Finite Group Databasebeta

Subgroups — #72223053

Order 3200.

968 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 243 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	#13102558	C₂	1	2	yes	maximal, complemented
2	#13102558	C₂	1	1	yes	maximal, characteristic, complemented
3	#13102557	C₂	1	2	yes	maximal, complemented
4	#13102556	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅²×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#87801	—	2	4	no	—
7	C₂×#87801	—	2	2	no	—
8	C₂×#87800	—	2	2	no	—
9	#11717194	C₂²	1	1	yes	characteristic
10	#11717194	C₂²	1	2	yes	—
11	#11717189	C₂²	1	1	yes	characteristic
12	C₄×C₅²×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅²×C₈	C₂²	1	2	yes	—
14	#11716688	—	2	4	no	—
15	#10598887	—	5	6	no	maximal
16	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
17	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
18	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
19	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
20	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
21	C₂×#6045	—	2	1	no	—
22	C₂×C₅²×C₈	D₄	1	4	yes	—
23	C₂×C₅²×C₈	D₄	1	2	yes	—
24	C₄²×C₅²	C₂³	1	1	yes	characteristic
25	C₄⋊#1039	—	2	2	no	—
26	#87801	—	4	8	no	—
27	#87801	—	4	4	no	—
28	#87800	—	4	4	no	—
29	#64984	—	5	6	no	—
30	#64984	—	5	12	no	—
31	#64983	—	5	12	no	—
32	#64982	—	5	6	no	—
33	C₅×#322	D₅	1	6	yes	complemented
34	C₂²×C₅⋊D₅	—	4	2	no	—
35	C₂²×C₅⋊D₅	—	4	1	no	—
36	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅²	C₂×D₄	1	2	yes	—
38	C₂×#1039	—	4	1	no	—
39	C₄⋊(C₅⋊D₅)	—	4	2	no	—
40	C₄⋊(C₅⋊D₅)	—	4	4	no	—
41	C₄⋊(C₅⋊D₅)	—	4	4	no	—
42	C₄⋊(C₅⋊D₅)	—	4	4	no	—
43	C₄⋊(C₅⋊D₅)	—	4	4	no	—
44	#6045	—	4	2	no	—
45	C₅²×C₈	—	2	4	no	—
46	C₅²×C₈	D₈	1	4	yes	complemented
47	C₂×D₄₀	—	10	24	no	—
48	C₂×D₄₀	—	10	12	no	—
49	C₂×#677	—	10	12	no	—
50	#4020	—	5	6	no	—
51	#4020	—	5	12	no	—
52	#4015	—	5	6	no	—
53	C₄×C₅×C₈	C₂×D₅	1	6	yes	—
54	C₄×C₅×C₈	C₂×D₅	1	12	yes	—
55	#3953	—	10	24	no	—
56	#1721	—	25	1	no	Sylow
57	C₂²×C₅²	#127	1	1	yes	characteristic
58	C₂×C₅⋊D₅	—	8	1	no	—
59	C₂×C₅⋊D₅	—	8	2	no	—
60	C₂×C₅⋊D₅	—	8	2	no	—
61	C₂×C₅⋊D₅	—	8	2	no	—
62	C₂×C₅⋊D₅	—	8	2	no	—
63	C₄×C₅²	C₂×D₈	1	2	yes	—
64	C₄×C₅²	AGL(1,ℤ/8)	1	4	yes	—
65	#1039	—	8	1	no	—
66	C₂×D₂₀	—	10	12	no	—
67	C₂×D₂₀	—	10	6	no	—
68	C₂×D₂₀	—	10	12	no	—
69	C₂×D₂₀	—	10	12	no	—
70	C₂×D₂₀	—	10	12	no	—
71	C₂×Dic₁₀	—	10	6	no	—
72	C₂×C₅×C₈	D₂₀	1	24	yes	—
73	C₂×C₅×C₈	D₂₀	1	12	yes	—
74	C₄²×C₅	C₂²×D₅	1	6	yes	—
75	C₄⋊Dic₅	—	10	12	no	—
76	D₄₀	—	20	48	no	—
77	D₄₀	—	20	24	no	—
78	#677	—	20	24	no	—
79	#493	—	25	2	no	—
80	#493	—	25	1	no	—
81	#492	—	25	1	no	—
82	#486	—	25	2	no	—
83	#322	C₅⋊D₅	1	1	yes	characteristic, complemented
84	C₂×C₅²	#493	1	1	yes	characteristic
85	C₂×C₅²	#496	1	2	yes	—
