Finite Group Databasebeta

Subgroups — #31823632

Order 17984.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582925	C₂	1	1	yes	maximal, characteristic, complemented
2	#22582907	C₂	1	1	yes	maximal, characteristic, complemented
3	#22582904	C₂	1	1	yes	maximal, characteristic, complemented
4	#22582876	C₂	1	1	yes	maximal, characteristic, complemented
5	#22582807	C₂	1	1	yes	maximal, characteristic, complemented
6	#22582795	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#17985139	—	2	1	no	—
9	#17985138	—	2	1	no	—
10	#17985138	C₂²	1	1	yes	characteristic
11	C₂×D₁₁₂₄	—	2	1	no	—
12	C₂×D₁₁₂₄	—	2	2	no	—
13	C₂×D₁₁₂₄	C₂²	1	1	yes	characteristic
14	C₂×D₁₁₂₄	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₂₈₁	—	2	2	no	—
16	D₄×D₂₈₁	—	2	1	no	—
17	#17985127	—	2	1	no	—
18	#17985125	—	2	1	no	—
19	D₂₂₄₈	—	2	1	no	—
20	#17985123	—	2	1	no	—
21	#17985122	C₂²	1	1	yes	characteristic, complemented
22	#41×C₂₈₁	C₂²	1	1	yes	characteristic
23	M₁₆×C₂₈₁	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₂₈₁	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₂₈₁	—	2	2	no	—
26	C₂²×D₂₈₁	—	2	1	no	—
27	C₂²×D₂₈₁	D₄	1	2	yes	—
28	Dic₅₆₂	D₄	1	1	yes	characteristic, complemented
29	D₁₁₂₄	—	2	2	no	—
30	D₁₁₂₄	—	2	1	no	—
31	D₁₁₂₄	—	2	1	no	—
32	D₁₁₂₄	—	2	1	no	—
33	D₁₁₂₄	—	2	1	no	—
34	D₁₁₂₄	—	4	4	no	—
35	D₁₁₂₄	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₂₈₁	—	2	2	no	—
37	C₂²⋊D₂₈₁	—	2	1	no	—
38	C₄×D₂₈₁	—	2	2	no	—
39	C₄×D₂₈₁	—	2	1	no	—
40	#14555886	—	2	1	no	—
41	Q₈×C₂₈₁	D₄	1	1	yes	characteristic, complemented
42	D₄×C₂₈₁	—	2	1	no	—
43	D₄×C₂₈₁	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₂₈₁	—	2	1	no	—
45	C₂×C₄×C₂₈₁	—	2	1	no	—
46	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
47	C₈×C₂₈₁	—	2	1	no	—
48	C₂²×C₂₈₁	—	2	1	no	—
49	C₂²×C₂₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×D₂₈₁	—	2	2	no	—
51	C₂×D₂₈₁	—	2	1	no	—
52	C₂×D₂₈₁	—	4	2	no	—
53	C₂×D₂₈₁	—	4	2	no	—
54	C₂×D₂₈₁	—	4	1	no	—
55	C₂×D₂₈₁	—	8	2	no	—
56	C₄×C₂₈₁	—	2	2	no	—
57	C₄×C₂₈₁	—	2	1	no	—
58	C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
59	C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
60	Dic₂₈₁	—	2	1	no	—
61	C₂×C₂₈₁	—	2	1	no	—
62	C₂×C₂₈₁	—	4	1	no	—
63	C₂×C₂₈₁	#120	1	1	yes	characteristic
64	D₂₈₁	—	4	2	no	—
65	D₂₈₁	—	4	1	no	—
66	D₂₈₁	—	8	1	no	—
67	C₂₈₁	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	281	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	281	1	no	—
70	AGL(1,ℤ/8)	—	281	1	no	—
71	AGL(1,ℤ/8)	—	281	1	no	—
72	#127	—	281	1	no	—
73	#104	—	281	1	no	—
74	#104	D₂₈₁	1	1	yes	characteristic, complemented
75	#100	—	281	1	no	—
76	#41	—	281	1	no	—
77	#41	—	562	1	no	—
78	#41	—	562	1	no	—
79	#41	C₂×D₂₈₁	1	1	yes	characteristic
80	C₂×D₄	—	281	1	no	—
81	C₂×D₄	—	281	1	no	—
82	C₂×D₄	—	562	1	no	—
83	C₂×D₄	—	562	2	no	—
84	C₂×D₄	—	562	2	no	—
85	C₂×D₄	—	562	1	no	—
86	SD₁₆	—	562	1	no	—
87	SD₁₆	—	562	1	no	—
88	D₈	—	562	1	no	—
89	D₈	—	562	1	no	—
90	M₁₆	—	281	1	no	—
91	M₁₆	C₂×D₂₈₁	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₂₈₁	1	1	yes	characteristic, complemented
93	C₂³	—	281	2	no	—
94	C₂³	—	562	2	no	—
95	C₂³	—	562	1	no	—
96	Q₈	—	281	1	no	—
97	Q₈	D₁₁₂₄	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	281	1	no	—
100	D₄	—	562	2	no	—
101	D₄	—	562	2	no	—
102	D₄	—	562	1	no	—
103	D₄	—	562	1	no	—
104	D₄	—	562	1	no	—
105	D₄	—	562	1	no	—
106	D₄	—	562	1	no	—
107	D₄	—	1124	4	no	—
108	D₄	D₁₁₂₄	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	562	2	no	—
112	C₂×C₄	—	562	1	no	—
113	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	562	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	562	1	no	—
118	C₂²	—	562	2	no	—
119	C₂²	—	1124	2	no	—
120	C₂²	—	1124	2	no	—
121	C₂²	—	1124	1	no	—
122	C₂²	—	2248	2	no	—
123	C₂²	D₄×D₂₈₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	562	1	no	—
127	C₄	D₄×D₂₈₁	1	1	yes	characteristic
128	C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	1124	2	no	—
132	C₂	—	1124	1	no	—
133	C₂	—	2248	1	no	—
134	C₂	#22582849	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22582849
commutator, cosocle	C₂×C₄×C₂₈₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₈₁
Fitting	#104×C₂₈₁	C₂
socle	C₂×C₂₈₁	#120
hypercenter	#104	D₂₈₁
hypocenter	C₂₈₁	#453
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #22582925	C₂	semidirect
2. #22582907	C₂	semidirect
3. #22582904	C₂	semidirect
4. #22582876	C₂	semidirect
5. #22582807	C₂	semidirect
6. #22582795	C₂	semidirect
7. #104×C₂₈₁	C₂	semidirect
14. C₂×D₁₁₂₄	C₂²	semidirect
21. #17985122	C₂²	semidirect
23. M₁₆×C₂₈₁	C₂²	semidirect
24. C₄²×C₂₈₁	C₂²	semidirect
28. Dic₅₆₂	D₄	semidirect
35. D₁₁₂₄	D₄	semidirect
41. Q₈×C₂₈₁	D₄	semidirect
43. D₄×C₂₈₁	D₄	semidirect
67. C₂₈₁	#453	semidirect
74. #104	D₂₈₁	semidirect
91. M₁₆	C₂×D₂₈₁	semidirect
92. C₄²	C₂×D₂₈₁	semidirect
97. Q₈	D₁₁₂₄	semidirect
108. D₄	D₁₁₂₄	semidirect

Maximal subgroups

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

subgroup	index
#22582925	2
#22582907	2
#22582904	2
#22582876	2
#22582807	2
#22582795	2
#104×C₂₈₁	2
#453	281