Finite Group Databasebeta

Subgroups — #13098390

Order 1600.

186 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₂²×C₈×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
2	#11716492	C₂	1	1	yes	maximal, characteristic
3	#11716486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87713	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716369	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716366	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716365	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₂₅	—	2	1	no	—
10	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₅	—	2	2	no	—
13	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
15	C₂²⋊Dic₂₅	—	2	1	no	—
16	#87720	—	2	1	no	—
17	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
19	#87718	—	2	1	no	—
20	#87713	—	2	2	no	—
21	#65403	—	5	1	no	maximal
22	C₂³×C₂₅	D₄	1	1	yes	characteristic
23	C₂²×D₂₅	—	2	1	no	—
24	C₂×C₄×C₂₅	—	2	1	no	—
25	C₂×C₄×C₂₅	—	2	1	no	—
26	C₂×C₄×C₂₅	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
28	C₂²⋊D₂₅	—	4	2	no	—
29	C₂×Dic₂₅	—	2	1	no	—
30	C₂×Dic₂₅	—	2	1	no	—
31	C₂×Dic₂₅	—	2	1	no	—
32	D₁₀₀	—	2	1	no	—
33	D₁₀₀	—	2	1	no	—
34	Dic₅₀	—	2	1	no	—
35	Dic₅₀	—	2	1	no	—
36	C₈×C₂₅	—	2	1	no	—
37	C₈×C₂₅	D₄	1	2	yes	complemented
38	C₂²×C₅×C₈	D₅	1	1	yes	characteristic
39	#4076	—	5	1	no	—
40	#4070	—	5	1	no	—
41	C₂×#677	—	5	1	no	—
42	#3953	—	5	1	no	—
43	#3949	—	5	1	no	—
44	#3948	—	5	1	no	—
45	C₂²×C₂₅	—	2	1	no	—
46	C₂²×C₂₅	—	2	1	no	—
47	C₂²×C₂₅	SD₁₆	1	2	yes	complemented
48	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
49	C₂×D₂₅	—	4	1	no	—
50	C₂×D₂₅	—	4	1	no	—
51	C₂×D₂₅	—	4	1	no	—
52	C₄×C₂₅	—	2	1	no	—
53	C₄×C₂₅	C₂×D₄	1	1	yes	characteristic
54	C₄×C₂₅	#41	1	1	yes	characteristic
55	Dic₂₅	—	4	1	no	—
56	Dic₂₅	—	4	1	no	—
57	Dic₂₅	—	4	1	no	—
58	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
59	C₂×C₂²⋊D₅	—	10	1	no	—
60	C₂×D₂₀	—	5	1	no	—
61	C₂×Dic₁₀	—	5	1	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
65	C₂²⋊Dic₅	—	10	1	no	—
66	#685	—	10	1	no	—
67	C₄⋊Dic₅	—	5	1	no	—
68	C₄⋊Dic₅	—	5	1	no	—
69	#683	—	10	1	no	—
70	#677	—	10	2	no	—
71	#465	—	25	1	no	Sylow
72	C₂×C₂₅	—	2	2	no	—
73	C₂×C₂₅	#121	1	1	yes	characteristic
74	C₂×C₂₅	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₂₅	#135	1	1	yes	characteristic
76	D₂₅	—	8	1	no	—
77	C₂³×C₅	D₂₀	1	1	yes	characteristic
78	C₂²×D₅	—	10	1	no	—
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
83	C₂²⋊D₅	—	20	2	no	—
84	C₂×Dic₅	—	10	1	no	—
85	C₂×Dic₅	—	10	1	no	—
86	C₂×Dic₅	—	10	1	no	—
87	D₂₀	—	10	1	no	—
88	D₂₀	—	10	1	no	—
89	Dic₁₀	—	10	1	no	—
90	Dic₁₀	—	10	1	no	—
91	C₅×C₈	—	2	1	no	—
92	C₅×C₈	C₂²⋊D₅	1	2	yes	—
93	C₂×SD₁₆	—	25	1	no	—
94	C₂²×C₈	D₂₅	1	1	yes	characteristic, complemented
