Subgroups — C₁₁₃×#1780 #28386576

Order 14464.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₈×C₁₁₃	C₂	1	6	yes	maximal, complemented
3	C₂×#41×C₁₁₃	—	2	12	no	—
4	C₂²×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
5	C₂²×D₄×C₁₁₃	C₄	1	3	yes	—
6	C₂³×C₄×C₁₁₃	C₂²	1	3	yes	—
7	C₂²×C₈×C₁₁₃	C₂²	1	4	yes	—
8	C₂²⋊C₈×C₁₁₃	—	2	12	no	—
9	C₂⁴×C₁₁₃	C₂×C₄	1	3	yes	—
10	#41×C₁₁₃	—	2	32	no	—
11	C₂×Q₈×C₁₁₃	—	2	6	no	—
12	C₂×D₄×C₁₁₃	—	2	12	no	—
13	C₂×D₄×C₁₁₃	—	2	6	no	—
14	C₂²×C₄×C₁₁₃	—	2	12	no	—
15	C₂²×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
16	C₂²×C₄×C₁₁₃	D₄	1	12	yes	—
17	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₁₃	—	2	12	no	—
19	C₂³×C₁₁₃	—	2	6	no	—
20	C₂³×C₁₁₃	—	2	12	no	—
21	C₂³×C₁₁₃	C₂²⋊C₄	1	6	yes	—
22	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
23	Q₈×C₁₁₃	—	2	8	no	—
24	D₄×C₁₁₃	—	2	24	no	—
25	C₂×C₄×C₁₁₃	—	2	6	no	—
26	C₂×C₄×C₁₁₃	—	2	24	no	—
27	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	6	yes	—
28	C₂×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
29	C₈×C₁₁₃	—	4	4	no	—
30	C₂²×C₁₁₃	—	2	6	no	—
31	C₂²×C₁₁₃	—	2	12	no	—
32	C₂²×C₁₁₃	—	4	12	no	—
33	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	3	yes	—
34	C₂²×C₁₁₃	#131	1	4	yes	—
35	C₄×C₁₁₃	—	2	6	no	—
36	C₄×C₁₁₃	#120	1	4	yes	—
37	C₂×C₁₁₃	—	4	6	no	—
38	C₂×C₁₁₃	#379	1	1	yes	characteristic
39	C₂×C₁₁₃	#408	1	6	yes	—
40	#1780	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₁₁₃	#1780	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×#41	C₂×C₁₁₃	1	1	yes	characteristic
43	C₂×C₂²⋊C₈	C₂×C₁₁₃	1	6	yes	complemented
44	C₂×#41	—	2	12	no	—
45	C₂²×Q₈	C₄×C₁₁₃	1	1	yes	characteristic
46	C₂²×D₄	C₄×C₁₁₃	1	3	yes	—
47	C₂³×C₄	C₂²×C₁₁₃	1	3	yes	—
48	C₂²×C₈	C₂²×C₁₁₃	1	4	yes	—
49	C₂²⋊C₈	—	2	12	no	—
50	C₂⁴	C₂×C₄×C₁₁₃	1	3	yes	—
51	#41	—	2	32	no	—
52	C₂×Q₈	—	2	6	no	—
53	C₂×D₄	—	2	12	no	—
54	C₂×D₄	—	2	6	no	—
55	C₂²×C₄	—	2	12	no	—
56	C₂²×C₄	C₂×C₄×C₁₁₃	1	3	yes	—
57	C₂²×C₄	D₄×C₁₁₃	1	12	yes	—
58	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
59	C₂×C₈	—	2	12	no	—
60	C₂³	—	2	6	no	—
61	C₂³	—	2	12	no	—
62	C₂³	C₂²⋊C₄×C₁₁₃	1	6	yes	—
63	C₂³	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
64	Q₈	—	2	8	no	—
65	D₄	—	2	24	no	—
66	C₂×C₄	—	2	6	no	—
67	C₂×C₄	—	2	24	no	—
68	C₂×C₄	C₂²⋊C₄×C₁₁₃	1	6	yes	—
69	C₂×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
70	C₈	—	4	4	no	—
71	C₂²	—	2	6	no	—
72	C₂²	—	2	12	no	—
73	C₂²	—	4	12	no	—
74	C₂²	C₂×C₂²⋊C₄×C₁₁₃	1	3	yes	—
75	C₂²	#131×C₁₁₃	1	4	yes	—
76	C₄	—	2	6	no	—
77	C₄	#120×C₁₁₃	1	4	yes	—
78	C₂	—	4	6	no	—
79	C₂	#379×C₁₁₃	1	1	yes	characteristic
80	C₂	#408×C₁₁₃	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₂³
commutator	C₂³	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₁₃
socle	C₂³×C₁₁₃	C₂²×C₄
prime radical	C₂²×#41×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊C₈×C₁₁₃	C₂	semidirect
40. #1780	C₁₁₃	direct (N × H)
41. C₁₁₃	#1780	direct (N × H)
43. C₂×C₂²⋊C₈	C₂×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₁₁₃	2
C₂×C₂²⋊C₈×C₁₁₃	2
#1780	113

Subgroups — C113×#1780 #28386576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#1780 #28386576