Subgroups — #26103141

Order 12032.

262 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal
2	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
4	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
5	C₄⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₈²×C₄₇	—	2	2	no	—
10	C₂×C₄₇⋊C₁₆	—	2	4	no	—
11	C₂×C₄₇⋊C₁₆	—	4	2	no	—
12	C₂×C₄₇⋊C₁₆	—	4	2	no	—
13	C₂²×C₈×C₄₇	—	2	1	no	—
14	C₂²×C₈×C₄₇	C₈	1	2	yes	—
15	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₄₇	—	2	1	no	—
19	C₄×C₈×C₄₇	—	2	1	no	—
20	C₄×C₈×C₄₇	—	2	1	no	—
21	C₄×C₈×C₄₇	C₈	1	2	yes	—
22	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₇	D₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₄₇	—	2	4	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₂×C₈×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	C₁₆	1	4	yes	complemented
34	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
38	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
39	C₄²×C₄₇	C₄²	1	1	yes	characteristic
40	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
43	C₄₇⋊C₁₆	—	4	4	no	—
44	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
58	C₈×C₄₇	—	2	4	no	—
59	C₈×C₄₇	—	2	2	no	—
60	C₈×C₄₇	—	2	4	no	—
61	C₈×C₄₇	#102	1	2	yes	—
62	C₈×C₄₇	#103	1	2	yes	—
63	#7151	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	#334	1	1	yes	characteristic
65	C₂²×C₄₇	#335	1	1	yes	characteristic
66	C₂²×C₄₇	#336	1	1	yes	characteristic
67	C₂²×C₄₇	C₄⋊C₁₆	1	2	yes	—
68	C₂²×C₄₇	#364	1	2	yes	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	#334	1	1	yes	characteristic
72	C₄×C₄₇	#335	1	1	yes	characteristic
73	C₄×C₄₇	#340	1	1	yes	characteristic
74	C₄×C₄₇	#341	1	1	yes	characteristic
75	C₄×C₄₇	C₄×C₁₆	1	1	yes	characteristic
76	C₄×C₄₇	#346	1	1	yes	characteristic
77	C₄×C₄₇	C₂²⋊C₁₆	1	2	yes	—
78	C₂×C₄⋊C₁₆	—	47	2	no	—
79	C₂×C₈²	D₄₇	1	1	yes	characteristic
80	C₂×C₄₇	#1277	1	1	yes	characteristic
81	C₂×C₄₇	#1367	1	2	yes	—
82	C₂×C₄₇	#1370	1	2	yes	—
83	C₂×C₄₇	#1374	1	1	yes	characteristic
84	C₂×C₄₇	#1376	1	1	yes	characteristic
85	C₂²×C₁₆	—	94	2	no	—
86	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
87	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	C₄⋊C₁₆	—	47	4	no	—
90	C₄⋊C₁₆	—	94	2	no	—
91	C₈²	—	2	2	no	—
92	C₄₇	#7151	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	C₄₇⋊C₈	1	2	yes	—
95	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
96	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
98	C₂×C₁₆	—	94	4	no	—
99	C₂×C₁₆	—	188	2	no	—
100	C₂×C₁₆	—	188	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
105	C₄×C₈	C₄×D₄₇	1	2	yes	—
106	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
107	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	#10625329	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	C₄₇⋊C₁₆	1	4	yes	complemented
117	C₂×C₈	#10625325	1	2	yes	—
118	C₂×C₈	D₃₇₆	1	1	yes	characteristic
119	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
120	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
121	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
122	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
123	C₄²	#10625329	1	1	yes	characteristic
124	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
125	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
126	C₁₆	—	188	4	no	—
127	C₂³	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
130	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
131	C₂×C₄	#13093935	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
134	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
135	C₂×C₄	#13093956	1	1	yes	characteristic
136	C₂×C₄	#13093958	1	1	yes	characteristic
137	C₂×C₄	#13093960	1	1	yes	characteristic
138	C₂×C₄	#13093960	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
140	C₂×C₄	#13093964	1	2	yes	—
141	C₈	—	2	4	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	4	no	—
144	C₈	#13093946	1	2	yes	—
145	C₈	#13093949	1	2	yes	—
146	C₂²	#15688874	1	1	yes	characteristic
147	C₂²	#15688876	1	1	yes	characteristic
148	C₂²	#15688878	1	1	yes	characteristic
149	C₂²	C₄⋊(C₄₇⋊C₁₆)	1	2	yes	—
150	C₂²	#15688944	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#15688874	1	1	yes	characteristic
154	C₄	#15688876	1	1	yes	characteristic
155	C₄	#15688886	1	1	yes	characteristic
156	C₄	#15688888	1	1	yes	characteristic
157	C₄	C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
158	C₄	#15688899	1	1	yes	characteristic
159	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	2	yes	—
160	C₂	#20241223	1	1	yes	characteristic
161	C₂	#20241452	1	2	yes	—
162	C₂	#20241458	1	2	yes	—
163	C₂	#20241466	1	1	yes	characteristic
164	C₂	#20241470	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₁₈₈
commutator	C₄×C₄₇	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#7151
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
33. C₂×C₈×C₄₇	C₁₆	semidirect
92. C₄₇	#7151	semidirect
116. C₂×C₈	C₄₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×C₈²×C₄₇	2
#7151	47