Finite Group Databasebeta

Subgroups — #46830327

Order 34624.

188 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₂×D₄×D₅₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#22575825	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22575824	C₂	1	1	yes	maximal, characteristic, complemented
4	#31772608	C₂	1	1	yes	maximal, characteristic, complemented
5	#31772548	C₂	1	1	yes	maximal, characteristic, complemented
6	#31772540	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₅₄₁	—	2	1	no	—
9	C₂×Dic₁₀₈₂	C₂²	1	1	yes	characteristic, complemented
10	C₂×D₂₁₆₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₅₄₁	—	2	1	no	—
12	D₄×D₅₄₁	—	2	2	no	—
13	D₄×D₅₄₁	—	2	2	no	—
14	C₂×C₄×D₅₄₁	C₂²	1	1	yes	characteristic
15	#22575825	—	2	2	no	—
16	#22575824	—	2	2	no	—
17	C₂×#16864513	C₂²	1	1	yes	characteristic, complemented
18	C₄⋊Dic₅₄₁	—	2	1	no	—
19	#22575813	—	2	1	no	—
20	C₂×D₄×C₅₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊C₄×C₅₄₁	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₅₄₁	—	2	4	no	—
24	C₂²×D₅₄₁	—	2	1	no	—
25	C₂²×D₅₄₁	—	4	2	no	—
26	C₂²×D₅₄₁	—	4	2	no	—
27	C₂²×D₅₄₁	D₄	1	1	yes	characteristic
28	Dic₁₀₈₂	—	2	1	no	—
29	Dic₁₀₈₂	—	2	1	no	—
30	D₂₁₆₄	—	2	1	no	—
31	D₂₁₆₄	D₄	1	2	yes	—
32	C₂²⋊D₅₄₁	—	2	4	no	—
33	C₂×Dic₅₄₁	—	2	1	no	—
34	C₂×Dic₅₄₁	D₄	1	1	yes	characteristic
35	C₄×D₅₄₁	—	2	1	no	—
36	C₄×D₅₄₁	—	2	1	no	—
37	#16864513	—	2	1	no	—
38	C₂³×C₅₄₁	—	2	1	no	—
39	D₄×C₅₄₁	—	2	1	no	—
40	D₄×C₅₄₁	D₄	1	2	yes	—
41	C₂×C₄×C₅₄₁	—	2	1	no	—
42	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
43	C₈×C₅₄₁	—	2	1	no	—
44	C₂²×C₅₄₁	—	2	2	no	—
45	C₂²×C₅₄₁	—	4	1	no	—
46	C₂²×C₅₄₁	—	4	1	no	—
47	C₂²×C₅₄₁	C₂×D₄	1	1	yes	characteristic
48	C₂×D₅₄₁	—	2	1	no	—
49	C₂×D₅₄₁	—	2	1	no	—
50	C₂×D₅₄₁	—	2	2	no	—
51	C₂×D₅₄₁	—	4	8	no	—
52	C₂×D₅₄₁	—	4	1	no	—
53	C₂×D₅₄₁	—	4	1	no	—
54	C₂×D₅₄₁	SD₁₆	1	2	yes	complemented
55	C₄×C₅₄₁	—	4	1	no	—
56	C₄×C₅₄₁	C₂×D₄	1	1	yes	characteristic
57	C₄×C₅₄₁	C₂×D₄	1	1	yes	characteristic
58	Dic₅₄₁	—	2	1	no	—
59	Dic₅₄₁	—	4	1	no	—
60	C₂×C₅₄₁	—	4	2	no	—
61	C₂×C₅₄₁	#120	1	1	yes	characteristic
62	C₂×C₅₄₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₅₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	D₅₄₁	—	2	2	no	—
65	D₅₄₁	—	4	2	no	—
66	C₅₄₁	#450	1	1	yes	characteristic, complemented, Sylow
67	#450	—	541	1	no	maximal, Sylow
68	C₂²×D₄	—	541	1	no	—
69	C₂×SD₁₆	—	541	1	no	—
70	C₂×SD₁₆	—	541	1	no	—
71	C₂²⋊Q₈	—	541	1	no	—
72	#102	—	541	1	no	—
