Subgroups — C₁₁₃×#7317 #43275337

Order 28928.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#1784	C₂	1	3	yes	maximal
2	C₂³×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
3	C₂×#101×C₁₁₃	—	2	6	no	—
4	#407×C₁₁₃	C₄	1	6	yes	—
5	C₂²×Q₈×C₁₁₃	—	4	6	no	—
6	C₂²×Q₈×C₁₁₃	D₄	1	3	yes	—
7	C₂²×Q₈×C₁₁₃	Q₈	1	1	yes	characteristic
8	C₂³×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
9	C₂×M₁₆×C₁₁₃	—	2	6	no	—
10	#101×C₁₁₃	—	4	12	no	—
11	C₂²⋊C₈×C₁₁₃	—	4	6	no	—
12	C₂⁴×C₁₁₃	C₄²	1	1	yes	characteristic
13	C₂×Q₈×C₁₁₃	—	2	12	no	—
14	C₂×Q₈×C₁₁₃	—	4	3	no	—
15	C₂×Q₈×C₁₁₃	—	4	3	no	—
16	C₂×Q₈×C₁₁₃	—	4	16	no	—
17	C₂²×C₄×C₁₁₃	—	2	6	no	—
18	C₂²×C₄×C₁₁₃	—	4	6	no	—
19	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	3	yes	—
20	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	3	yes	—
21	M₁₆×C₁₁₃	—	4	12	no	—
22	C₂×C₈×C₁₁₃	—	4	6	no	—
23	C₂³×C₁₁₃	—	2	3	no	—
24	C₂³×C₁₁₃	—	4	2	no	—
25	C₂³×C₁₁₃	#95	1	1	yes	characteristic
26	Q₈×C₁₁₃	—	4	12	no	—
27	Q₈×C₁₁₃	—	4	4	no	—
28	C₂×C₄×C₁₁₃	—	2	6	no	—
29	C₂×C₄×C₁₁₃	—	2	3	no	—
30	C₂×C₄×C₁₁₃	—	2	6	no	—
31	C₂×C₄×C₁₁₃	—	4	12	no	—
32	C₂×C₄×C₁₁₃	#99	1	6	yes	—
33	C₈×C₁₁₃	—	8	6	no	—
34	C₂²×C₁₁₃	—	2	6	no	—
35	C₂²×C₁₁₃	—	4	1	no	—
36	C₂²×C₁₁₃	—	4	3	no	—
37	C₂²×C₁₁₃	—	4	1	no	—
38	C₂²×C₁₁₃	#342	1	3	yes	—
39	C₄×C₁₁₃	—	2	6	no	—
40	C₄×C₁₁₃	—	4	3	no	—
41	#7317	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₁₁₃	—	2	3	no	—
43	C₂×C₁₁₃	—	4	2	no	—
44	C₂×C₁₁₃	#1304	1	1	yes	characteristic
45	#1784	C₂×C₁₁₃	1	3	yes	—
46	C₁₁₃	#7317	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
48	C₂×#101	—	2	6	no	—
49	#407	C₄×C₁₁₃	1	6	yes	—
50	C₂²×Q₈	—	4	6	no	—
51	C₂²×Q₈	D₄×C₁₁₃	1	3	yes	—
52	C₂²×Q₈	Q₈×C₁₁₃	1	1	yes	characteristic
53	C₂³×C₄	C₂×C₄×C₁₁₃	1	3	yes	—
54	C₂×M₁₆	—	2	6	no	—
55	#101	—	4	12	no	—
56	C₂²⋊C₈	—	4	6	no	—
57	C₂⁴	C₄²×C₁₁₃	1	1	yes	characteristic
58	C₂×Q₈	—	2	12	no	—
59	C₂×Q₈	—	4	3	no	—
60	C₂×Q₈	—	4	3	no	—
61	C₂×Q₈	—	4	16	no	—
62	C₂²×C₄	—	2	6	no	—
63	C₂²×C₄	—	4	6	no	—
64	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	3	yes	—
65	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	3	yes	—
66	M₁₆	—	4	12	no	—
67	C₂×C₈	—	4	6	no	—
68	C₂³	—	2	3	no	—
69	C₂³	—	4	2	no	—
70	C₂³	#95×C₁₁₃	1	1	yes	characteristic
71	Q₈	—	4	12	no	—
72	Q₈	—	4	4	no	—
73	C₂×C₄	—	2	6	no	—
74	C₂×C₄	—	2	3	no	—
75	C₂×C₄	—	2	6	no	—
76	C₂×C₄	—	4	12	no	—
77	C₂×C₄	#99×C₁₁₃	1	6	yes	—
78	C₈	—	8	6	no	—
79	C₂²	—	2	6	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	4	3	no	—
82	C₂²	—	4	1	no	—
83	C₂²	#342×C₁₁₃	1	3	yes	—
84	C₄	—	2	6	no	—
85	C₄	—	4	3	no	—
86	C₂	—	2	3	no	—
87	C₂	—	4	2	no	—
88	C₂	C₁₁₃×#1304	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₁₃	#1304
commutator	C₂⁴	C₄²×C₁₁₃
Frattini, cosocle	C₂³×Q₈	C₂²×C₁₁₃
prime radical	C₂⁴×C₁₁₃	C₄²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁₃ ◁ C₂³×C₁₁₃ ◁ C₂³×Q₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
41. #7317	C₁₁₃	direct (N × H)
46. C₁₁₃	#7317	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁₃×#1784	2
#7317	113

Subgroups — C113×#7317 #43275337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#7317 #43275337