Subgroups — #28392604

Order 14464.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298780	C₂	1	3	yes	maximal
2	#20298397	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813670	C₂²	1	1	yes	characteristic
5	#16813653	—	2	3	no	—
6	#16813550	—	2	3	no	—
7	#16813541	C₂²	1	3	yes	—
8	C₄⋊Q₈×C₁₁₃	—	2	3	no	—
9	C₄⋊Q₈×C₁₁₃	C₂²	1	3	yes	—
10	C₄×Q₈×C₁₁₃	—	2	3	no	—
11	#14268648	—	4	1	no	—
12	#14268647	—	2	3	no	—
13	Q₈×D₁₁₃	—	4	1	no	—
14	#14268637	—	4	3	no	—
15	#14268634	—	4	3	no	—
16	#14268631	—	2	3	no	—
17	C₂×Q₈×C₁₁₃	—	2	1	no	—
18	C₂×Q₈×C₁₁₃	—	2	1	no	—
19	C₂×Q₈×C₁₁₃	D₄	1	3	yes	—
20	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₁₃	—	2	3	no	—
22	C₄⋊C₄×C₁₁₃	—	2	6	no	—
23	C₄²×C₁₁₃	—	2	3	no	—
24	C₄²×C₁₁₃	D₄	1	3	yes	—
25	Q₈×C₁₁₃	—	2	4	no	—
26	Q₈×C₁₁₃	—	4	1	no	—
27	Q₈×C₁₁₃	—	4	3	no	—
28	Q₈×C₁₁₃	—	8	1	no	—
29	C₂×C₄×C₁₁₃	—	2	3	no	—
30	C₂×C₄×C₁₁₃	—	2	3	no	—
31	C₂×C₄×C₁₁₃	C₂×D₄	1	3	yes	—
32	C₂²⋊D₁₁₃	—	4	1	no	—
33	D₄₅₂	—	2	3	no	—
34	C₄×D₁₁₃	—	4	3	no	—
35	Dic₂₂₆	—	2	3	no	—
36	#11744751	—	4	3	no	—
37	C₂²×C₁₁₃	#120	1	1	yes	characteristic
38	C₂×D₁₁₃	—	4	1	no	—
39	C₄×C₁₁₃	—	2	6	no	—
40	C₄×C₁₁₃	—	2	3	no	—
41	C₄×C₁₁₃	—	4	3	no	—
42	Dic₁₁₃	—	4	1	no	—
43	C₂×C₁₁₃	—	2	1	no	—
44	C₂×C₁₁₃	B(4,𝔽₂)	1	1	yes	characteristic
45	D₁₁₃	—	8	1	no	—
46	#2205	—	113	1	no	maximal, Sylow
47	C₁₁₃	#2205	1	1	yes	characteristic, complemented, Sylow
48	Q₈²	D₁₁₃	1	1	yes	characteristic, complemented
49	#456	—	113	3	no	—
50	#356	—	113	3	no	—
51	2₋¹⁺⁴	—	113	1	no	—
52	#137	—	226	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂×D₁₁₃	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	226	3	no	—
57	#101	—	113	3	no	—
58	#41	—	452	1	no	—
59	#41	—	226	3	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	452	1	no	—
63	C₂×Q₈	C₂²⋊D₁₁₃	1	3	yes	—
64	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
65	Q₁₆	—	452	3	no	—
66	SD₁₆	—	452	3	no	—
67	M₁₆	—	226	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₁₁₃	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	3	no	—
75	Q₈	—	8	1	no	—
76	Q₈	—	226	3	no	—
77	D₄	—	452	1	no	—
78	D₄	—	226	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	452	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	3	yes	—
83	C₈	—	452	3	no	—
84	C₂²	—	452	1	no	—
85	C₂²	#16813591	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	452	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	904	1	no	—
92	C₂	#20298785	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20298785
commutator, cosocle	C₂×Q₈×C₁₁₃	C₂³
Frattini	C₂×Q₈	C₂²×D₁₁₃
Fitting	Q₈²×C₁₁₃	C₂
socle	C₂×C₁₁₃	B(4,𝔽₂)
prime radical	#16813670	C₂²
hypercenter	Q₈²	D₁₁₃
hypocenter	C₁₁₃	#2205
LCS	G ▷ C₂×Q₈×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #20298397	C₂	semidirect
3. Q₈²×C₁₁₃	C₂	semidirect
47. C₁₁₃	#2205	semidirect
48. Q₈²	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20298780	2
#20298397	2
Q₈²×C₁₁₃	2
#2205	113