Subgroups — #28383444

Order 14272.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₂₃⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₂₃⋊#95	C₂	1	2	yes	maximal
4	C₂₂₃⋊#95	C₂	1	1	yes	maximal, characteristic
5	#95×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
7	C₂²×Dic₂₂₃	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₂₃	C₂²	1	1	yes	characteristic
9	Dic₂₂₃⋊C₄	—	2	4	no	—
10	C₄×Dic₂₂₃	C₄	1	4	yes	complemented
11	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₂₃	—	2	2	no	—
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
18	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
19	C₂×Dic₂₂₃	D₄	1	2	yes	—
20	C₂×Dic₂₂₃	Q₈	1	2	yes	—
21	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	—	2	2	no	—
24	C₂×C₄×C₂₂₃	—	2	2	no	—
25	C₂×C₄×C₂₂₃	—	2	2	no	—
26	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₂₂₃	#41	1	2	yes	—
31	C₂²×C₂₂₃	#41	1	2	yes	—
32	C₄×C₂₂₃	—	2	2	no	—
33	C₄×C₂₂₃	—	4	2	no	—
34	Dic₂₂₃	—	2	2	no	—
35	Dic₂₂₃	—	4	2	no	—
36	Dic₂₂₃	C₄⋊C₄	1	4	yes	—
37	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₂₂₃	#117	1	1	yes	characteristic
39	C₂×C₂₂₃	#117	1	1	yes	characteristic
40	C₂×C₂₂₃	#124	1	2	yes	—
41	C₂×C₂₂₃	#125	1	2	yes	—
42	C₂₂₃	#382	1	1	yes	characteristic, complemented, Sylow
43	#382	—	223	1	no	maximal, Sylow
44	C₂×C₄⋊C₄	—	223	2	no	—
45	C₂×C₄²	—	223	1	no	—
46	#95	—	223	1	no	—
47	#95	—	223	2	no	—
48	#95	D₂₂₃	1	1	yes	characteristic
49	C₂²×C₄	—	223	2	no	—
50	C₂²×C₄	—	223	1	no	—
51	C₂²×C₄	—	223	1	no	—
52	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
53	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
54	C₄⋊C₄	—	446	4	no	—
55	C₄²	—	223	4	no	—
56	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	446	2	no	—
62	C₂×C₄	—	446	2	no	—
63	C₂×C₄	—	446	2	no	—
64	C₂×C₄	—	446	2	no	—
65	C₂×C₄	—	223	2	no	—
66	C₂×C₄	—	223	2	no	—
67	C₂×C₄	—	223	2	no	—
68	C₂×C₄	—	223	2	no	—
69	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
70	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
71	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
72	C₂²	Q₈×D₂₂₃	1	1	yes	characteristic
73	C₂²	#16813169	1	2	yes	—
74	C₂²	#16813170	1	2	yes	—
75	C₄	—	2	2	no	—
76	C₄	—	4	2	no	—
77	C₄	—	892	2	no	—
78	C₄	—	446	2	no	—
79	C₄	—	223	4	no	—
80	C₂	C₄⋊C₄×D₂₂₃	1	1	yes	characteristic
81	C₂	#20297260	1	1	yes	characteristic
82	C₂	#20297261	1	1	yes	characteristic
83	C₂	#20297290	1	2	yes	—
84	C₂	#20297294	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₂₃
commutator	C₂²×C₂₂₃	C₂²×C₄
Fitting	#95×C₂₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₂₃	C₂³
hypercenter	#95	D₂₂₃
hypocenter	C₂₂₃	#382
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₂₃	C₄	semidirect
42. C₂₂₃	#382	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₂₃⋊C₄	2
C₂×C₄×Dic₂₂₃	2
C₂₂₃⋊#95	2
#95×C₂₂₃	2
#382	223