Subgroups — #28383431

Order 14272.

194 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	2	yes	maximal
3	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
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	C₄	1	8	yes	complemented
10	C₄×Dic₂₂₃	—	2	4	no	—
11	C₄×Dic₂₂₃	C₄	1	4	yes	complemented
12	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	C₂×C₄	1	4	yes	—
17	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
18	C₂×Dic₂₂₃	C₂×C₄	1	4	yes	—
19	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
20	C₂×Dic₂₂₃	D₄	1	2	yes	—
21	C₂×Dic₂₂₃	Q₈	1	2	yes	—
22	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
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	C₂×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
28	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₂₂₃	C₂²×C₄	1	2	yes	—
30	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₂₂₃	#41	1	2	yes	—
33	C₄×C₂₂₃	—	2	2	no	—
34	C₄×C₂₂₃	—	2	4	no	—
35	Dic₂₂₃	—	2	4	no	—
36	Dic₂₂₃	C₄²	1	4	yes	complemented
37	Dic₂₂₃	C₄⋊C₄	1	4	yes	complemented
38	C₂×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₂₂₃	#117	1	1	yes	characteristic
41	C₂×C₂₂₃	C₄×D₄	1	2	yes	—
42	C₂×C₂₂₃	C₄×Q₈	1	2	yes	—
43	C₂₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₄×C₄⋊C₄	—	223	1	no	maximal, Sylow
45	C₂×C₄⋊C₄	—	223	2	no	—
46	C₂×C₄²	—	223	2	no	—
47	C₂×C₄²	—	223	1	no	—
48	#95	—	223	1	no	—
49	#95	D₂₂₃	1	1	yes	characteristic
50	C₂²×C₄	—	223	2	no	—
51	C₂²×C₄	—	223	1	no	—
52	C₂²×C₄	—	223	1	no	—
53	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
55	C₄⋊C₄	—	223	8	no	—
56	C₄²	—	446	4	no	—
57	C₄²	—	223	4	no	—
58	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	446	2	no	—
63	C₂×C₄	—	446	2	no	—
64	C₂×C₄	—	223	4	no	—
65	C₂×C₄	—	223	2	no	—
66	C₂×C₄	—	223	2	no	—
67	C₂×C₄	—	223	4	no	—
68	C₂×C₄	—	223	2	no	—
69	C₂×C₄	—	223	2	no	—
70	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
71	C₂×C₄	C₄×D₂₂₃	1	4	yes	—
72	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
73	C₂²	C₂×C₄×D₂₂₃	1	2	yes	—
74	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
75	C₂²	Q₈×D₂₂₃	1	1	yes	characteristic
76	C₂²	#16813169	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	—	2	4	no	—
79	C₄	—	446	4	no	—
80	C₄	—	223	4	no	—
81	C₄	—	223	4	no	—
82	C₂	C₄²×D₂₂₃	1	1	yes	characteristic
83	C₂	C₄⋊C₄×D₂₂₃	1	1	yes	characteristic
84	C₂	#20297260	1	1	yes	characteristic
85	C₂	(C₂²⋊D₂₂₃)⋊C₄	1	2	yes	—
86	C₂	#20297269	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂×C₄²
Fitting	#95×C₂₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₂₃	C₂³
hypercenter	#95	D₂₂₃
hypocenter	C₂₂₃	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
9. Dic₂₂₃⋊C₄	C₄	semidirect
11. C₄×Dic₂₂₃	C₄	semidirect
36. Dic₂₂₃	C₄²	semidirect
37. Dic₂₂₃	C₄⋊C₄	semidirect
43. C₂₂₃	C₄×C₄⋊C₄	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
C₄×C₄⋊C₄	223