Finite Group Databasebeta

Subgroups — #13100409

Order 1600.

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

normal subgroups only

This group has 194 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	#11716921	C₂	1	1	yes	maximal, characteristic
2	#11716665	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3930	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716584	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716583	C₂	1	1	yes	maximal, characteristic, complemented
7	D₅×#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#87873	C₄	1	1	yes	characteristic
9	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
10	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
12	#87769	C₄	1	1	yes	characteristic
13	#87768	C₄	1	1	yes	characteristic
14	#87768	C₄	1	1	yes	characteristic
15	C₅²×C₁₆	C₂²	1	1	yes	characteristic
16	#87756	C₂²	1	1	yes	characteristic
17	C₅×#672	C₂²	1	1	yes	characteristic
18	C₅×#672	C₂²	1	1	yes	characteristic
19	#65196	—	5	1	no	maximal
20	#65190	—	5	1	no	maximal
21	C₄×C₅⋊D₅	C₂×C₄	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	#6039	C₈	1	1	yes	characteristic
25	#6038	C₈	1	1	yes	characteristic
26	#6038	C₈	1	1	yes	characteristic
27	#6037	C₈	1	1	yes	characteristic
28	C₅²×C₈	C₂³	1	1	yes	characteristic
29	#6029	C₂×C₄	1	1	yes	characteristic
30	C₅×#168	C₂×C₄	1	1	yes	characteristic
31	C₅×#168	C₂×C₄	1	1	yes	characteristic
32	#4047	—	5	1	no	—
33	#4047	—	5	1	no	—
34	C₅×M₃₂	—	5	1	no	—
35	C₂×C₅×C₁₆	—	5	1	no	—
36	#3944	—	5	1	no	—
37	C₂×#672	—	5	1	no	—
38	#3930	—	5	1	no	—
39	#3930	—	5	1	no	—
40	#3930	—	5	1	no	—
41	#3930	—	10	2	no	—
42	#3930	D₅	1	1	yes	characteristic, complemented
43	D₅×C₁₆	—	5	1	no	—
44	D₅×C₁₆	—	5	1	no	—
45	D₅×C₁₆	—	5	1	no	—
46	D₅×C₁₆	D₅	1	1	yes	characteristic, complemented
47	C₂×C₅⋊D₅	C₂×C₈	1	1	yes	characteristic
48	C₂×C₅×D₅	C₂×C₈	1	1	yes	characteristic
49	C₂×C₅×D₅	C₂×C₈	1	1	yes	characteristic
50	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
51	#1039	C₂×C₈	1	1	yes	characteristic
52	C₅×Dic₅	C₂×C₈	1	1	yes	characteristic
53	C₅×Dic₅	C₂×C₈	1	1	yes	characteristic
54	#709	—	5	1	no	—
55	#709	—	5	1	no	—
56	C₂×C₅×C₈	—	5	1	no	—
57	C₂×C₅×C₈	—	5	1	no	—
58	#681	—	5	1	no	—
59	#681	—	5	1	no	—
60	#676	—	5	1	no	—
61	#676	—	5	1	no	—
62	#676	—	5	1	no	—
63	#676	—	5	1	no	—
64	C₈×D₅	—	5	1	no	—
65	C₈×D₅	—	5	1	no	—
66	C₈×D₅	—	10	2	no	—
67	C₈×D₅	C₂×D₅	1	1	yes	characteristic
68	C₈×D₅	C₂×D₅	1	1	yes	characteristic
69	C₅×C₁₆	—	2	2	no	—
70	C₅×C₁₆	—	5	1	no	—
71	C₅×C₁₆	—	5	1	no	—
72	C₅×C₁₆	C₂×D₅	1	1	yes	characteristic
73	C₅×C₁₆	C₂×D₅	1	1	yes	characteristic
74	#672	—	5	1	no	—
75	#672	—	5	1	no	—
76	#672	—	10	2	no	—
77	#672	C₂×D₅	1	1	yes	characteristic
78	#672	C₂×D₅	1	1	yes	characteristic
79	#504	—	25	1	no	Sylow
80	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
81	C₅⋊D₅	—	2	1	no	—
82	C₅×D₅	—	2	1	no	—
83	C₅×D₅	—	2	1	no	—
84	C₂×C₄×C₅	—	5	1	no	—
85	C₂×C₄×C₅	—	5	1	no	—
86	C₂²⋊D₅	—	5	1	no	—
87	C₂²⋊D₅	—	5	1	no	—
88	C₂²⋊D₅	—	5	1	no	—
89	C₂²⋊D₅	—	5	1	no	—
90	D₂₀	—	5	1	no	—
91	D₂₀	—	5	1	no	—
92	C₄×D₅	—	5	1	no	—
