Finite Group Databasebeta

Subgroups — #13102707

Order 1600.

496 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 125 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×#87801	C₂	1	1	yes	maximal, characteristic
2	C₂×#87800	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717195	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716688	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅⋊D₅	—	2	1	no	—
11	C₂×#6045	C₂²	1	1	yes	characteristic
12	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
14	C₄²×C₅²	C₂²	1	1	yes	characteristic
15	#87808	—	2	1	no	—
16	#87808	—	2	1	no	—
17	#87808	—	2	1	no	—
18	C₄⋊#1039	C₂²	1	1	yes	characteristic
19	#87801	—	2	2	no	—
20	#87800	—	2	2	no	—
21	#65133	—	5	6	no	maximal
22	C₂²×C₅⋊D₅	—	2	1	no	—
23	C₂²×C₅⋊D₅	—	2	1	no	—
24	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
25	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
26	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
27	C₂×#1039	—	2	1	no	—
28	C₂×#1039	—	2	1	no	—
29	C₄⋊(C₅⋊D₅)	—	2	1	no	—
30	C₄⋊(C₅⋊D₅)	—	2	1	no	—
31	C₄⋊(C₅⋊D₅)	—	2	1	no	—
32	C₄⋊(C₅⋊D₅)	D₄	1	2	yes	—
33	C₄×C₅⋊D₅	—	4	2	no	—
34	#6045	—	2	1	no	—
35	#6045	—	2	1	no	—
36	C₅²×C₈	—	2	1	no	—
37	C₅²×C₈	—	2	1	no	—
38	C₂×D₄₀	—	5	6	no	—
39	C₂×#677	—	5	6	no	—
40	#4021	—	5	6	no	—
41	C₄×D₂₀	—	5	6	no	—
42	C₅×C₄⋊C₈	D₅	1	6	yes	complemented
43	#3953	—	5	6	no	—
44	#3948	—	5	6	no	—
45	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₅⋊D₅	—	2	2	no	—
47	C₂×C₅⋊D₅	—	4	1	no	—
48	C₂×C₅⋊D₅	—	4	1	no	—
49	C₂×C₅⋊D₅	—	4	1	no	—
50	C₂×C₅⋊D₅	—	4	1	no	—
51	C₂×C₅⋊D₅	—	4	1	no	—
52	C₄×C₅²	—	2	1	no	—
53	C₄×C₅²	—	2	1	no	—
54	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
55	C₄×C₅²	#41	1	1	yes	characteristic
56	#1039	—	4	1	no	—
57	#1039	—	4	1	no	—
58	C₂×D₂₀	—	5	6	no	—
59	C₂×D₂₀	—	5	6	no	—
60	C₂×C₄×D₅	—	10	6	no	—
61	C₂×Dic₁₀	—	5	6	no	—
62	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
63	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
64	C₄²×C₅	C₂×D₅	1	6	yes	—
65	#685	—	10	6	no	—
66	#685	—	10	6	no	—
67	#685	—	10	6	no	—
68	C₄⋊Dic₅	—	5	6	no	—
69	D₄₀	—	10	12	no	—
70	#677	—	10	12	no	—
71	#463	—	25	1	no	Sylow
72	C₂×C₅²	#121	1	1	yes	characteristic
73	C₂×C₅²	#135	1	1	yes	characteristic
74	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₅⋊D₅	—	4	2	no	—
76	C₅⋊D₅	—	8	1	no	—
77	C₂²×D₅	—	10	6	no	—
78	C₂²×D₅	—	10	6	no	—
79	C₂×C₄×C₅	D₂₀	1	6	yes	—
80	C₂×C₄×C₅	D₂₀	1	6	yes	—
81	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
82	C₂×Dic₅	—	10	6	no	—
83	C₂×Dic₅	—	10	6	no	—
84	D₂₀	—	5	12	no	—
85	D₂₀	—	10	6	no	—
86	D₂₀	—	10	6	no	—
87	D₂₀	—	10	6	no	—
88	C₄×D₅	—	20	12	no	—
