Finite Group Databasebeta

Subgroups — #13103577

Order 1600.

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

normal subgroups only

This group has 201 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	#11716955	C₂	1	1	yes	maximal, characteristic
2	C₄×#6032	C₂	1	1	yes	maximal, characteristic
3	C₅×#4004	C₂	1	1	yes	maximal, characteristic
4	#11716760	C₂	1	1	yes	maximal, characteristic
5	#11716759	C₂	1	1	yes	maximal, characteristic
6	#11716752	C₂	1	1	yes	maximal, characteristic
7	#11716752	C₂	1	1	yes	maximal, characteristic
8	C₂×#6039	C₄	1	1	yes	characteristic
9	C₂×#6032	C₂²	1	1	yes	characteristic
10	C₂×#6032	C₂²	1	1	yes	characteristic
11	C₂×C₅×#170	C₂²	1	1	yes	characteristic
12	C₂×C₅×#170	C₂²	1	1	yes	characteristic
13	#87806	C₄	1	1	yes	characteristic
14	C₅×#683	C₄	1	1	yes	characteristic
15	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	#87780	C₄	1	1	yes	characteristic
17	Dic₅²	C₂²	1	1	yes	characteristic
18	Dic₅²	C₂²	1	1	yes	characteristic
19	#65707	—	5	1	no	maximal
20	#65120	—	5	1	no	maximal
21	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
22	C₂×#1039	C₂×C₄	1	1	yes	characteristic
23	C₂×#1039	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
28	#6039	—	2	2	no	—
29	#6032	—	2	1	no	—
30	#6032	Q₈	1	2	yes	complemented
31	C₅×#170	—	2	1	no	—
32	C₅×#170	—	2	1	no	—
33	#4033	—	5	1	no	—
34	C₄×Dic₁₀	—	5	1	no	—
35	#4004	—	5	1	no	—
36	#4004	D₅	1	1	yes	characteristic
37	#4002	—	5	1	no	—
38	#4002	—	5	1	no	—
39	C₄⋊#170	—	5	1	no	—
40	C₄×#170	—	5	1	no	—
41	C₅×C₄⋊C₈	—	5	1	no	—
42	#3947	—	5	1	no	—
43	#3947	—	5	1	no	—
44	#3946	—	5	1	no	—
45	#3946	—	5	1	no	—
46	C₄×#168	—	5	1	no	—
47	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
48	C₄×C₅²	—	2	1	no	—
49	#1039	—	2	1	no	—
50	#1039	—	2	1	no	—
51	C₅×Dic₅	—	2	1	no	—
52	C₅×Dic₅	—	2	1	no	—
53	C₅×Dic₅	M₁₆	1	2	yes	—
54	C₅×Dic₅	C₂×Q₈	1	1	yes	characteristic
55	C₅×Dic₅	#41	1	1	yes	characteristic
56	C₂×Dic₁₀	—	5	1	no	—
57	C₂×Dic₁₀	—	5	1	no	—
58	C₂×#170	—	5	1	no	—
59	C₂×#170	—	5	1	no	—
60	C₂×#170	C₂×D₅	1	1	yes	characteristic
61	C₂×#170	C₂×D₅	1	1	yes	characteristic
62	C₂×C₅×C₈	—	5	1	no	—
63	C₂×C₅×C₈	—	5	1	no	—
64	C₅×C₄⋊C₄	—	5	1	no	—
65	C₄²×C₅	—	5	1	no	—
66	C₄⋊Dic₅	—	5	1	no	—
67	#683	—	5	1	no	—
68	#683	—	5	1	no	—
69	#683	—	5	1	no	—
70	#683	—	20	1	no	—
71	#683	AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄×Dic₅	—	5	1	no	—
73	C₄×Dic₅	—	5	1	no	—
74	C₄×Dic₅	—	5	1	no	—
75	C₄×Dic₅	—	5	1	no	—
76	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
77	C₂×#168	—	5	1	no	—
78	C₂×#168	—	5	1	no	—
79	#446	—	25	1	no	Sylow
80	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₅²	#131	1	1	yes	characteristic
83	C₂×C₄×C₅	—	4	1	no	—
84	C₂×C₄×C₅	—	5	1	no	—
85	C₂×C₄×C₅	—	5	1	no	—
86	C₂×C₄×C₅	—	5	1	no	—
87	C₂×C₄×C₅	—	5	1	no	—
88	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
90	C₂×Dic₅	—	5	1	no	—
91	C₂×Dic₅	—	5	1	no	—
92	C₂×Dic₅	—	5	1	no	—
93	C₂×Dic₅	—	5	1	no	—
