Finite Group Databasebeta

Subgroups — #13103589

Order 1600.

214 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 74 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	D₅⋊#681	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#700	C₂	1	1	yes	maximal, characteristic
3	D₅×#681	C₂	1	1	yes	maximal, characteristic
4	#11716745	C₂	1	2	yes	maximal, complemented
5	#11716744	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
7	#87853	—	2	1	no	—
8	#87849	C₂²	1	1	yes	characteristic
9	#87849	C₂²	1	1	yes	characteristic
10	D₅⋊#168	—	2	1	no	—
11	C₂×C₅×#170	—	2	1	no	—
12	C₅×#700	—	2	1	no	—
13	C₅×#700	C₂²	1	2	yes	—
14	#87804	C₄	1	1	yes	characteristic
15	C₅×#681	C₄	1	1	yes	characteristic
16	#87768	C₄	1	2	yes	—
17	D₅×#168	C₂²	1	2	yes	—
18	#65729	—	5	1	no	maximal
19	#65413	—	5	1	no	maximal
20	C₂²×C₅×D₅	Q₈	1	1	yes	characteristic
21	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×Dic₅	D₄	1	1	yes	characteristic
23	C₄×C₅×D₅	C₂×C₄	1	1	yes	characteristic
24	C₄×C₅×D₅	D₄	1	1	yes	characteristic
25	C₄×C₅×D₅	Q₈	1	1	yes	characteristic
26	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
27	#6032	—	2	1	no	—
28	#6032	—	2	1	no	—
29	C₅×#170	—	2	2	no	—
30	#6029	C₂×C₄	1	2	yes	—
31	C₅×#168	C₂×C₄	1	2	yes	—
32	D₅⋊M₁₆	—	5	1	no	—
33	C₂×#700	D₅	1	1	yes	characteristic
34	C₂×C₅×M₁₆	—	5	1	no	—
35	C₂×#681	—	5	1	no	—
36	C₂×#681	—	5	1	no	—
37	D₅×M₁₆	—	5	1	no	—
38	#3996	—	5	2	no	—
39	#3995	—	5	2	no	—
40	#3954	—	5	2	no	—
41	#3954	—	5	2	no	—
42	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅×D₅	—	4	1	no	—
44	C₂×C₅×D₅	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₅×D₅	C₂×Q₈	1	1	yes	characteristic
46	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
48	C₅×Dic₅	C₄⋊C₄	1	1	yes	characteristic
49	C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	—	5	1	no	—
51	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
52	#704	—	10	1	no	—
53	C₂×#170	—	2	1	no	—
54	#700	—	2	1	no	—
55	#700	—	5	1	no	—
56	#700	—	5	1	no	—
57	#700	C₂×D₅	1	2	yes	—
58	D₅⋊C₈	—	10	1	no	—
59	C₅×M₁₆	—	5	1	no	—
60	C₅×M₁₆	—	5	2	no	—
61	C₅×M₁₆	—	10	1	no	—
62	C₂×C₅×C₈	—	10	1	no	—
63	#681	—	5	1	no	—
64	#681	—	5	1	no	—
65	#681	—	5	2	no	—
66	#681	—	5	1	no	—
67	#681	—	5	1	no	—
68	#681	—	10	1	no	—
69	#681	—	20	1	no	—
70	#681	AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₂×#168	—	5	2	no	—
72	C₂×#168	—	10	1	no	—
73	#676	—	5	2	no	—
74	C₈×D₅	—	5	2	no	—
75	#430	—	25	1	no	Sylow
76	C₂×C₅²	—	2	1	no	—
77	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₅×D₅	—	2	1	no	—
79	C₅×D₅	—	2	1	no	—
80	C₂³×C₅	—	5	1	no	—
81	C₂²×D₅	Dic₁₀	1	1	yes	characteristic
82	C₂×C₄×C₅	—	4	1	no	—
83	C₂×C₄×C₅	—	5	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	C₂²⋊D₅	1	1	yes	characteristic
91	C₄×D₅	Dic₁₀	1	1	yes	characteristic