89	Dic₁₀	—	10	6	no	—
90	Dic₁₀	—	10	6	no	—
91	C₅×C₈	—	2	6	no	—
92	C₅×C₈	—	2	6	no	—
93	C₂×SD₁₆	—	25	1	no	—
94	C₂×D₈	—	25	1	no	—
95	#124	—	25	1	no	—
96	C₄×D₄	—	25	1	no	—
97	C₄⋊C₈	C₅⋊D₅	1	1	yes	characteristic, complemented
98	#103	—	25	1	no	—
99	#102	—	25	1	no	—
100	C₅²	#463	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₅	C₂×D₂₀	1	6	yes	—
102	C₂×D₅	—	10	12	no	—
103	C₂×D₅	—	20	6	no	—
104	C₂×D₅	—	20	6	no	—
105	C₂×D₅	—	20	6	no	—
106	C₂×D₅	—	20	6	no	—
107	C₂×D₅	—	20	6	no	—
108	C₄×C₅	—	2	6	no	—
109	C₄×C₅	—	2	6	no	—
110	C₄×C₅	D₄×D₅	1	6	yes	—
111	C₄×C₅	#713	1	6	yes	—
112	Dic₅	—	20	6	no	—
113	Dic₅	—	20	6	no	—
114	C₂×Q₈	—	25	1	no	—
115	C₂×D₄	—	25	1	no	—
116	C₂×D₄	—	25	1	no	—
117	C₂²×C₄	—	50	1	no	—
118	SD₁₆	—	50	2	no	—
119	D₈	—	50	2	no	—
120	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
121	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
122	C₄⋊C₄	—	25	1	no	—
123	C₂²⋊C₄	—	50	1	no	—
124	C₂²⋊C₄	—	50	1	no	—
125	C₂²⋊C₄	—	50	1	no	—
126	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
127	C₂×C₅	#4041	1	6	yes	—
128	C₂×C₅	#4050	1	6	yes	—
129	C₂×C₅	#4054	1	6	yes	—
130	D₅	—	20	12	no	—
131	D₅	—	40	6	no	—
132	C₂³	—	50	1	no	—
133	C₂³	—	50	1	no	—
134	Q₈	—	50	1	no	—
135	Q₈	—	50	1	no	—
136	D₄	—	25	2	no	—
137	D₄	—	50	1	no	—
138	D₄	—	50	1	no	—
139	D₄	—	50	1	no	—
140	C₂×C₄	—	50	1	no	—
141	C₂×C₄	—	50	1	no	—
142	C₂×C₄	—	100	2	no	—
143	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
144	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₅	#65133	1	6	yes	complemented
149	C₂²	—	50	2	no	—
150	C₂²	—	100	1	no	—
151	C₂²	—	100	1	no	—
152	C₂²	—	100	1	no	—
153	C₂²	—	100	1	no	—
154	C₂²	—	100	1	no	—
155	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	100	1	no	—
159	C₄	—	100	1	no	—
160	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
161	C₄	#87904	1	1	yes	characteristic
162	C₂	—	100	2	no	—
163	C₂	—	200	1	no	—
164	C₂	#11717215	1	1	yes	characteristic
165	C₂	#11717224	1	1	yes	characteristic
166	C₂	#11717228	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×C₄⋊C₈	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×#87801	C₂
hypercenter	C₄⋊C₈	C₅⋊D₅
hypocenter	C₅²	#463
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#87800	C₂	semidirect
3. #11717195	C₂	semidirect
4. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
5. C₅²×C₄⋊C₈	C₂	semidirect
6. #11716688	C₂	semidirect
7. #11716683	C₂	semidirect
42. C₅×C₄⋊C₈	D₅	semidirect
97. C₄⋊C₈	C₅⋊D₅	semidirect
100. C₅²	#463	semidirect
148. C₅	#65133	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87801	2
C₂×#87800	2
#11717195	2
C₄×C₄⋊(C₅⋊D₅)	2
C₅²×C₄⋊C₈	2
#11716688	2
#11716683	2
#65133	5