Finite Group Databasebeta

Subgroups — #13101034

Order 1600.

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

normal subgroups only

This group has 220 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	#11717037	C₂	1	1	yes	maximal, characteristic
2	#11716991	C₂	1	1	yes	maximal, characteristic
3	#11716682	C₂	1	1	yes	maximal, characteristic
4	C₅×#3950	C₂	1	1	yes	maximal, characteristic
5	C₅×#3949	C₂	1	1	yes	maximal, characteristic
6	#11716617	C₂	1	1	yes	maximal, characteristic
7	#11716615	C₂	1	1	yes	maximal, characteristic
8	C₂×#6039	—	2	1	no	—
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	C₄×#1039	C₂²	1	1	yes	characteristic
11	C₂×#6029	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	#87781	—	2	1	no	—
17	#87780	—	2	1	no	—
18	#87780	—	2	1	no	—
19	#65165	—	5	1	no	maximal
20	#65145	—	5	1	no	maximal
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×#1039	D₄	1	1	yes	characteristic
23	C₂×#1039	D₄	1	1	yes	characteristic
24	C₂×C₅×Dic₅	—	2	1	no	—
25	C₂×C₅×Dic₅	—	2	1	no	—
26	C₂×C₅×Dic₅	—	2	1	no	—
27	C₂×C₅×Dic₅	—	2	1	no	—
28	#6039	—	4	2	no	—
29	C₅²×C₈	Q₈	1	2	yes	complemented
30	#6029	Q₈	1	2	yes	complemented
31	#4036	—	5	1	no	—
32	#4036	—	5	1	no	—
33	#4034	—	5	1	no	—
34	#4034	—	5	1	no	—
35	C₅×#107	—	5	1	no	—
36	C₅×#106	—	5	1	no	—
37	#3950	D₅	1	1	yes	characteristic
38	#3949	D₅	1	1	yes	characteristic
39	#3947	—	5	1	no	—
40	#3947	—	5	1	no	—
41	#3947	—	10	2	no	—
42	#3940	—	5	1	no	—
43	#3940	—	5	1	no	—
44	#3939	—	5	1	no	—
45	#3939	—	5	1	no	—
46	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
47	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
48	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
49	#1039	—	2	1	no	—
50	#1039	—	2	1	no	—
51	C₅×Dic₅	—	4	1	no	—
52	C₅×Dic₅	—	4	1	no	—
53	C₅×Dic₅	—	4	1	no	—
54	C₅×Dic₅	—	4	1	no	—
55	C₂×Dic₁₀	—	10	1	no	—
56	C₂×Dic₁₀	—	10	1	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
60	C₅×C₄⋊C₄	—	5	1	no	—
61	C₅×C₄⋊C₄	—	5	1	no	—
62	C₅×C₄⋊C₄	—	5	1	no	—
63	C₅×C₄⋊C₄	—	5	1	no	—
64	C₄⋊Dic₅	—	10	1	no	—
65	C₄⋊Dic₅	—	10	1	no	—
66	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
67	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
68	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
69	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
70	#683	—	10	1	no	—
71	#683	—	10	1	no	—
72	#683	—	10	1	no	—
73	#683	—	10	1	no	—
74	C₄×Dic₅	—	5	1	no	—
75	C₄×Dic₅	—	5	1	no	—
76	C₄×Dic₅	—	10	2	no	—
77	C₂×#168	—	5	1	no	—
78	C₂×#168	—	5	1	no	—
79	C₂×#168	—	10	2	no	—
80	#501	—	25	1	no	Sylow
81	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₅²	#137	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	10	1	no	—
86	C₂×C₄×C₅	—	10	1	no	—
87	C₂×C₄×C₅	—	10	1	no	—
88	C₂×C₄×C₅	—	10	1	no	—
89	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
91	C₂×Dic₅	—	2	1	no	—
92	C₂×Dic₅	—	2	1	no	—
93	C₂×Dic₅	—	2	1	no	—
94	C₂×Dic₅	—	2	1	no	—
95	C₂×Dic₅	—	5	1	no	—
96	C₂×Dic₅	—	5	1	no	—
97	C₂×Dic₅	—	5	1	no	—
98	C₂×Dic₅	—	5	1	no	—
99	C₂×Dic₅	—	10	2	no	—
100	C₂×Dic₅	—	10	2	no	—
