Finite Group Databasebeta

Subgroups — #46855170

Order 35008.

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	#31776048	C₂	1	1	yes	maximal, characteristic, complemented
2	#31776030	C₂	1	1	yes	maximal, characteristic, complemented
3	#31776027	C₂	1	1	yes	maximal, characteristic, complemented
4	#31775999	C₂	1	1	yes	maximal, characteristic, complemented
5	#31775930	C₂	1	1	yes	maximal, characteristic, complemented
6	#31775918	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₅₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#22576777	—	2	1	no	—
9	#22576776	—	2	1	no	—
10	#22576776	C₂²	1	1	yes	characteristic
11	C₂×D₂₁₈₈	—	2	1	no	—
12	C₂×D₂₁₈₈	—	2	2	no	—
13	C₂×D₂₁₈₈	C₂²	1	1	yes	characteristic
14	C₂×D₂₁₈₈	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₅₄₇	—	2	2	no	—
16	D₄×D₅₄₇	—	2	1	no	—
17	#22576765	—	2	1	no	—
18	#22576763	—	2	1	no	—
19	D₄₃₇₆	—	2	1	no	—
20	#22576761	—	2	1	no	—
21	#22576760	C₂²	1	1	yes	characteristic, complemented
22	#41×C₅₄₇	C₂²	1	1	yes	characteristic
23	M₁₆×C₅₄₇	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₅₄₇	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₅₄₇	—	2	2	no	—
26	C₂²×D₅₄₇	—	2	1	no	—
27	C₂²×D₅₄₇	D₄	1	2	yes	—
28	Dic₁₀₉₄	D₄	1	1	yes	characteristic, complemented
29	D₂₁₈₈	—	2	2	no	—
30	D₂₁₈₈	—	2	1	no	—
31	D₂₁₈₈	—	2	1	no	—
32	D₂₁₈₈	—	2	1	no	—
33	D₂₁₈₈	—	2	1	no	—
34	D₂₁₈₈	—	4	4	no	—
35	D₂₁₈₈	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₅₄₇	—	2	2	no	—
37	C₂²⋊D₅₄₇	—	2	1	no	—
38	C₄×D₅₄₇	—	2	2	no	—
39	C₄×D₅₄₇	—	2	1	no	—
40	C₅₄₇⋊C₈	—	2	1	no	—
41	Q₈×C₅₄₇	D₄	1	1	yes	characteristic, complemented
42	D₄×C₅₄₇	—	2	1	no	—
43	D₄×C₅₄₇	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₅₄₇	—	2	1	no	—
45	C₂×C₄×C₅₄₇	—	2	1	no	—
46	C₂×C₄×C₅₄₇	C₂³	1	1	yes	characteristic
47	C₈×C₅₄₇	—	2	1	no	—
48	C₂²×C₅₄₇	—	2	1	no	—
49	C₂²×C₅₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×D₅₄₇	—	2	2	no	—
51	C₂×D₅₄₇	—	2	1	no	—
52	C₂×D₅₄₇	—	4	2	no	—
53	C₂×D₅₄₇	—	4	2	no	—
54	C₂×D₅₄₇	—	4	1	no	—
55	C₂×D₅₄₇	—	8	2	no	—
56	C₄×C₅₄₇	—	2	2	no	—
57	C₄×C₅₄₇	—	2	1	no	—
58	C₄×C₅₄₇	C₂×D₄	1	1	yes	characteristic
59	C₄×C₅₄₇	C₂×D₄	1	1	yes	characteristic
60	Dic₅₄₇	—	2	1	no	—
61	C₂×C₅₄₇	—	2	1	no	—
62	C₂×C₅₄₇	—	4	1	no	—
63	C₂×C₅₄₇	#120	1	1	yes	characteristic
64	D₅₄₇	—	4	2	no	—
65	D₅₄₇	—	4	1	no	—
66	D₅₄₇	—	8	1	no	—
67	C₅₄₇	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	547	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	547	1	no	—
70	AGL(1,ℤ/8)	—	547	1	no	—
71	AGL(1,ℤ/8)	—	547	1	no	—
72	#127	—	547	1	no	—
