Subgroups — #26106717

Order 12032.

354 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243074	C₂	1	1	yes	maximal, characteristic
2	#20241671	C₂	1	1	yes	maximal, characteristic
3	#20241218	C₂	1	2	yes	maximal
4	#20241216	C₂	1	2	yes	maximal
5	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093935	C₂²	1	1	yes	characteristic
8	C₂×#13093934	C₄	1	2	yes	—
9	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
10	#15688969	C₂²	1	1	yes	characteristic
11	#15688844	C₄	1	2	yes	complemented
12	C₈×C₄₇⋊C₈	C₄	1	2	yes	complemented
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#322	C₄	1	2	yes	complemented
16	C₈²×C₄₇	C₄	1	2	yes	complemented
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
21	#13093935	—	2	2	no	—
22	#13093934	—	2	4	no	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₈×Dic₄₇	C₈	1	8	yes	complemented
25	C₄₇⋊#95	—	2	2	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₂²⋊C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	#41	1	2	yes	—
38	C₂²×Dic₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	4	no	—
40	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
46	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₄₇	#41	1	2	yes	—
49	C₄×Dic₄₇	—	2	2	no	—
50	C₄×Dic₄₇	C₂×C₈	1	4	yes	—
51	C₂×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₂×C₄₇⋊C₈	—	2	4	no	—
54	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
55	C₂³×C₄₇	#117	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	4	2	no	—
57	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#117	1	2	yes	—
59	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#117	1	2	yes	—
61	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	4	4	no	—
65	C₈×C₄₇	—	4	4	no	—
66	C₈×C₄₇	C₄×C₈	1	2	yes	—
67	C₈×C₄₇	C₄×C₈	1	2	yes	—
68	C₈×C₄₇	#97	1	2	yes	—
69	C₈×C₄₇	#97	1	2	yes	—
70	C₄₇⋊C₈	—	2	4	no	—
71	C₄₇⋊C₈	—	4	4	no	—
72	#8137	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	#376	1	1	yes	characteristic
75	C₂²×C₄₇	#431	1	2	yes	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
79	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
80	C₄×C₄₇	#431	1	2	yes	—
81	Dic₄₇	—	4	4	no	—
82	#1726	—	47	1	no	—
83	#1450	—	47	1	no	—
84	#1450	D₄₇	1	1	yes	characteristic
85	#1275	—	47	2	no	—
86	#1274	—	47	2	no	—
87	C₂×C₄₇	—	2	2	no	—
88	C₂×C₄₇	#1744	1	1	yes	characteristic
89	C₂×C₄₇	#1747	1	2	yes	—
90	#431	—	47	2	no	—
91	#431	C₂×D₄₇	1	2	yes	—
92	C₂×#97	—	47	2	no	—
93	C₂×#97	—	47	1	no	—
94	C₂×C₄×C₈	—	47	2	no	—
95	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
96	#376	—	47	1	no	—
97	#322	—	47	2	no	—
98	#322	Dic₄₇	1	2	yes	complemented
99	C₈²	—	47	2	no	—
100	C₈²	Dic₄₇	1	2	yes	complemented
101	C₄₇	#8137	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	47	2	no	—
103	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
105	C₂×C₄²	—	47	2	no	—
106	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	94	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	94	2	no	—
111	#97	—	94	4	no	—
112	#97	—	94	2	no	—
113	C₄×C₈	—	47	8	no	—
114	C₄×C₈	—	47	4	no	—
115	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
116	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
118	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
120	#95	—	94	2	no	—
121	C₂²×C₄	—	94	2	no	—
122	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
123	C₂²×C₄	#10625350	1	2	yes	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	47	4	no	—
126	C₂×C₈	—	94	2	no	—
127	C₂×C₈	—	94	2	no	—
128	C₂×C₈	—	94	4	no	—
129	C₂×C₈	C₈×D₄₇	1	4	yes	—
130	C₂×C₈	#10625324	1	4	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	C₄²	—	47	4	no	—
136	C₄²	—	94	2	no	—
137	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₄²	#10625350	1	2	yes	—
139	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
140	C₂³	#13093981	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	94	4	no	—
143	C₂×C₄	—	188	4	no	—
144	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
145	C₂×C₄	#13093981	1	2	yes	—
146	C₂×C₄	#13093981	1	1	yes	characteristic
147	C₂×C₄	#13093983	1	2	yes	—
148	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
149	C₂×C₄	C₂×#10625324	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	94	4	no	—
152	C₈	—	188	4	no	—
153	C₈	C₈×Dic₄₇	1	2	yes	—
154	C₈	C₈×Dic₄₇	1	2	yes	—
155	C₈	#13093934	1	2	yes	—
156	C₈	#13093934	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#15688969	1	1	yes	characteristic
159	C₂²	#15689174	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	188	4	no	—
163	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
164	C₄	C₂×#13093934	1	1	yes	characteristic
165	C₄	#15689174	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#20243133	1	1	yes	characteristic
168	C₂	#20243143	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	#376
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#8137
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
11. #15688844	C₄	semidirect
12. C₈×C₄₇⋊C₈	C₄	semidirect
15. C₄₇×#322	C₄	semidirect
16. C₈²×C₄₇	C₄	semidirect
24. C₈×Dic₄₇	C₈	semidirect
98. #322	Dic₄₇	semidirect
100. C₈²	Dic₄₇	semidirect
101. C₄₇	#8137	semidirect

Maximal subgroups

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

subgroup	index
#20243074	2
#20241671	2
#20241218	2
#20241216	2
C₄₇×#1450	2
#8137	47