93	C₄×D₅	—	5	1	no	—
94	C₄×D₅	—	10	2	no	—
95	C₄×D₅	C₄×D₅	1	1	yes	characteristic
96	C₄×D₅	C₄×D₅	1	1	yes	characteristic
97	Dic₁₀	—	5	1	no	—
98	Dic₁₀	—	5	1	no	—
99	C₅×C₈	—	2	2	no	—
100	C₅×C₈	—	5	1	no	—
101	C₅×C₈	—	5	1	no	—
102	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
103	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
104	#168	—	5	1	no	—
105	#168	—	5	1	no	—
106	#168	—	10	2	no	—
107	#168	C₄×D₅	1	1	yes	characteristic
108	#168	C₄×D₅	1	1	yes	characteristic
109	#131	—	25	1	no	—
110	M₃₂	—	5	1	no	—
111	M₃₂	—	25	1	no	—
112	M₃₂	—	25	1	no	—
113	C₂×C₁₆	—	5	1	no	—
114	C₂×C₁₆	—	25	1	no	—
115	C₂×C₁₆	—	25	1	no	—
116	C₅²	#504	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₅	—	5	1	no	—
118	C₂²×C₅	—	5	1	no	—
119	C₂×D₅	—	5	1	no	—
120	C₂×D₅	—	5	1	no	—
121	C₂×D₅	—	10	2	no	—
122	C₂×D₅	C₈×D₅	1	1	yes	characteristic
123	C₂×D₅	C₈×D₅	1	1	yes	characteristic
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	5	1	no	—
126	C₄×C₅	—	5	1	no	—
127	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
128	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
129	Dic₅	—	5	1	no	—
130	Dic₅	—	5	1	no	—
131	Dic₅	—	10	2	no	—
132	Dic₅	C₈×D₅	1	1	yes	characteristic
133	Dic₅	C₈×D₅	1	1	yes	characteristic
134	#41	—	25	1	no	—
135	M₁₆	—	25	1	no	—
136	M₁₆	—	25	1	no	—
137	M₁₆	—	25	1	no	—
138	C₂×C₈	—	5	1	no	—
139	C₂×C₈	—	5	1	no	—
140	C₂×C₈	—	25	1	no	—
141	C₁₆	—	5	1	no	—
142	C₁₆	—	5	1	no	—
143	C₁₆	—	25	1	no	—
144	C₁₆	D₅²	1	1	yes	characteristic
145	C₂×C₅	—	2	2	no	—
146	C₂×C₅	—	10	1	no	—
147	C₂×C₅	—	10	1	no	—
148	C₂×C₅	C₂×C₈×D₅	1	1	yes	characteristic
149	C₂×C₅	C₂×C₈×D₅	1	1	yes	characteristic
150	D₅	—	2	1	no	—
151	D₅	—	2	1	no	—
152	D₅	—	10	1	no	—
153	D₅	—	10	1	no	—
154	D₅	—	20	2	no	—
155	Q₈	—	25	1	no	—
156	D₄	—	25	1	no	—
157	D₄	—	25	1	no	—
158	D₄	—	25	1	no	—
159	C₂×C₄	—	5	1	no	—
160	C₂×C₄	—	5	1	no	—
161	C₂×C₄	—	25	1	no	—
162	C₈	—	5	1	no	—
163	C₈	—	5	1	no	—
164	C₈	—	25	1	no	—
165	C₈	C₂×D₅²	1	1	yes	characteristic
166	C₅	—	2	2	no	—
167	C₅	#65190	1	1	yes	characteristic, complemented
168	C₅	#65196	1	1	yes	characteristic, complemented
169	C₂²	—	5	1	no	—
170	C₂²	—	5	1	no	—
171	C₂²	—	25	1	no	—
172	C₄	—	5	1	no	—
173	C₄	—	5	1	no	—
174	C₄	—	25	1	no	—
175	C₄	C₄×D₅²	1	1	yes	characteristic
176	C₂	—	10	1	no	—
177	C₂	—	10	1	no	—
178	C₂	—	50	1	no	—
179	C₂	C₈×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×D₅²
commutator, socle	C₂×C₅²	C₂²×C₈
Fitting	C₅²×C₁₆	C₂²
prime radical	#87873	C₄
cosocle	C₅²×C₈	C₂³
hypercenter	C₁₆	D₅²
hypocenter	C₅²	#504
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
2. #11716665	C₂	semidirect
3. C₅×#3930	C₂	semidirect
4. C₅×D₅×C₁₆	C₂	semidirect
5. #11716584	C₂	semidirect
6. #11716583	C₂	semidirect
7. D₅×#672	C₂	semidirect
42. #3930	D₅	semidirect
46. D₅×C₁₆	D₅	semidirect
116. C₅²	#504	semidirect
167. C₅	#65190	semidirect
168. C₅	#65196	semidirect

Maximal subgroups

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

subgroup	index
#11716921	2
#11716665	2
C₅×#3930	2
C₅×D₅×C₁₆	2
#11716584	2
#11716583	2
D₅×#672	2
#65196	5
#65190	5