73	#102	D₅₄₁	1	1	yes	characteristic, complemented
74	C₂²⋊C₈	—	541	1	no	—
75	C₂⁴	—	1082	1	no	—
76	C₂×Q₈	—	541	1	no	—
77	C₂×D₄	—	541	1	no	—
78	C₂×D₄	—	1082	1	no	—
79	C₂×D₄	—	1082	2	no	—
80	C₂×D₄	—	1082	2	no	—
81	C₂×D₄	C₂×D₅₄₁	1	1	yes	characteristic
82	C₂²×C₄	—	541	1	no	—
83	SD₁₆	—	1082	2	no	—
84	SD₁₆	—	1082	2	no	—
85	C₂×C₈	—	541	1	no	—
86	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic, complemented
87	C₄⋊C₄	—	1082	1	no	—
88	C₄⋊C₄	C₂×D₅₄₁	1	1	yes	characteristic, complemented
89	C₂²⋊C₄	—	1082	1	no	—
90	C₂³	—	2	1	no	—
91	C₂³	—	541	1	no	—
92	C₂³	—	1082	4	no	—
93	C₂³	—	1082	1	no	—
94	C₂³	—	2164	2	no	—
95	C₂³	—	2164	2	no	—
96	Q₈	—	1082	1	no	—
97	Q₈	—	1082	1	no	—
98	D₄	—	2	1	no	—
99	D₄	—	541	2	no	—
100	D₄	—	1082	4	no	—
101	D₄	—	1082	1	no	—
102	D₄	D₂₁₆₄	1	2	yes	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	541	1	no	—
105	C₂×C₄	—	1082	1	no	—
106	C₂×C₄	—	1082	1	no	—
107	C₂×C₄	—	1082	1	no	—
108	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	—	1082	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	541	2	no	—
115	C₂²	—	1082	1	no	—
116	C₂²	—	1082	1	no	—
117	C₂²	—	1082	2	no	—
118	C₂²	—	2164	8	no	—
119	C₂²	—	2164	1	no	—
120	C₂²	—	2164	1	no	—
121	C₂²	D₄×D₅₄₁	1	1	yes	characteristic
122	C₄	—	4	1	no	—
123	C₄	—	1082	1	no	—
124	C₄	—	2164	1	no	—
125	C₄	D₄×D₅₄₁	1	1	yes	characteristic
126	C₄	C₂×D₂₁₆₄	1	1	yes	characteristic
127	C₂	—	4	2	no	—
128	C₂	—	1082	2	no	—
129	C₂	—	2164	2	no	—
130	C₂	#31772598	1	1	yes	characteristic
131	C₂	SD₁₆×D₅₄₁	1	1	yes	characteristic
132	C₂	#31772652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₄₁
commutator, cosocle	C₂×C₄×C₅₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	#102×C₅₄₁	C₂
socle	C₂²×C₅₄₁	C₂×D₄
prime radical	C₂×D₄×D₅₄₁	C₂
hypercenter	#102	D₅₄₁
hypocenter	C₅₄₁	#450
LCS	G ▷ C₂×C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
2. C₂×#22575825	C₂	semidirect
3. C₂×#22575824	C₂	semidirect
4. #31772608	C₂	semidirect
5. #31772548	C₂	semidirect
6. #31772540	C₂	semidirect
7. #102×C₅₄₁	C₂	semidirect
9. C₂×Dic₁₀₈₂	C₂²	semidirect
17. C₂×#16864513	C₂²	semidirect
21. C₂×C₈×C₅₄₁	C₂²	semidirect
22. C₄⋊C₄×C₅₄₁	C₂²	semidirect
54. C₂×D₅₄₁	SD₁₆	semidirect
66. C₅₄₁	#450	semidirect
73. #102	D₅₄₁	semidirect
86. C₂×C₈	C₂×D₅₄₁	semidirect
88. C₄⋊C₄	C₂×D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₅₄₁	2
C₂×#22575825	2
C₂×#22575824	2
#31772608	2
#31772548	2
#31772540	2
#102×C₅₄₁	2
#450	541