101	Dic₁₀	—	20	2	no	—
102	Dic₁₀	—	20	2	no	—
103	C₅×C₈	—	2	4	no	—
104	C₅×C₈	Dic₁₀	1	2	yes	complemented
105	C₅×C₈	Dic₁₀	1	2	yes	complemented
106	#168	—	5	2	no	—
107	#168	—	5	2	no	—
108	#168	—	10	4	no	—
109	C₄⋊Q₈	—	25	1	no	—
110	#125	—	25	1	no	—
111	#107	—	5	1	no	—
112	#107	—	25	1	no	—
113	#106	—	5	1	no	—
114	#106	—	25	1	no	—
115	#97	—	25	1	no	—
116	C₅²	#501	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₅	—	2	2	no	—
118	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
119	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
120	C₄×C₅	—	2	2	no	—
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	20	1	no	—
123	C₄×C₅	—	20	1	no	—
124	C₄×C₅	—	20	1	no	—
125	C₄×C₅	—	20	1	no	—
126	C₄×C₅	C₂×Dic₁₀	1	1	yes	characteristic
127	C₄×C₅	C₂×Dic₁₀	1	1	yes	characteristic
128	C₄×C₅	Q₈×D₅	1	1	yes	characteristic
129	C₄×C₅	Q₈×D₅	1	1	yes	characteristic
130	Dic₅	—	4	1	no	—
131	Dic₅	—	4	1	no	—
132	Dic₅	—	4	1	no	—
133	Dic₅	—	4	1	no	—
134	Dic₅	—	10	1	no	—
135	Dic₅	—	10	1	no	—
136	Dic₅	—	10	1	no	—
137	Dic₅	—	10	1	no	—
138	Dic₅	—	20	2	no	—
139	Dic₅	—	20	2	no	—
140	C₂×Q₈	—	50	1	no	—
141	C₂×C₈	—	25	1	no	—
142	C₂×C₈	D₅²	1	1	yes	characteristic
143	C₄⋊C₄	—	5	1	no	—
144	C₄⋊C₄	—	5	1	no	—
145	C₄⋊C₄	—	5	1	no	—
146	C₄⋊C₄	—	5	1	no	—
147	C₄⋊C₄	—	50	1	no	—
148	C₄⋊C₄	—	50	1	no	—
149	C₄⋊C₄	—	50	1	no	—
150	C₄²	—	25	1	no	—
151	C₂×C₅	—	2	2	no	—
152	C₂×C₅	—	2	2	no	—
153	C₂×C₅	—	2	2	no	—
154	C₂×C₅	#4034	1	1	yes	characteristic
155	C₂×C₅	#4034	1	1	yes	characteristic
156	C₂×C₅	#4057	1	1	yes	characteristic
157	C₂×C₅	#4060	1	1	yes	characteristic
158	C₂×C₅	#4061	1	1	yes	characteristic
159	C₂×C₅	#4064	1	1	yes	characteristic
160	Q₈	—	100	2	no	—
161	C₂×C₄	—	10	1	no	—
162	C₂×C₄	—	10	1	no	—
163	C₂×C₄	—	10	1	no	—
164	C₂×C₄	—	10	1	no	—
165	C₂×C₄	—	25	1	no	—
166	C₂×C₄	—	25	1	no	—
167	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
168	C₈	—	25	2	no	—
169	C₈	#6039	1	2	yes	complemented
170	C₅	—	2	2	no	—
171	C₅	#65145	1	1	yes	characteristic, complemented
172	C₅	#65165	1	1	yes	characteristic, complemented
173	C₂²	#87877	1	1	yes	characteristic
174	C₄	—	20	1	no	—
175	C₄	—	20	1	no	—
176	C₄	—	20	1	no	—
177	C₄	—	20	1	no	—
178	C₄	—	50	1	no	—
179	C₄	—	50	1	no	—
180	C₄	#87872	1	1	yes	characteristic
181	C₄	C₂×#6039	1	1	yes	characteristic
182	C₂	#11716916	1	1	yes	characteristic
183	C₂	#11716929	1	1	yes	characteristic
184	C₂	#11717037	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#87877
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₂×C₅²×C₈	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#501
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
29. C₅²×C₈	Q₈	semidirect
30. #6029	Q₈	semidirect
104. C₅×C₈	Dic₁₀	semidirect
105. C₅×C₈	Dic₁₀	semidirect
116. C₅²	#501	semidirect
169. C₈	#6039	semidirect
171. C₅	#65145	semidirect
172. C₅	#65165	semidirect

Maximal subgroups

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

subgroup	index
#11717037	2
#11716991	2
#11716682	2
C₅×#3950	2
C₅×#3949	2
#11716617	2
#11716615	2
#65165	5
#65145	5