Subgroups — #28383341

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297263	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	—	2	1	no	—
5	C₄⋊Dic₂₂₃	C₄	1	2	yes	complemented
6	C₂²⋊Dic₂₂₃	—	2	1	no	—
7	C₄×Dic₂₂₃	—	2	1	no	—
8	C₄×Dic₂₂₃	—	2	1	no	—
9	C₄×Dic₂₂₃	C₄	1	2	yes	complemented
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	M₁₆×C₂₂₃	—	2	1	no	—
12	M₁₆×C₂₂₃	C₄	1	2	yes	complemented
13	C₂×C₈×C₂₂₃	—	2	1	no	—
14	C₂×Dic₂₂₃	—	2	4	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄×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	Q₈	1	1	yes	characteristic
24	C₈×C₂₂₃	—	2	2	no	—
25	C₂²×C₂₂₃	—	2	2	no	—
26	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₂₂₃	C₄²	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
33	Dic₂₂₃	—	2	4	no	—
34	Dic₂₂₃	—	4	2	no	—
35	C₂×C₂₂₃	—	2	1	no	—
36	C₂×C₂₂₃	—	2	1	no	—
37	C₂×C₂₂₃	#95	1	1	yes	characteristic
38	C₂×C₂₂₃	#104	1	2	yes	—
39	C₂₂₃	#339	1	1	yes	characteristic, complemented, Sylow
40	#339	—	223	1	no	maximal, Sylow
41	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic
42	#117	—	223	1	no	—
43	C₂×C₄²	—	223	1	no	—
44	C₂²×C₄	—	446	1	no	—
45	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
46	M₁₆	—	2	1	no	—
47	M₁₆	Dic₂₂₃	1	2	yes	complemented
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	223	2	no	—
50	C₂²⋊C₄	—	446	1	no	—
51	C₄²	—	446	1	no	—
52	C₄²	—	446	1	no	—
53	C₄²	—	223	2	no	—
54	C₂³	D₈₉₂	1	1	yes	characteristic
55	C₂×C₄	—	446	2	no	—
56	C₂×C₄	—	446	4	no	—
57	C₂×C₄	—	446	2	no	—
58	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
59	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
63	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
67	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
68	C₂²	C₄⋊Dic₂₂₃	1	1	yes	characteristic
69	C₄	—	892	2	no	—
70	C₄	—	446	4	no	—
71	C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
72	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
73	C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
74	C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	C₂₂₃⋊#95	1	1	yes	characteristic
78	C₂	#20297230	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂×M₁₆×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	#339
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊Dic₂₂₃	C₄	semidirect
9. C₄×Dic₂₂₃	C₄	semidirect
12. M₁₆×C₂₂₃	C₄	semidirect
39. C₂₂₃	#339	semidirect
47. M₁₆	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297263	2
C₂×C₄×Dic₂₂₃	2
C₂×M₁₆×C₂₂₃	2
#339	223