94	C₂×Dic₅	—	20	1	no	—
95	C₂×Dic₅	—	20	1	no	—
96	C₂×Dic₅	C₄×D₅	1	1	yes	characteristic
97	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
100	Dic₁₀	—	10	2	no	—
101	Dic₁₀	—	10	2	no	—
102	#170	—	2	1	no	—
103	#170	—	2	1	no	—
104	#170	—	5	2	no	—
105	#170	—	10	1	no	—
106	C₅×C₈	—	10	1	no	—
107	C₅×C₈	—	10	1	no	—
108	#168	—	5	2	no	—
109	#168	—	10	1	no	—
110	C₄×Q₈	—	25	1	no	—
111	C₄⋊C₈	—	5	1	no	—
112	C₄⋊C₈	—	25	1	no	—
113	C₄⋊C₈	—	25	1	no	—
114	#97	—	25	1	no	—
115	#97	—	25	1	no	—
116	C₄×C₈	—	25	1	no	—
117	C₅²	#446	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₅	—	4	1	no	—
119	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
120	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₄×C₅	—	2	1	no	—
122	C₄×C₅	—	2	1	no	—
123	C₄×C₅	—	5	2	no	—
124	C₄×C₅	—	5	1	no	—
125	C₄×C₅	—	5	1	no	—
126	C₄×C₅	—	8	1	no	—
127	C₄×C₅	—	10	1	no	—
128	C₄×C₅	—	10	1	no	—
129	Dic₅	—	2	1	no	—
130	Dic₅	—	2	1	no	—
131	Dic₅	—	10	1	no	—
132	Dic₅	—	10	1	no	—
133	Dic₅	—	10	1	no	—
134	Dic₅	—	10	1	no	—
135	Dic₅	—	20	2	no	—
136	Dic₅	—	40	1	no	—
137	Dic₅	#676	1	2	yes	—
138	Dic₅	#711	1	1	yes	characteristic
139	Dic₅	Q₈×D₅	1	1	yes	characteristic
140	C₂×Q₈	—	25	1	no	—
141	C₂×C₈	—	5	1	no	—
142	C₂×C₈	—	5	1	no	—
143	C₂×C₈	—	25	1	no	—
144	C₂×C₈	—	25	1	no	—
145	C₄⋊C₄	—	5	1	no	—
146	C₄⋊C₄	—	25	1	no	—
147	C₄⋊C₄	—	25	1	no	—
148	C₄²	—	5	1	no	—
149	C₄²	—	25	1	no	—
150	C₄²	—	25	1	no	—
151	C₂×C₅	—	4	1	no	—
152	C₂×C₅	—	4	1	no	—
153	C₂×C₅	—	4	1	no	—
154	C₂×C₅	#4033	1	1	yes	characteristic
155	C₂×C₅	C₂×#676	1	1	yes	characteristic
156	C₂×C₅	#4053	1	1	yes	characteristic
157	C₂×C₅	D₅⋊M₁₆	1	1	yes	characteristic
158	C₂×C₅	#4131	1	1	yes	characteristic
159	C₂×C₅	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
160	Q₈	—	50	2	no	—
161	C₂×C₄	—	5	1	no	—
162	C₂×C₄	—	5	1	no	—
163	C₂×C₄	—	5	1	no	—
164	C₂×C₄	—	5	1	no	—
165	C₂×C₄	—	25	1	no	—
166	C₂×C₄	—	25	1	no	—
167	C₂×C₄	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₈	—	10	1	no	—
169	C₈	—	10	1	no	—
170	C₈	—	25	2	no	—
171	C₈	—	50	1	no	—
172	C₅	—	4	1	no	—
173	C₅	#65120	1	1	yes	characteristic, complemented
174	C₅	#65707	1	1	yes	characteristic, complemented
175	C₂²	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	5	1	no	—
178	C₄	—	5	1	no	—
179	C₄	—	5	2	no	—
180	C₄	—	10	1	no	—
181	C₄	—	10	1	no	—
182	C₄	—	50	1	no	—
183	C₄	—	50	1	no	—
184	C₂	#1042⋊Q₈	1	1	yes	characteristic
185	C₂	D₅⋊#676	1	1	yes	characteristic
186	C₂	#11717345	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₄×C₅²	C₂×C₄
cosocle	C₂×C₅×Dic₅	C₂³
hypercenter	C₂×C₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	#446
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
30. #6032	Q₈	semidirect
117. C₅²	#446	semidirect
173. C₅	#65120	semidirect
174. C₅	#65707	semidirect

Maximal subgroups

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

subgroup	index
#11716955	2
C₄×#6032	2
C₅×#4004	2
#11716760	2
#11716759	2
#11716752	2
#65707	5
#65120	5