Finite Group Databasebeta

Subgroups — #13103598

Order 1600.

207 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 68 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×#87767	C₂	1	1	yes	maximal, characteristic
2	#11716841	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3998	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716749	C₂	1	2	yes	maximal, complemented
5	#11716747	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₄	1	1	yes	characteristic
7	C₂×C₅×#168	C₄	1	1	yes	characteristic
8	C₂×C₅×#168	C₂²	1	1	yes	characteristic
9	#87767	C₄	1	1	yes	characteristic
10	#87767	C₄	1	1	yes	characteristic
11	#87767	C₂²	1	1	yes	characteristic
12	#87767	C₂²	1	1	yes	characteristic
13	#87763	C₂²	1	2	yes	complemented
14	C₅×#674	C₂²	1	2	yes	complemented
15	D₅⋊M₃₂	—	5	1	no	maximal
16	D₅×M₃₂	—	5	1	no	maximal
17	C₂²×C₅⋊D₅	C₈	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×#1039	C₈	1	1	yes	characteristic
20	C₄×C₅⋊D₅	C₈	1	1	yes	characteristic
21	C₄×C₅⋊D₅	C₈	1	1	yes	characteristic
22	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
24	C₅×#168	C₂×C₄	1	1	yes	characteristic
25	C₅×#168	C₂×C₄	1	1	yes	characteristic
26	C₅×#168	C₂×C₄	1	1	yes	characteristic
27	C₅×#168	C₂³	1	1	yes	characteristic
28	C₂×C₈×D₅	—	5	1	no	—
29	C₂×C₈×D₅	—	5	1	no	—
30	#3998	—	5	1	no	—
31	#3998	D₅	1	1	yes	characteristic, complemented
32	#3990	—	5	2	no	—
33	D₅⋊C₁₆	—	5	2	no	—
34	C₅×M₃₂	—	5	1	no	—
35	#3944	—	5	1	no	—
36	#3930	—	5	2	no	—
37	D₅×C₁₆	—	5	2	no	—
38	C₂²×C₅²	C₂×C₈	1	1	yes	characteristic
39	C₂×C₅⋊D₅	—	2	2	no	—
40	C₂×C₅⋊D₅	C₂×C₈	1	1	yes	characteristic
41	C₂×C₅⋊D₅	C₂×C₈	1	1	yes	characteristic
42	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
43	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
44	#1039	C₂×C₈	1	1	yes	characteristic
45	#1039	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×D₅	—	5	1	no	—
47	C₂×C₄×D₅	—	5	1	no	—
48	C₂×C₄×D₅	—	20	1	no	—
49	C₂×C₅×C₈	—	5	1	no	—
50	C₂×C₅×C₈	—	5	1	no	—
51	C₂×#168	AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂×#168	C₂×D₅	1	1	yes	characteristic
53	C₈×D₅	—	5	1	no	—
54	C₈×D₅	—	5	1	no	—
55	C₈×D₅	—	5	1	no	—
56	C₈×D₅	—	5	1	no	—
57	C₈×D₅	—	5	1	no	—
58	C₈×D₅	—	5	1	no	—
59	C₈×D₅	—	5	1	no	—
60	C₈×D₅	—	5	1	no	—
61	#674	—	5	2	no	—
62	#674	C₂×D₅	1	2	yes	complemented
63	C₅×C₁₆	—	5	2	no	—
64	#672	—	5	2	no	—
65	C₂×M₃₂	—	25	1	no	Sylow
66	C₂×C₅²	—	2	1	no	—
67	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
68	C₅⋊D₅	—	2	1	no	—
69	C₅⋊D₅	M₃₂	1	2	yes	complemented
70	C₂²×D₅	—	5	1	no	—
71	C₂²×D₅	—	5	1	no	—
72	C₂²×D₅	—	20	1	no	—
73	C₂×C₄×C₅	—	4	1	no	—
74	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₂×Dic₅	—	5	1	no	—
77	C₂×Dic₅	—	5	1	no	—
78	C₂×Dic₅	—	20	1	no	—
79	C₄×D₅	—	5	1	no	—
80	C₄×D₅	—	5	1	no	—
81	C₄×D₅	—	5	1	no	—
82	C₄×D₅	—	5	1	no	—
83	C₄×D₅	—	5	1	no	—
84	C₄×D₅	—	5	1	no	—
85	C₄×D₅	—	5	1	no	—
86	C₄×D₅	—	5	1	no	—
87	C₄×D₅	—	20	1	no	—
88	C₄×D₅	—	20	1	no	—
89	C₄×D₅	—	20	1	no	—
90	C₄×D₅	—	20	1	no	—
91	C₅×C₈	—	5	1	no	—
92	C₅×C₈	—	5	1	no	—
