Subgroups — #28383334

Order 14272.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	C₄	1	2	yes	—
5	C₂×C₂₂₃⋊C₈	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	—	2	1	no	—
7	C₂×C₂₂₃⋊C₈	C₄	1	2	yes	complemented
8	C₄×Dic₂₂₃	—	2	1	no	—
9	C₄×Dic₂₂₃	—	2	1	no	—
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	—	2	1	no	—
12	C₂×C₈×C₂₂₃	—	2	1	no	—
13	C₂×C₈×C₂₂₃	C₄	1	2	yes	complemented
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	C₈	1	4	yes	complemented
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
25	C₈×C₂₂₃	—	2	2	no	—
26	C₂²×C₂₂₃	C₄²	1	1	yes	characteristic
27	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₂₂₃	C₂×C₈	1	2	yes	—
30	C₂²×C₂₂₃	M₁₆	1	2	yes	—
31	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
35	Dic₂₂₃	—	2	4	no	—
36	C₂×C₂₂₃	#95	1	1	yes	characteristic
37	C₂×C₂₂₃	C₄×C₈	1	1	yes	characteristic
38	C₂×C₂₂₃	#97	1	1	yes	characteristic
39	C₂×C₂₂₃	C₂²⋊C₈	1	2	yes	—
40	C₂×C₂₂₃	C₄⋊C₈	1	2	yes	—
41	C₂₂₃	#336	1	1	yes	characteristic, complemented, Sylow
42	#336	—	223	1	no	maximal, Sylow
43	C₂²×C₈	—	223	1	no	—
44	C₂²×C₈	D₂₂₃	1	1	yes	characteristic
45	C₂×C₄²	—	223	1	no	—
46	C₂²×C₄	—	223	2	no	—
47	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	446	1	no	—
51	C₂×C₈	—	446	1	no	—
52	C₂×C₈	—	223	2	no	—
53	C₂×C₈	Dic₂₂₃	1	2	yes	complemented
54	C₄²	—	446	1	no	—
55	C₄²	—	446	1	no	—
56	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
57	C₂×C₄	—	446	2	no	—
58	C₂×C₄	—	446	2	no	—
59	C₂×C₄	—	223	4	no	—
60	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
61	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
64	C₂×C₄	D₈₉₂	1	1	yes	characteristic
65	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	446	2	no	—
68	C₂²	C₄×Dic₂₂₃	1	1	yes	characteristic
69	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
70	C₂²	Dic₂₂₃⋊C₄	1	1	yes	characteristic
71	C₂²	C₈×D₂₂₃	1	2	yes	—
72	C₂²	#16813153	1	2	yes	—
73	C₄	—	446	4	no	—
74	C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
75	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
76	C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
77	C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
78	C₂	C₂₂₃⋊#95	1	1	yes	characteristic
79	C₂	C₈×Dic₂₂₃	1	1	yes	characteristic
80	C₂	#20297214	1	1	yes	characteristic
81	C₂	(C₂×D₂₂₃)⋊C₈	1	2	yes	—
82	C₂	Dic₂₂₃⋊C₈	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₄×C₈
Fitting	C₂²×C₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂²×C₈	D₂₂₃
hypocenter	C₂₂₃	#336
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
7. C₂×C₂₂₃⋊C₈	C₄	semidirect
13. C₂×C₈×C₂₂₃	C₄	semidirect
16. C₂×Dic₂₂₃	C₈	semidirect
41. C₂₂₃	#336	semidirect
53. C₂×C₈	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
C₂×C₄×Dic₂₂₃	2
C₂²×C₈×C₂₂₃	2
#336	223