Subgroups — #26103450

Order 12032.

238 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688944	C₂	1	1	yes	maximal, characteristic
2	#20243545	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	—	2	1	no	—
5	C₂×#13093956	—	2	1	no	—
6	C₂×#13093956	—	2	1	no	—
7	#15688993	C₄	1	2	yes	—
8	#15688944	—	2	1	no	—
9	#15688944	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₄₇	—	2	1	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#364	—	2	1	no	—
13	C₄₇×#364	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₄₇	—	2	2	no	—
15	C₂×C₄⋊Dic₄₇	—	4	2	no	—
16	#13093964	—	2	2	no	—
17	#13093964	—	4	1	no	—
18	C₂×C₄₇⋊C₁₆	—	4	1	no	—
19	#13093956	—	4	1	no	—
20	#13093956	—	4	1	no	—
21	#13093956	—	4	1	no	—
22	#13093956	—	4	1	no	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₄₇	—	2	2	no	—
27	M₃₂×C₄₇	—	4	1	no	—
28	C₂×C₁₆×C₄₇	—	4	1	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	—	2	1	no	—
34	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
41	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
42	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
47	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
48	C₄²×C₄₇	—	2	1	no	—
49	C₄²×C₄₇	C₄²	1	1	yes	characteristic
50	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄⋊Dic₄₇	—	4	2	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	C₄⋊Dic₄₇	—	8	4	no	—
54	C₁₆×C₄₇	—	4	2	no	—
55	C₄₇⋊C₁₆	—	4	2	no	—
56	C₂³×C₄₇	#100	1	1	yes	characteristic
57	C₂×C₄×C₄₇	—	2	4	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
66	C₂×Dic₄₇	—	8	4	no	—
67	C₂×Dic₄₇	—	8	2	no	—
68	C₈×C₄₇	#102	1	1	yes	characteristic
69	C₈×C₄₇	#102	1	1	yes	characteristic
70	C₈×C₄₇	#103	1	1	yes	characteristic
71	C₈×C₄₇	#103	1	1	yes	characteristic
72	C₈×C₄₇	#106	1	1	yes	characteristic
73	C₈×C₄₇	#106	1	1	yes	characteristic
74	C₈×C₄₇	#107	1	1	yes	characteristic
75	C₈×C₄₇	#107	1	1	yes	characteristic
76	#7268	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	2	no	—
78	C₂²×C₄₇	#331	1	1	yes	characteristic
79	C₂²×C₄₇	#333	1	1	yes	characteristic
80	C₂²×C₄₇	#343	1	1	yes	characteristic
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	#332	1	1	yes	characteristic
84	C₄×C₄₇	#332	1	1	yes	characteristic
85	C₄×C₄₇	#340	1	1	yes	characteristic
86	C₄×C₄₇	#340	1	1	yes	characteristic
87	Dic₄₇	—	16	2	no	—
88	C₂×#364	—	47	1	no	—
89	C₂×#364	D₄₇	1	1	yes	characteristic
90	#1844	—	47	1	no	—
91	C₂×C₄₇	—	2	1	no	—
92	C₂×C₄₇	—	2	1	no	—
93	C₂×C₄₇	#1296	1	1	yes	characteristic
94	C₂×C₄₇	#1359	1	2	yes	—
95	C₂×M₃₂	—	2	1	no	—
96	C₂×M₃₂	—	94	1	no	—
97	C₂×#106	—	94	1	no	—
98	C₂×#106	—	94	1	no	—
99	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
100	#384	—	47	2	no	—
101	#364	—	2	1	no	—
102	#364	—	47	2	no	—
103	#364	—	94	1	no	—
104	#364	Dic₄₇	1	2	yes	complemented
105	C₄₇	#7268	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
107	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	94	2	no	—
109	C₂×C₄⋊C₄	—	188	2	no	—
110	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
111	M₃₂	—	2	2	no	—
112	M₃₂	—	4	1	no	—
113	M₃₂	—	94	2	no	—
114	M₃₂	—	188	1	no	—
115	C₂×C₁₆	—	4	1	no	—
116	C₂×C₁₆	—	188	1	no	—
117	#106	—	188	1	no	—
118	#106	—	188	1	no	—
119	#106	—	188	1	no	—
120	#106	—	188	1	no	—
121	C₄×C₈	Dic₉₄	1	1	yes	characteristic
122	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
123	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	188	2	no	—
127	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
128	C₂×C₈	#10625325	1	1	yes	characteristic
129	C₂×C₈	#10625325	1	1	yes	characteristic
130	C₂×C₈	D₃₇₆	1	1	yes	characteristic
131	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
132	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
134	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
135	C₂×C₈	#10625334	1	1	yes	characteristic
136	C₂×C₈	#10625335	1	1	yes	characteristic
137	C₂×C₈	#10625336	1	1	yes	characteristic
138	C₂×C₈	#10625337	1	1	yes	characteristic
139	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
140	C₄⋊C₄	—	376	4	no	—
141	C₄⋊C₄	—	188	2	no	—
142	C₄⋊C₄	—	188	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
145	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
146	C₁₆	—	4	2	no	—
147	C₁₆	—	188	2	no	—
148	C₂³	#13093940	1	1	yes	characteristic
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	376	4	no	—
153	C₂×C₄	—	376	2	no	—
154	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
155	C₂×C₄	#13093942	1	1	yes	characteristic
156	C₂×C₄	#13093944	1	1	yes	characteristic
157	C₂×C₄	#13093946	1	1	yes	characteristic
158	C₂×C₄	#13093947	1	1	yes	characteristic
159	C₂×C₄	#13093949	1	1	yes	characteristic
160	C₈	#13093945	1	1	yes	characteristic
161	C₈	#13093946	1	1	yes	characteristic
162	C₈	#13093948	1	1	yes	characteristic
163	C₈	#13093949	1	1	yes	characteristic
164	C₈	#13093955	1	1	yes	characteristic
165	C₈	#13093956	1	1	yes	characteristic
166	C₈	#13093957	1	1	yes	characteristic
167	C₈	#13093958	1	1	yes	characteristic
168	C₂²	—	2	2	no	—
169	C₂²	#15688866	1	1	yes	characteristic
170	C₂²	#15688871	1	1	yes	characteristic
171	C₂²	#15688891	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	752	2	no	—
175	C₄	#15688868	1	1	yes	characteristic
176	C₄	#15688870	1	1	yes	characteristic
177	C₄	#15688885	1	1	yes	characteristic
178	C₄	#15688886	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#20241272	1	1	yes	characteristic
182	C₂	#20241428	1	2	yes	—

	subgroup	G/H
center	C₂²	#15688866
commutator	C₄²×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	#331
prime radical	C₂³×C₄₇	#100
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	#7268
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #15688944	C₄	semidirect
13. C₄₇×#364	C₄	semidirect
104. #364	Dic₄₇	semidirect
105. C₄₇	#7268	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688944	2
#20243545	2
C₂×C₄₇×#364	2
#7268	47