Subgroups — #43252965

Order 28544.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383948	C₂	1	1	yes	maximal, characteristic
2	#28383379	C₂	1	1	yes	maximal, characteristic
3	#356×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297296	C₄	1	1	yes	characteristic
5	#20297228	—	2	1	no	—
6	#20297228	—	2	1	no	—
7	#20297223	—	2	1	no	—
8	#20297215	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	#101×C₂₂₃	—	2	1	no	—
11	#16813154	—	4	1	no	—
12	C₂×C₂₂₃⋊C₈	—	2	1	no	—
13	C₄⋊Dic₂₂₃	—	2	1	no	—
14	Dic₂₂₃⋊C₄	—	4	1	no	—
15	C₂×Q₈×C₂₂₃	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₂₂₃	D₄	1	1	yes	characteristic
17	M₁₆×C₂₂₃	—	4	1	no	—
18	C₄⋊C₄×C₂₂₃	—	4	1	no	—
19	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₂₃	—	4	1	no	—
21	C₂₂₃⋊C₈	—	4	1	no	—
22	C₂₂₃⋊C₈	—	8	1	no	—
23	Q₈×C₂₂₃	—	4	1	no	—
24	Q₈×C₂₂₃	—	4	1	no	—
25	C₂×C₄×C₂₂₃	—	2	1	no	—
26	C₂×C₄×C₂₂₃	—	2	1	no	—
27	C₂×C₄×C₂₂₃	—	2	1	no	—
28	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₂₃	—	8	1	no	—
30	C₂²×C₂₂₃	#99	1	1	yes	characteristic
31	C₄×C₂₂₃	—	2	1	no	—
32	C₄×C₂₂₃	—	4	1	no	—
33	C₄×C₂₂₃	—	4	1	no	—
34	C₄×C₂₂₃	—	4	1	no	—
35	Dic₂₂₃	—	8	1	no	—
36	C₂×C₂₂₃	—	2	1	no	—
37	C₂×C₂₂₃	#354	1	1	yes	characteristic
38	C₂₂₃	#1413	1	1	yes	characteristic, complemented, Sylow
39	#1413	—	223	1	no	maximal, Sylow
40	#491	—	223	1	no	—
41	#356	—	223	1	no	—
42	#356	D₂₂₃	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₂₂₃	1	1	yes	characteristic
44	#125	—	223	1	no	—
45	#103	—	446	1	no	—
46	#103	—	446	1	no	—
47	#101	—	2	1	no	—
48	#101	—	446	1	no	—
49	#97	—	223	1	no	—
50	C₂×Q₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
51	C₂×Q₈	D₈₉₂	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	892	1	no	—
54	C₂×C₈	—	446	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	892	1	no	—
57	C₄⋊C₄	—	446	1	no	—
58	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	892	1	no	—
65	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	892	1	no	—
68	C₈	—	1784	1	no	—
69	C₂²	#20297218	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	1784	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#28383374	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383374
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₂₃
Fitting	#356×C₂₂₃	C₂
socle	C₂×C₂₂₃	#354
prime radical	C₂²×C₂₂₃	#99
cosocle	C₄⋊Q₈×C₂₂₃	C₂²
hypercenter	#356	D₂₂₃
hypocenter	C₂₂₃	#1413
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
38. C₂₂₃	#1413	semidirect

Maximal subgroups

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

subgroup	index
#28383948	2
#28383379	2
#356×C₂₂₃	2
#1413	223