86	C₅⋊D₅	—	16	1	no	—
87	C₅⋊D₅	—	16	2	no	—
88	C₂²×D₅	—	20	12	no	—
89	C₂²×D₅	—	20	6	no	—
90	C₂×C₄×C₅	C₂×D₂₀	1	6	yes	—
91	C₂×C₄×C₅	C₂×D₂₀	1	12	yes	—
92	C₂×Dic₅	—	20	6	no	—
93	D₂₀	—	20	12	no	—
94	D₂₀	—	20	24	no	—
95	D₂₀	—	20	24	no	—
96	D₂₀	—	20	24	no	—
97	D₂₀	—	20	24	no	—
98	Dic₁₀	—	20	12	no	—
99	C₅×C₈	—	2	24	no	—
100	C₅×C₈	D₄₀	1	24	yes	complemented
101	C₂×SD₁₆	—	50	2	no	—
102	C₂×D₈	—	50	4	no	—
103	C₂×D₈	—	50	2	no	—
104	C₄⋊Q₈	—	25	1	no	—
105	#127	—	25	2	no	—
106	#127	—	25	1	no	—
107	#102	—	50	4	no	—
108	C₄×C₈	C₂×C₅⋊D₅	1	2	yes	—
109	C₄×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
110	C₅²	#1721	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₅	#4020	1	6	yes	—
112	C₂×D₅	—	40	6	no	—
113	C₂×D₅	—	40	12	no	—
114	C₂×D₅	—	40	12	no	—
115	C₂×D₅	—	40	12	no	—
116	C₂×D₅	—	40	12	no	—
117	C₄×C₅	C₂×D₄₀	1	12	yes	—
118	C₄×C₅	#4054	1	24	yes	—
119	Dic₅	—	40	6	no	—
120	C₂×Q₈	—	50	1	no	—
121	C₂×D₄	—	50	2	no	—
122	C₂×D₄	—	50	2	no	—
123	C₂×D₄	—	50	2	no	—
124	C₂×D₄	—	50	2	no	—
125	C₂×D₄	—	50	1	no	—
126	SD₁₆	—	100	4	no	—
127	D₈	—	100	8	no	—
128	D₈	—	100	4	no	—
129	C₂×C₈	C₄⋊(C₅⋊D₅)	1	4	yes	—
130	C₂×C₈	C₄⋊(C₅⋊D₅)	1	2	yes	—
131	C₄⋊C₄	—	50	2	no	—
132	C₄²	C₂²×C₅⋊D₅	1	1	yes	characteristic
133	C₂×C₅	#64984	1	6	yes	—
134	C₂×C₅	#65001	1	12	yes	—
135	D₅	—	80	6	no	—
136	D₅	—	80	12	no	—
137	C₂³	—	100	2	no	—
138	C₂³	—	100	1	no	—
139	Q₈	—	100	2	no	—
140	D₄	—	100	4	no	—
141	D₄	—	100	2	no	—
142	D₄	—	100	4	no	—
143	D₄	—	100	4	no	—
144	D₄	—	100	4	no	—
145	C₂×C₄	—	100	1	no	—
146	C₂×C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄⋊(C₅⋊D₅)	1	2	yes	—
148	C₈	—	2	4	no	—
149	C₈	#87801	1	4	yes	complemented
150	C₅	#10598887	1	6	yes	complemented
151	C₂²	—	200	2	no	—
152	C₂²	—	200	1	no	—
153	C₂²	—	200	2	no	—
154	C₂²	—	200	2	no	—
155	C₂²	—	200	2	no	—
156	C₂²	#11717194	1	1	yes	characteristic
157	C₄	—	200	1	no	—
158	C₄	C₂×#87801	1	2	yes	—
159	C₄	#11717228	1	4	yes	—
160	C₂	—	400	2	no	—
161	C₂	—	400	1	no	—
162	C₂	#13102558	1	1	yes	characteristic
163	C₂	#13102576	1	2	yes	—

	subgroup	G/H
center	C₂²	#11717194
commutator, cosocle	C₄²×C₅²	C₂³
Frattini	C₄²	C₂²×C₅⋊D₅
Fitting	C₅²×#322	C₂
socle	C₂²×C₅²	#127
hypercenter	#322	C₅⋊D₅
hypocenter	C₅²	#1721
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #13102558	C₂	semidirect
2. #13102558	C₂	semidirect
3. #13102557	C₂	semidirect
4. #13102556	C₂	semidirect
5. C₅²×#322	C₂	semidirect
33. C₅×#322	D₅	semidirect
46. C₅²×C₈	D₈	semidirect
83. #322	C₅⋊D₅	semidirect
100. C₅×C₈	D₄₀	semidirect
110. C₅²	#1721	semidirect
149. C₈	#87801	semidirect
150. C₅	#10598887	semidirect

Maximal subgroups

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

subgroup	index
#13102558	2
#13102557	2
#13102556	2
C₅²×#322	2
#10598887	5