92	C₄×D₅	C₄×D₅	1	1	yes	characteristic
93	C₄×D₅	C₂²⋊D₅	1	1	yes	characteristic
94	C₄×D₅	C₂²×D₅	1	1	yes	characteristic
95	#170	—	2	2	no	—
96	#170	—	10	1	no	—
97	#170	—	10	1	no	—
98	C₅×C₈	—	5	2	no	—
99	C₅×C₈	—	10	2	no	—
100	#168	—	5	2	no	—
101	#168	—	5	2	no	—
102	#168	—	10	1	no	—
103	#168	—	10	1	no	—
104	#168	—	20	2	no	—
105	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
106	C₂×M₁₆	—	5	1	no	—
107	C₂×M₁₆	—	25	1	no	—
108	C₂×M₁₆	—	25	1	no	—
109	#108	—	25	2	no	—
110	#108	—	25	2	no	—
111	C₅²	#430	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₅	—	4	1	no	—
113	C₂²×C₅	—	5	1	no	—
114	C₂²×C₅	—	5	1	no	—
115	C₂²×C₅	—	20	1	no	—
116	C₂²×C₅	#683	1	1	yes	characteristic
117	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₂×D₅	—	4	1	no	—
119	C₂×D₅	#683	1	1	yes	characteristic
120	C₂×D₅	C₂×Dic₁₀	1	1	yes	characteristic
121	C₄×C₅	—	4	1	no	—
122	C₄×C₅	—	4	1	no	—
123	C₄×C₅	—	5	1	no	—
124	C₄×C₅	—	5	1	no	—
125	C₄×C₅	#683	1	1	yes	characteristic
126	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
128	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
129	Dic₅	#683	1	1	yes	characteristic
130	Dic₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
131	C₂²×C₄	—	5	1	no	—
132	M₁₆	—	5	1	no	—
133	M₁₆	—	5	2	no	—
134	M₁₆	—	10	1	no	—
135	M₁₆	—	25	2	no	—
136	M₁₆	—	25	1	no	—
137	M₁₆	—	25	1	no	—
138	M₁₆	—	25	1	no	—
139	M₁₆	—	50	1	no	—
140	C₂×C₈	—	10	1	no	—
141	C₂×C₈	—	25	2	no	—
142	C₂×C₈	—	50	1	no	—
143	C₂×C₅	—	2	1	no	—
144	C₂×C₅	—	2	1	no	—
145	C₂×C₅	—	4	1	no	—
146	C₂×C₅	—	8	1	no	—
147	C₂×C₅	—	10	1	no	—
148	C₂×C₅	—	10	1	no	—
149	C₂×C₅	C₂×#683	1	1	yes	characteristic
150	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
151	D₅	—	2	1	no	—
152	D₅	—	2	1	no	—
153	C₂³	—	5	1	no	—
154	C₂×C₄	—	5	1	no	—
155	C₂×C₄	—	5	1	no	—
156	C₂×C₄	—	5	1	no	—
157	C₂×C₄	—	5	1	no	—
158	C₂×C₄	—	5	1	no	—
159	C₂×C₄	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₈	—	5	2	no	—
161	C₈	—	10	2	no	—
162	C₈	—	25	2	no	—
163	C₈	—	50	1	no	—
164	C₈	—	50	1	no	—
165	C₅	—	4	1	no	—
166	C₅	#65413	1	1	yes	characteristic, complemented
167	C₅	#65729	1	1	yes	characteristic, complemented
168	C₂²	—	5	1	no	—
169	C₂²	—	5	1	no	—
170	C₂²	—	20	1	no	—
171	C₂²	#87825	1	1	yes	characteristic
172	C₄	—	5	1	no	—
173	C₄	—	5	1	no	—
174	C₄	#87825	1	1	yes	characteristic
175	C₄	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	10	1	no	—
178	C₂	—	10	1	no	—
179	C₂	C₂×#87825	1	1	yes	characteristic

Complements

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

N	H ≅ G/N	type
4. #11716745	C₂	semidirect
5. #11716744	C₂	semidirect
111. C₅²	#430	semidirect
166. C₅	#65413	semidirect
167. C₅	#65729	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊#681	2
C₂×C₅×#700	2
D₅×#681	2
#11716745	2
#11716744	2
#65729	5
#65413	5