Subgroups — #28384186

Order 14272.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297301	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297289	C₂	1	4	yes	maximal, complemented
3	#20297279	C₂	1	2	yes	maximal, complemented
4	#20297279	C₂	1	4	yes	maximal, complemented
5	#20297266	C₂	1	2	yes	maximal, complemented
6	#20297261	C₂	1	1	yes	maximal, characteristic, complemented
7	#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈₉₂	C₂²	1	2	yes	—
9	C₂×D₈₉₂	C₂²	1	4	yes	—
10	C₂×D₈₉₂	C₂²	1	4	yes	—
11	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₂₃	C₂²	1	4	yes	—
13	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
14	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
15	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
16	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
17	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
20	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
21	C₂²×D₂₂₃	C₂³	1	1	yes	characteristic
22	C₂²×D₂₂₃	C₂³	1	4	yes	—
23	D₈₉₂	—	2	4	no	—
24	D₈₉₂	—	4	4	no	—
25	D₈₉₂	—	4	4	no	—
26	C₂×Dic₂₂₃	C₂³	1	2	yes	—
27	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
28	C₄×D₂₂₃	—	2	2	no	—
29	C₄×D₂₂₃	—	4	4	no	—
30	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
31	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
32	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
34	C₂×D₂₂₃	—	2	1	no	—
35	C₂×D₂₂₃	—	2	4	no	—
36	C₂×D₂₂₃	—	2	4	no	—
37	C₂×D₂₂₃	—	2	2	no	—
38	C₂×D₂₂₃	—	2	4	no	—
39	C₄×C₂₂₃	—	2	4	no	—
40	C₄×C₂₂₃	—	2	2	no	—
41	C₄×C₂₂₃	#41	1	2	yes	—
42	Dic₂₂₃	—	2	1	no	—
43	Dic₂₂₃	—	2	2	no	—
44	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
45	C₂×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
46	D₂₂₃	—	4	1	no	—
47	D₂₂₃	—	4	4	no	—
48	C₂₂₃	#540	1	1	yes	characteristic, complemented, Sylow
49	#540	—	223	1	no	maximal, Sylow
50	#127	—	223	1	no	—
51	#125	D₂₂₃	1	1	yes	characteristic, complemented
52	#123	—	223	4	no	—
53	#121	—	223	4	no	—
54	#121	—	223	2	no	—
55	C₄×D₄	—	223	2	no	—
56	#117	—	223	1	no	—
57	C₂×D₄	—	223	4	no	—
58	C₂×D₄	—	223	4	no	—
59	C₂×D₄	—	223	2	no	—
60	C₂²×C₄	—	223	4	no	—
61	C₂²×C₄	—	223	1	no	—
62	C₄⋊C₄	—	223	2	no	—
63	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
64	C₄⋊C₄	C₂×D₂₂₃	1	4	yes	—
65	C₂²⋊C₄	—	223	2	no	—
66	C₂²⋊C₄	—	223	4	no	—
67	C₂²⋊C₄	—	223	4	no	—
68	C₄²	—	223	1	no	—
69	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
70	C₂³	—	223	1	no	—
71	C₂³	—	223	4	no	—
72	D₄	—	892	4	no	—
73	D₄	—	892	4	no	—
74	D₄	—	446	4	no	—
75	C₂×C₄	—	892	4	no	—
76	C₂×C₄	—	446	2	no	—
77	C₂×C₄	—	223	2	no	—
78	C₂×C₄	—	223	1	no	—
79	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
80	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
81	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
82	C₂²	—	446	1	no	—
83	C₂²	—	446	4	no	—
84	C₂²	—	446	4	no	—
85	C₂²	—	446	2	no	—
86	C₂²	—	446	4	no	—
87	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	—	446	1	no	—
91	C₄	—	446	2	no	—
92	C₄	#16813171	1	2	yes	—
93	C₂	—	892	1	no	—
94	C₂	—	892	4	no	—
95	C₂	C₂×#16813171	1	1	yes	characteristic
96	C₂	#20297350	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#125×C₂₂₃	C₂
hypercenter	#125	D₂₂₃
hypocenter	C₂₂₃	#540
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #20297301	C₂	semidirect
2. #20297289	C₂	semidirect
3. #20297279	C₂	semidirect
4. #20297279	C₂	semidirect
5. #20297266	C₂	semidirect
6. #20297261	C₂	semidirect
7. #125×C₂₂₃	C₂	semidirect
48. C₂₂₃	#540	semidirect
51. #125	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297301	2
#20297289	2
#20297279	2
#20297266	2
#20297261	2
#125×C₂₂₃	2
#540	223