73	#104	—	547	1	no	—
74	#104	D₅₄₇	1	1	yes	characteristic, complemented
75	#100	—	547	1	no	—
76	#41	—	547	1	no	—
77	#41	—	1094	1	no	—
78	#41	—	1094	1	no	—
79	#41	C₂×D₅₄₇	1	1	yes	characteristic
80	C₂×D₄	—	547	1	no	—
81	C₂×D₄	—	547	1	no	—
82	C₂×D₄	—	1094	1	no	—
83	C₂×D₄	—	1094	2	no	—
84	C₂×D₄	—	1094	2	no	—
85	C₂×D₄	—	1094	1	no	—
86	SD₁₆	—	1094	1	no	—
87	SD₁₆	—	1094	1	no	—
88	D₈	—	1094	1	no	—
89	D₈	—	1094	1	no	—
90	M₁₆	—	547	1	no	—
91	M₁₆	C₂×D₅₄₇	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₅₄₇	1	1	yes	characteristic, complemented
93	C₂³	—	547	2	no	—
94	C₂³	—	1094	2	no	—
95	C₂³	—	1094	1	no	—
96	Q₈	—	547	1	no	—
97	Q₈	D₂₁₈₈	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	547	1	no	—
100	D₄	—	1094	2	no	—
101	D₄	—	1094	2	no	—
102	D₄	—	1094	1	no	—
103	D₄	—	1094	1	no	—
104	D₄	—	1094	1	no	—
105	D₄	—	1094	1	no	—
106	D₄	—	1094	1	no	—
107	D₄	—	2188	4	no	—
108	D₄	D₂₁₈₈	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	1094	2	no	—
112	C₂×C₄	—	1094	1	no	—
113	C₂×C₄	C₂²×D₅₄₇	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	1094	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	4376	2	no	—
118	C₂²	—	1094	1	no	—
119	C₂²	—	1094	2	no	—
120	C₂²	—	2188	2	no	—
121	C₂²	—	2188	2	no	—
122	C₂²	—	2188	1	no	—
123	C₂²	D₄×D₅₄₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	1094	1	no	—
127	C₄	D₄×D₅₄₇	1	1	yes	characteristic
128	C₄	C₂×D₂₁₈₈	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4376	1	no	—
132	C₂	—	2188	1	no	—
133	C₂	—	2188	2	no	—
134	C₂	#31775972	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31775972
commutator, cosocle	C₂×C₄×C₅₄₇	C₂³
Frattini	C₂×C₄	C₂²×D₅₄₇
Fitting	#104×C₅₄₇	C₂
socle	C₂×C₅₄₇	#120
hypercenter	#104	D₅₄₇
hypocenter	C₅₄₇	#453
LCS	G ▷ C₂×C₄×C₅₄₇ ▷ C₂×C₅₄₇ ▷ C₅₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #31776048	C₂	semidirect
2. #31776030	C₂	semidirect
3. #31776027	C₂	semidirect
4. #31775999	C₂	semidirect
5. #31775930	C₂	semidirect
6. #31775918	C₂	semidirect
7. #104×C₅₄₇	C₂	semidirect
14. C₂×D₂₁₈₈	C₂²	semidirect
21. #22576760	C₂²	semidirect
23. M₁₆×C₅₄₇	C₂²	semidirect
24. C₄²×C₅₄₇	C₂²	semidirect
28. Dic₁₀₉₄	D₄	semidirect
35. D₂₁₈₈	D₄	semidirect
41. Q₈×C₅₄₇	D₄	semidirect
43. D₄×C₅₄₇	D₄	semidirect
67. C₅₄₇	#453	semidirect
74. #104	D₅₄₇	semidirect
91. M₁₆	C₂×D₅₄₇	semidirect
92. C₄²	C₂×D₅₄₇	semidirect
97. Q₈	D₂₁₈₈	semidirect
108. D₄	D₂₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
#31776048	2
#31776030	2
#31776027	2
#31775999	2
#31775930	2
#31775918	2
#104×C₅₄₇	2
#453	547