93	C₅×C₈	—	5	1	no	—
94	C₅×C₈	—	5	1	no	—
95	#168	C₄×D₅	1	1	yes	characteristic
96	#168	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
97	#168	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	#168	C₂²×D₅	1	1	yes	characteristic
99	C₂²×C₈	—	25	1	no	—
100	M₃₂	—	5	1	no	—
101	M₃₂	—	25	2	no	—
102	M₃₂	—	25	1	no	—
103	C₂×C₁₆	—	25	2	no	—
104	C₅²	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₅	—	4	1	no	—
106	C₂²×C₅	C₈×D₅	1	1	yes	characteristic
107	C₂²×C₅	D₅⋊C₈	1	1	yes	characteristic
108	C₂×D₅	—	5	1	no	—
109	C₂×D₅	—	5	1	no	—
110	C₂×D₅	—	5	1	no	—
111	C₂×D₅	—	5	1	no	—
112	C₂×D₅	—	10	2	no	—
113	C₂×D₅	—	10	2	no	—
114	C₂×D₅	—	20	1	no	—
115	C₂×D₅	—	20	1	no	—
116	C₂×D₅	—	20	4	no	—
117	C₄×C₅	—	4	1	no	—
118	C₄×C₅	—	4	1	no	—
119	C₄×C₅	C₈×D₅	1	1	yes	characteristic
120	C₄×C₅	D₅⋊C₈	1	1	yes	characteristic
121	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
122	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
123	Dic₅	—	5	1	no	—
124	Dic₅	—	5	1	no	—
125	Dic₅	—	5	1	no	—
126	Dic₅	—	5	1	no	—
127	Dic₅	—	20	1	no	—
128	Dic₅	—	20	1	no	—
129	C₂²×C₄	—	25	1	no	—
130	C₂×C₈	—	5	1	no	—
131	C₂×C₈	—	5	1	no	—
132	C₂×C₈	—	25	1	no	—
133	C₂×C₈	—	25	1	no	—
134	C₂×C₈	—	25	1	no	—
135	C₂×C₈	—	25	1	no	—
136	C₁₆	—	5	2	no	—
137	C₁₆	—	25	2	no	—
138	C₂×C₅	—	2	1	no	—
139	C₂×C₅	—	2	1	no	—
140	C₂×C₅	—	4	2	no	—
141	C₂×C₅	—	4	1	no	—
142	C₂×C₅	C₂×C₈×D₅	1	1	yes	characteristic
143	C₂×C₅	C₂×D₅⋊C₈	1	1	yes	characteristic
144	D₅	—	5	2	no	—
145	D₅	—	5	2	no	—
146	D₅	—	10	1	no	—
147	D₅	—	10	1	no	—
148	D₅	—	20	2	no	—
149	D₅	—	20	2	no	—
150	C₂³	—	25	1	no	—
151	C₂×C₄	—	25	1	no	—
152	C₂×C₄	—	25	1	no	—
153	C₂×C₄	—	25	1	no	—
154	C₂×C₄	—	25	1	no	—
155	C₂×C₄	—	25	1	no	—
156	C₂×C₄	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₈	—	5	1	no	—
158	C₈	—	5	1	no	—
159	C₈	—	5	1	no	—
160	C₈	—	5	1	no	—
161	C₅	—	4	1	no	—
162	C₅	D₅×M₃₂	1	1	yes	characteristic, complemented
163	C₅	D₅⋊M₃₂	1	1	yes	characteristic, complemented
164	C₂²	—	25	1	no	—
165	C₂²	—	25	1	no	—
166	C₂²	—	50	2	no	—
167	C₂²	#87827	1	1	yes	characteristic
168	C₄	—	25	1	no	—
169	C₄	—	25	1	no	—
170	C₄	#87827	1	1	yes	characteristic
171	C₄	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	25	2	no	—
174	C₂	—	50	1	no	—
175	C₂	C₂×#87827	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₅×AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₂²×C₈
Fitting	C₂×C₄×C₅²	C₂×C₄
prime radical	C₂²×C₅⋊D₅	C₈
cosocle	C₅×#168	C₂³
hypercenter	C₂×C₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	C₂×M₃₂
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. #11716841	C₂	semidirect
3. C₅×#3998	C₂	semidirect
4. #11716749	C₂	semidirect
5. #11716747	C₂	semidirect
13. #87763	C₂²	semidirect
14. C₅×#674	C₂²	semidirect
31. #3998	D₅	semidirect
62. #674	C₂×D₅	semidirect
69. C₅⋊D₅	M₃₂	semidirect
104. C₅²	C₂×M₃₂	semidirect
162. C₅	D₅×M₃₂	semidirect
163. C₅	D₅⋊M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87767	2
#11716841	2
C₅×#3998	2
#11716749	2
#11716747	2
D₅⋊M₃₂	5
D₅×M₃₂	5