Subgroups — #26109971

Order 12032.

294 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	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20243552	C₂	1	1	yes	maximal, characteristic
3	#20241247	C₂	1	4	yes	maximal
4	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093960	—	2	1	no	—
6	C₂×#13093960	—	2	1	no	—
7	C₂×#13093958	—	2	1	no	—
8	C₂×#13093956	—	2	1	no	—
9	#15689147	C₂²	1	2	yes	—
10	#15688989	C₂²	1	2	yes	—
11	C₄₇×#431	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#322	C₄	1	2	yes	complemented
14	C₈²×C₄₇	C₄	1	2	yes	complemented
15	C₂×#10625329	—	2	2	no	—
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	#13093960	—	4	2	no	—
18	#13093960	—	4	2	no	—
19	#13093958	—	4	2	no	—
20	#13093956	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₄₇⋊#95	—	2	4	no	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₄₇	—	2	2	no	—
27	C₂²⋊C₈×C₄₇	—	2	2	no	—
28	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	#41	1	2	yes	—
35	C₂²×Dic₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	—	2	4	no	—
37	C₂×C₈×C₄₇	C₄⋊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	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	D₈	1	2	yes	—
42	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
43	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
44	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
45	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₄₇	#41	1	2	yes	—
48	C₄⋊Dic₄₇	—	4	2	no	—
49	C₄⋊Dic₄₇	—	4	2	no	—
50	#10625329	—	4	2	no	—
51	#10625329	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂³×C₄₇	#124	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	4	2	no	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#117	1	2	yes	—
57	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#125	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₄₇	—	8	4	no	—
64	C₂×Dic₄₇	—	8	2	no	—
65	C₈×C₄₇	—	4	4	no	—
66	C₈×C₄₇	#106	1	2	yes	—
67	C₈×C₄₇	#106	1	2	yes	—
68	C₈×C₄₇	#107	1	2	yes	—
69	C₈×C₄₇	#107	1	2	yes	—
70	C₄₇⋊C₈	—	8	2	no	—
71	#8793	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	#382	1	1	yes	characteristic
74	C₂²×C₄₇	#486	1	1	yes	characteristic
75	C₂²×C₄₇	#487	1	1	yes	characteristic
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
79	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
80	C₄×C₄₇	#486	1	1	yes	characteristic
81	C₄×C₄₇	#487	1	1	yes	characteristic
82	Dic₄₇	—	16	2	no	—
83	#1848	—	47	1	no	—
84	#1846	—	47	1	no	—
85	#1450	D₄₇	1	1	yes	characteristic
86	#1286	—	47	4	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	#1833	1	2	yes	—
90	C₂×C₄₇	#1834	1	1	yes	characteristic
91	#431	C₂×D₄₇	1	2	yes	—
92	C₂×#108	—	94	1	no	—
93	C₂×#108	—	94	1	no	—
94	C₂×#107	—	94	1	no	—
95	C₂×#106	—	94	1	no	—
96	#423	—	47	2	no	—
97	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
98	#382	—	47	2	no	—
99	#322	Dic₄₇	1	2	yes	complemented
100	C₈²	Dic₄₇	1	2	yes	complemented
101	C₄₇	#8793	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	94	2	no	—
103	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
104	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	94	2	no	—
106	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
107	#108	—	188	2	no	—
108	#108	—	188	2	no	—
109	#107	—	188	2	no	—
110	#106	—	188	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	94	4	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₉₄	1	1	yes	characteristic
115	C₄×C₈	Dic₉₄	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
117	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
119	#95	—	94	4	no	—
120	C₂²×C₄	—	188	2	no	—
121	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
122	C₂²×C₄	#10625350	1	2	yes	—
123	M₁₆	—	188	2	no	—
124	M₁₆	—	188	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	188	2	no	—
127	C₂×C₈	#10625325	1	2	yes	—
128	C₂×C₈	#10625325	1	2	yes	—
129	C₂×C₈	D₃₇₆	1	2	yes	—
130	C₂×C₈	Dic₁₈₈	1	2	yes	—
131	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
132	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
135	C₄⋊C₄	—	188	2	no	—
136	C₄⋊C₄	—	188	2	no	—
137	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
138	C₄²	#10625350	1	2	yes	—
139	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
140	C₂³	#13094012	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	376	4	no	—
143	C₂×C₄	—	376	2	no	—
144	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
145	C₂×C₄	#13093983	1	2	yes	—
146	C₂×C₄	#13094012	1	1	yes	characteristic
147	C₂×C₄	#13094016	1	2	yes	—
148	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
149	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
150	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	376	2	no	—
154	C₈	#13093956	1	2	yes	—
155	C₈	#13093956	1	2	yes	—
156	C₈	#13093958	1	2	yes	—
157	C₈	#13093958	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#15688989	1	1	yes	characteristic
160	C₂²	#15689470	1	1	yes	characteristic
161	C₂²	#15689474	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	752	2	no	—
165	C₄	C₂×#13093956	1	1	yes	characteristic
166	C₄	C₂×#13093958	1	1	yes	characteristic
167	C₄	#15689470	1	1	yes	characteristic
168	C₄	#15689474	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#20243512	1	2	yes	—
172	C₂	#20243516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688989
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#8793
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
13. C₄₇×#322	C₄	semidirect
14. C₈²×C₄₇	C₄	semidirect
99. #322	Dic₄₇	semidirect
100. C₈²	Dic₄₇	semidirect
101. C₄₇	#8793	semidirect

Maximal subgroups

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

subgroup	index
#20243560	2
#20243552	2
#20241247	2
C₄₇×#1450	2
#8793	47