Subgroups — #28383543

Order 14272.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813154	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₂₂₃)⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₂₂₃	C₄	1	1	yes	characteristic
6	C₂²×Dic₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₂₃	—	2	2	no	—
8	C₂×C₄×D₂₂₃	—	2	2	no	—
9	C₂×C₄×D₂₂₃	C₄	1	2	yes	—
10	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
11	#16813154	—	2	2	no	—
12	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	M₁₆×C₂₂₃	—	2	2	no	—
15	C₂×C₈×C₂₂₃	C₂²	1	2	yes	complemented
16	C₂²×D₂₂₃	—	2	2	no	—
17	C₂²×D₂₂₃	—	2	4	no	—
18	C₂²×D₂₂₃	C₂×C₄	1	2	yes	—
19	C₂×Dic₂₂₃	—	2	2	no	—
20	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
21	C₄×D₂₂₃	—	2	4	no	—
22	C₄×D₂₂₃	—	2	2	no	—
23	C₄×D₂₂₃	—	2	2	no	—
24	C₂₂₃⋊C₈	—	2	2	no	—
25	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
28	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
29	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₈×C₂₂₃	—	2	2	no	—
31	C₂²×C₂₂₃	—	2	2	no	—
32	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₂₂₃	—	2	8	no	—
35	C₂×D₂₂₃	—	2	2	no	—
36	C₂×D₂₂₃	—	2	2	no	—
37	C₂×D₂₂₃	M₁₆	1	4	yes	complemented
38	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
39	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
40	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
41	Dic₂₂₃	—	2	2	no	—
42	C₂×C₂₂₃	—	2	2	no	—
43	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₂₂₃	C₂×M₁₆	1	2	yes	—
45	D₂₂₃	—	2	4	no	—
46	C₂₂₃	#407	1	1	yes	characteristic, complemented, Sylow
47	#407	—	223	1	no	maximal, Sylow
48	C₂³×C₄	—	223	1	no	—
49	C₂×M₁₆	—	223	1	no	—
50	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic, complemented
51	C₂²⋊C₈	—	223	4	no	—
52	C₂⁴	—	223	1	no	—
53	C₂²×C₄	—	446	2	no	—
54	C₂²×C₄	—	446	2	no	—
55	C₂²×C₄	—	223	1	no	—
56	C₂²×C₄	—	223	2	no	—
57	C₂²×C₄	—	223	2	no	—
58	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
59	M₁₆	—	2	2	no	—
60	M₁₆	—	446	2	no	—
61	C₂×C₈	—	223	2	no	—
62	C₂×C₈	C₂×D₂₂₃	1	2	yes	complemented
63	C₂³	—	446	4	no	—
64	C₂³	—	446	2	no	—
65	C₂³	—	223	2	no	—
66	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
67	C₂×C₄	—	446	2	no	—
68	C₂×C₄	—	446	4	no	—
69	C₂×C₄	—	446	2	no	—
70	C₂×C₄	—	446	2	no	—
71	C₂×C₄	—	223	2	no	—
72	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
74	C₂×C₄	D₈₉₂	1	2	yes	—
75	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	446	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	446	8	no	—
80	C₂²	—	446	2	no	—
81	C₂²	—	446	2	no	—
82	C₂²	—	223	4	no	—
83	C₂²	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
84	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
85	C₄	—	446	2	no	—
86	C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
87	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
88	C₄	C₂×D₈₉₂	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	446	4	no	—
91	C₂	C₂×(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
92	C₂	M₁₆×D₂₂₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂×M₁₆×C₂₂₃	C₂
prime radical	C₂³×D₂₂₃	C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	#407
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16813154	C₂	semidirect
3. (C₂×D₂₂₃)⋊C₈	C₂	semidirect
4. C₂×M₁₆×C₂₂₃	C₂	semidirect
12. C₂×C₂₂₃⋊C₈	C₂²	semidirect
15. C₂×C₈×C₂₂₃	C₂²	semidirect
37. C₂×D₂₂₃	M₁₆	semidirect
46. C₂₂₃	#407	semidirect
50. C₂×M₁₆	D₂₂₃	semidirect
62. C₂×C₈	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₂₂₃	2
C₂×#16813154	2
(C₂×D₂₂₃)⋊C₈	2
C₂×M₁₆×C₂₂₃	2
#407	223