Subgroups — #28383574

Order 14272.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813170	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297227	C₂	1	2	yes	maximal, complemented
4	#20297224	C₂	1	2	yes	maximal, complemented
5	#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813170	—	2	2	no	—
7	#16813170	C₄	1	4	yes	complemented
8	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
9	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₂₂₃	—	2	1	no	—
11	C₂×C₄×D₂₂₃	—	2	1	no	—
12	C₂×C₂₂₃⋊C₈	—	2	2	no	—
13	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
16	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
17	C₄²×C₂₂₃	—	2	1	no	—
18	C₂²×D₂₂₃	—	2	1	no	—
19	Dic₄₄₆	—	2	1	no	—
20	Dic₄₄₆	C₂×C₄	1	2	yes	—
21	D₈₉₂	—	2	1	no	—
22	D₈₉₂	C₂×C₄	1	2	yes	—
23	C₂²⋊D₂₂₃	—	2	4	no	—
24	C₂×Dic₂₂₃	—	2	1	no	—
25	C₄×D₂₂₃	—	2	4	no	—
26	C₂₂₃⋊C₈	—	2	2	no	—
27	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	—	2	2	no	—
29	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
32	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₂₂₃	—	2	2	no	—
34	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×D₂₂₃	—	2	2	no	—
37	C₂×D₂₂₃	—	4	2	no	—
38	C₄×C₂₂₃	—	4	2	no	—
39	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
40	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
42	Dic₂₂₃	—	2	2	no	—
43	C₂×C₂₂₃	—	2	2	no	—
44	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₂₂₃	#135	1	2	yes	—
46	D₂₂₃	—	4	2	no	—
47	C₂₂₃	#416	1	1	yes	characteristic, complemented, Sylow
48	#416	—	223	1	no	maximal, Sylow
49	C₂×#41	—	223	1	no	—
50	C₂²×C₈	—	223	1	no	—
51	#117	D₂₂₃	1	1	yes	characteristic, complemented
52	#103	—	223	2	no	—
53	#102	—	223	2	no	—
54	#41	—	446	2	no	—
55	#41	—	223	4	no	—
56	C₂×Q₈	—	223	1	no	—
57	C₂×D₄	—	446	1	no	—
58	C₂×D₄	—	223	1	no	—
59	C₂²×C₄	—	446	1	no	—
60	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
61	C₂×C₈	—	446	2	no	—
62	C₂×C₈	—	223	2	no	—
63	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	—	2	1	no	—
66	C₂³	—	446	1	no	—
67	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
68	Q₈	—	446	1	no	—
69	Q₈	—	223	2	no	—
70	D₄	—	446	4	no	—
71	D₄	—	446	1	no	—
72	D₄	—	223	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	446	1	no	—
75	C₂×C₄	—	446	4	no	—
76	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
77	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
78	C₂×C₄	D₈₉₂	1	2	yes	—
79	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
80	C₈	—	446	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	892	2	no	—
83	C₂²	—	446	2	no	—
84	C₂²	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	—	446	2	no	—
88	C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
89	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
90	C₄	C₂×D₈₉₂	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	892	2	no	—
93	C₂	C₂×(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
94	C₂	#20297324	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#117×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×#16813170	C₂
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#117	D₂₂₃
hypocenter	C₂₂₃	#416
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₂₃⋊C₈	C₂	semidirect
3. #20297227	C₂	semidirect
4. #20297224	C₂	semidirect
5. #117×C₂₂₃	C₂	semidirect
7. #16813170	C₄	semidirect
47. C₂₂₃	#416	semidirect
51. #117	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813170	2
C₂²×C₂₂₃⋊C₈	2
#20297227	2
#20297224	2
#117×C₂₂₃	2
#416	223