₄₇	1	4	yes	—
104	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
106	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
107	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
108	#117	—	94	8	no	—
109	#117	—	188	4	no	—
110	C₂×C₄⋊C₄	—	94	4	no	—
111	C₂×C₂²⋊C₄	—	188	2	no	—
112	C₂×C₄²	—	188	2	no	—
113	C₂×C₁₆	—	2	8	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
116	#107	—	47	8	no	—
117	#107	—	94	4	no	—
118	#106	—	94	8	no	—
119	C₂⁴	D₃₇₆	1	1	yes	characteristic
120	C₂²×C₄	—	188	4	no	—
121	C₂²×C₄	#10625325	1	4	yes	—
122	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
123	C₂²×C₄	Dic₁₈₈	1	2	yes	—
124	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
125	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
126	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
127	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
128	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
129	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
130	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
131	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
132	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
134	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
135	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
136	C₄⋊C₄	—	94	8	no	—
137	C₄⋊C₄	—	188	4	no	—
138	C₂²⋊C₄	—	188	8	no	—
139	C₄²	—	188	8	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	—	2	4	no	—
142	C₂³	#13093946	1	4	yes	—
143	C₂³	#13093958	1	2	yes	—
144	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
145	C₂×C₄	—	376	8	no	—
146	C₂×C₄	—	188	8	no	—
147	C₂×C₄	C₄₇⋊#95	1	4	yes	—
148	C₂×C₄	#13093946	1	4	yes	—
149	C₂×C₄	#13093949	1	8	yes	—
150	C₂×C₄	#13093956	1	4	yes	—
151	C₂×C₄	#13093958	1	2	yes	—
152	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
154	C₂×C₄	C₂×#10625325	1	2	yes	—
155	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
156	C₈	C₄₇⋊#95	1	4	yes	—
157	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
158	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
159	C₈	C₂×Dic₄₇⋊C₄	1	2	yes	—
160	C₂²	—	2	8	no	—
161	C₂²	—	2	4	no	—
162	C₂²	#15688886	1	4	yes	—
163	C₂²	C₂×#13093946	1	2	yes	—
164	C₂²	C₂×#13093958	1	1	yes	characteristic
165	C₂²	#15689555	1	4	yes	—
166	C₄	—	376	8	no	—
167	C₄	#15688886	1	4	yes	—
168	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
169	C₄	C₂×#13093949	1	2	yes	—
170	C₄	C₂×#13093956	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	C₂×#15688886	1	1	yes	characteristic
173	C₂	#20245096	1	4	yes	—
174	C₂	#20245172	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₃₇₆
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂³×C₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂³×C₁₆	D₄₇
hypocenter	C₄₇	#12556
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20241480	C₂	semidirect
6. #15689161	C₄	semidirect
11. C₂²×C₁₆×C₄₇	C₄	semidirect
94. C₂²×C₁₆	Dic₄₇	semidirect
100. C₄₇	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
#20241480	2
C₂³×C₁₆×C₄₇	2
#12556	47