Subgroups — #26131168

Order 12032.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	2	yes	maximal
2	#20241482	C₂	1	4	yes	maximal, complemented
3	C₂²×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	—	2	2	no	—
5	C₂×#13093960	—	2	2	no	—
6	C₂×#13093960	—	2	4	no	—
7	C₂×#13093960	C₄	1	8	yes	—
8	C₂×#13093960	C₂²	1	4	yes	—
9	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₄₇	—	2	4	no	—
11	C₂×M₃₂×C₄₇	C₄	1	4	yes	—
12	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
13	C₂×#10625329	—	2	8	no	—
14	C₂×#10625329	—	2	4	no	—
15	C₂×#10625329	—	4	2	no	—
16	C₂×#10625329	—	4	4	no	—
17	C₂²×C₄₇⋊C₈	—	4	2	no	—
18	#13093960	—	2	8	no	—
19	#13093960	—	2	4	no	—
20	#13093960	C₂×C₄	1	8	yes	—
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	M₃₂×C₄₇	—	2	8	no	—
29	C₂×C₁₆×C₄₇	—	2	4	no	—
30	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
31	C₂⁴×C₄₇	Q₁₆	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	2	yes	—
35	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₄²	1	4	yes	—
39	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	#10625329	—	2	8	no	—
48	#10625329	—	4	8	no	—
49	#10625329	—	4	4	no	—
50	C₂×C₄₇⋊C₈	—	4	8	no	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₁₆×C₄₇	—	2	4	no	—
53	C₂³×C₄₇	—	4	2	no	—
54	C₂³×C₄₇	#103	1	4	yes	—
55	C₂³×C₄₇	#107	1	2	yes	—
56	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#95	1	4	yes	—
58	C₂×C₄×C₄₇	#102	1	4	yes	—
59	C₂×C₄×C₄₇	#102	1	4	yes	—
60	C₂×C₄×C₄₇	#103	1	4	yes	—
61	C₂×C₄×C₄₇	#106	1	4	yes	—
62	C₂×C₄×C₄₇	#107	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
67	C₈×C₄₇	#95	1	4	yes	—
68	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₄₇⋊C₈	—	4	8	no	—
72	#12559	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	4	no	—
74	C₂²×C₄₇	—	2	2	no	—
75	C₂²×C₄₇	—	4	4	no	—
76	C₂²×C₄₇	#340	1	4	yes	—
77	C₂²×C₄₇	C₂×#103	1	2	yes	—
78	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
79	C₄×C₄₇	#340	1	4	yes	—
80	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
81	C₄×C₄₇	C₂×#102	1	2	yes	—
82	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
83	C₂²×M₃₂	D₄₇	1	1	yes	characteristic
84	C₂²×#108	—	47	2	no	—
85	#1381	—	47	4	no	—
86	C₂×C₄₇	—	2	4	no	—
87	C₂×C₄₇	—	2	2	no	—
88	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
89	C₂×C₄₇	#2161	1	2	yes	—
90	C₂²×M₁₆	—	94	2	no	—
91	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
92	C₂×M₃₂	—	2	4	no	—
93	C₂×M₃₂	Dic₄₇	1	4	yes	—
94	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
95	C₂×#108	—	47	8	no	—
96	C₂×#108	—	47	4	no	—
97	C₂×#108	—	94	2	no	—
98	C₂×#108	—	94	4	no	—
99	C₄₇	#12559	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
101	C₂×M₁₆	—	94	8	no	—
102	C₂×M₁₆	—	94	4	no	—
103	C₂×M₁₆	—	188	4	no	—
104	C₂×M₁₆	—	188	2	no	—
105	C₂²×C₈	—	188	2	no	—
106	C₂²×C₈	Dic₉₄	1	2	yes	—
107	C₂²×C₈	C₄×D₄₇	1	4	yes	—
108	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
110	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
111	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
112	M₃₂	—	2	8	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
115	#108	—	47	8	no	—
116	#108	—	94	8	no	—
117	#108	—	94	4	no	—
118	C₂⁴	Dic₁₈₈	1	1	yes	characteristic
119	C₂²×C₄	#10625325	1	4	yes	—
120	C₂²×C₄	D₃₇₆	1	2	yes	—
121	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
122	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
123	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
124	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
125	M₁₆	—	94	8	no	—
126	M₁₆	—	188	8	no	—
127	M₁₆	—	188	4	no	—
128	C₂×C₈	—	188	8	no	—
129	C₂×C₈	—	188	4	no	—
130	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
131	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
132	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
133	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
134	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
135	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
137	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	#13093949	1	4	yes	—
142	C₂³	#13093958	1	2	yes	—
143	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
144	C₂×C₄	C₄₇⋊#95	1	4	yes	—
145	C₂×C₄	#13093946	1	4	yes	—
146	C₂×C₄	#13093946	1	4	yes	—
147	C₂×C₄	#13093949	1	4	yes	—
148	C₂×C₄	#13093956	1	4	yes	—
149	C₂×C₄	#13093958	1	2	yes	—
150	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
152	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
153	C₂×C₄	C₂×#10625325	1	2	yes	—
154	C₈	—	188	8	no	—
155	C₈	C₄₇⋊#95	1	4	yes	—
156	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
157	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
158	C₈	C₂×Dic₄₇⋊C₄	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	#15688886	1	4	yes	—
163	C₂²	C₂×#13093949	1	2	yes	—
164	C₂²	C₂×#13093958	1	1	yes	characteristic
165	C₄	#15688886	1	4	yes	—
166	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
167	C₄	C₂×#13093946	1	2	yes	—
168	C₄	C₂×#13093956	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	2	4	no	—
171	C₂	C₂×#15688886	1	1	yes	characteristic
172	C₂	#20245183	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₇₆
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂⁴×C₄₇	Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×M₃₂	D₄₇
hypocenter	C₄₇	#12559
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20241482	C₂	semidirect
99. C₄₇	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093960	2
#20241482	2
C₂²×M₃₂×C₄₇	2
#12559	47