Subgroups — #28383636

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₂₂₃)⋊C₈	C₂	1	2	yes	maximal
4	#20297214	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
11	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
12	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₄×D₂₂₃	—	2	2	no	—
20	C₄×D₂₂₃	—	2	2	no	—
21	C₄×D₂₂₃	—	2	1	no	—
22	C₄×D₂₂₃	—	2	1	no	—
23	C₂₂₃⋊C₈	—	2	2	no	—
24	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
27	C₈×C₂₂₃	—	2	2	no	—
28	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₂₂₃	—	2	1	no	—
30	C₂×D₂₂₃	—	2	1	no	—
31	C₂×D₂₂₃	M₁₆	1	2	yes	—
32	C₄×C₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	M₁₆	1	2	yes	—
34	C₄×C₂₂₃	#41	1	1	yes	characteristic
35	C₄×C₂₂₃	#41	1	1	yes	characteristic
36	Dic₂₂₃	—	2	1	no	—
37	Dic₂₂₃	—	2	1	no	—
38	Dic₂₂₃	—	2	1	no	—
39	C₂×C₂₂₃	#117	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
42	D₂₂₃	—	2	2	no	—
43	C₂₂₃	#432	1	1	yes	characteristic, complemented, Sylow
44	#432	—	223	1	no	maximal, Sylow
45	C₂×C₄²	—	223	1	no	—
46	C₄⋊C₈	—	223	1	no	—
47	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic, complemented
48	C₂²⋊C₈	—	223	2	no	—
49	#97	—	223	2	no	—
50	C₂²×C₄	—	223	1	no	—
51	C₂²×C₄	—	223	1	no	—
52	C₂²×C₄	—	223	1	no	—
53	C₂×C₈	—	223	2	no	—
54	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
55	C₄²	—	223	1	no	—
56	C₄²	—	223	1	no	—
57	C₄²	—	223	1	no	—
58	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
59	C₂³	—	223	1	no	—
60	C₂×C₄	—	446	2	no	—
61	C₂×C₄	—	446	2	no	—
62	C₂×C₄	—	446	1	no	—
63	C₂×C₄	—	446	1	no	—
64	C₂×C₄	—	223	1	no	—
65	C₂×C₄	—	223	1	no	—
66	C₂×C₄	—	223	1	no	—
67	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
68	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	446	2	no	—
72	C₂²	—	446	1	no	—
73	C₂²	—	446	1	no	—
74	C₂²	—	223	2	no	—
75	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	446	1	no	—
78	C₄	—	446	1	no	—
79	C₄	—	446	1	no	—
80	C₄	#16813153	1	2	yes	—
81	C₄	#16813169	1	1	yes	characteristic
82	C₄	#16813171	1	1	yes	characteristic
83	C₂	—	446	2	no	—
84	C₂	#20297262	1	1	yes	characteristic
85	C₂	M₁₆×D₂₂₃	1	1	yes	characteristic
86	C₂	C₂×#16813153	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₄⋊C₈×C₂₂₃	C₂
prime radical	C₂²×D₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#432
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
4. #20297214	C₂	semidirect
5. C₄⋊C₈×C₂₂₃	C₂	semidirect
43. C₂₂₃	#432	semidirect
47. C₄⋊C₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₂₂₃	2
C₄⋊(C₂₂₃⋊C₈)	2
(C₂×D₂₂₃)⋊C₈	2
#20297214	2
C₄⋊C₈×C₂₂₃	2
#432	223