95	C₂²⋊Q₈	—	25	1	no	—
96	#121	—	25	1	no	—
97	#106	—	25	1	no	—
98	#103	—	25	1	no	—
99	#102	—	25	1	no	—
100	C₂₅	#465	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₅	—	2	1	no	—
102	C₂²×C₅	—	2	1	no	—
103	C₂²×C₅	#677	1	2	yes	—
104	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
105	C₂×D₅	—	20	1	no	—
106	C₂×D₅	—	20	1	no	—
107	C₂×D₅	—	20	1	no	—
108	C₄×C₅	—	2	1	no	—
109	C₄×C₅	#709	1	1	yes	characteristic
110	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
111	Dic₅	—	20	1	no	—
112	Dic₅	—	20	1	no	—
113	Dic₅	—	20	1	no	—
114	C₂×Q₈	—	25	1	no	—
115	C₂×D₄	—	25	1	no	—
116	C₂×D₄	—	50	1	no	—
117	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
118	SD₁₆	—	50	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	C₂×D₂₅	1	1	yes	characteristic
121	C₂×C₈	C₂×D₂₅	1	1	yes	characteristic
122	C₄⋊C₄	—	25	1	no	—
123	C₄⋊C₄	—	25	1	no	—
124	C₄⋊C₄	—	50	1	no	—
125	C₂²⋊C₄	—	50	1	no	—
126	C₂²⋊C₄	—	50	1	no	—
127	C₂×C₅	—	2	2	no	—
128	C₂×C₅	C₂×#677	1	1	yes	characteristic
129	C₂×C₅	#4050	1	1	yes	characteristic
130	C₂×C₅	#4076	1	1	yes	characteristic
131	D₅	—	40	1	no	—
132	C₂³	—	50	1	no	—
133	C₂³	D₁₀₀	1	1	yes	characteristic
134	Q₈	—	50	1	no	—
135	Q₈	—	50	1	no	—
136	D₄	—	50	1	no	—
137	D₄	—	50	1	no	—
138	D₄	—	100	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	50	1	no	—
142	C₂×C₄	—	50	1	no	—
143	C₂×C₄	—	50	1	no	—
144	C₂×C₄	D₁₀₀	1	1	yes	characteristic
145	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	C₂²⋊D₂₅	1	2	yes	complemented
148	C₅	#65403	1	1	yes	characteristic
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	100	1	no	—
152	C₂²	—	100	1	no	—
153	C₂²	—	100	1	no	—
154	C₂²	#87713	1	2	yes	complemented
155	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	100	1	no	—
158	C₄	—	100	1	no	—
159	C₄	—	100	1	no	—
160	C₄	#87744	1	1	yes	characteristic
161	C₄	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
162	C₂	—	2	2	no	—
163	C₂	—	200	1	no	—
164	C₂	C₂×#87713	1	1	yes	characteristic
165	C₂	#11716467	1	1	yes	characteristic
166	C₂	#11716492	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₀
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂²×C₈×C₂₅	C₂
socle	C₂²×C₅	C₂×D₂₀
prime radical	#11716492	C₂
hypercenter	C₂²×C₈	D₂₅
hypocenter	C₂₅	#465
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×C₂₅	C₂	semidirect
3. #11716486	C₂	semidirect
4. C₂×#87713	C₂	semidirect
5. #11716369	C₂	semidirect
6. #11716366	C₂	semidirect
7. #11716365	C₂	semidirect
37. C₈×C₂₅	D₄	semidirect
47. C₂²×C₂₅	SD₁₆	semidirect
94. C₂²×C₈	D₂₅	semidirect
100. C₂₅	#465	semidirect
147. C₈	C₂²⋊D₂₅	semidirect
154. C₂²	#87713	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×C₂₅	2
#11716492	2
#11716486	2
C₂×#87713	2
#11716369	2
#11716366	2
#11716365	2
#65403	5