Finite Group Databasebeta

Subgroups — #13104107

Order 1600.

270 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×Dic₅²	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#6034	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#6033	C₂	1	1	yes	maximal, characteristic
4	#11716783	C₂	1	2	yes	maximal, complemented
5	#11716778	C₂	1	2	yes	maximal, complemented
6	C₂²×#1039	C₂²	1	1	yes	characteristic
7	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic
8	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic
9	C₂×#6034	C₂²	1	2	yes	—
10	C₂×#6033	C₂²	1	2	yes	—
11	Dic₅²	C₄	1	1	yes	characteristic
12	Dic₅²	C₄	1	1	yes	characteristic
13	Dic₅²	C₂²	1	1	yes	characteristic
14	Dic₅²	C₂²	1	1	yes	characteristic
15	#65759	—	5	1	no	maximal
16	#65757	—	5	1	no	maximal
17	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	—	2	2	no	—
19	C₂×#1039	C₂×C₄	1	1	yes	characteristic
20	C₂×#1039	C₂³	1	1	yes	characteristic
21	C₂×C₅×Dic₅	—	2	2	no	—
22	C₂×C₅×Dic₅	—	2	2	no	—
23	C₂×C₅×Dic₅	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	#6034	—	2	2	no	—
28	#6033	—	2	2	no	—
29	C₂×C₄×Dic₅	—	5	1	no	—
30	C₂×C₄×Dic₅	—	5	1	no	—
31	C₂²⋊#170	—	5	1	no	—
32	C₂²⋊#170	—	5	1	no	—
33	C₂²⋊#170	—	5	1	no	—
34	C₂²⋊#170	—	5	1	no	—
35	C₂²⋊#170	—	10	2	no	—
36	#4004	—	5	2	no	—
37	#4002	—	5	2	no	—
38	#4002	—	5	2	no	—
39	C₄⋊#170	—	5	2	no	—
40	C₂²×C₅²	—	2	1	no	—
41	C₂²×C₅²	—	2	1	no	—
42	C₂²×C₅²	M₁₆	1	2	yes	—
43	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
44	#1039	—	2	1	no	—
45	#1039	#41	1	2	yes	—
46	C₅×Dic₅	—	2	1	no	—
47	C₅×Dic₅	—	2	1	no	—
48	C₅×Dic₅	—	2	1	no	—
49	C₅×Dic₅	M₁₆	1	2	yes	—
50	C₂²×C₄×C₅	—	5	1	no	—
51	C₂²×C₄×C₅	—	5	1	no	—
52	C₂²×Dic₅	—	5	1	no	—
53	C₂²×Dic₅	—	5	1	no	—
54	C₂²×Dic₅	—	10	2	no	—
55	C₂²×Dic₅	AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂²×Dic₅	AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×#170	—	5	2	no	—
58	C₂×#170	—	5	2	no	—
59	C₂×#170	—	5	2	no	—
60	C₂×#170	—	5	2	no	—
61	C₂×#170	—	10	4	no	—
62	C₄×Dic₅	—	5	1	no	—
63	C₄×Dic₅	—	5	1	no	—
64	C₄×Dic₅	—	5	1	no	—
65	C₄×Dic₅	—	5	1	no	—
66	C₄×Dic₅	—	5	1	no	—
67	C₄×Dic₅	—	5	1	no	—
68	C₄×Dic₅	—	5	1	no	—
69	C₄×Dic₅	—	5	1	no	—
70	#432	—	25	1	no	Sylow
71	C₂×C₅²	—	2	2	no	—
72	C₂×C₅²	#117	1	1	yes	characteristic
73	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
75	C₂³×C₅	—	2	2	no	—
76	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂×C₄×C₅	—	5	1	no	—
79	C₂×C₄×C₅	—	5	1	no	—
80	C₂×C₄×C₅	—	5	1	no	—
81	C₂×C₄×C₅	—	5	1	no	—
82	C₂×C₄×C₅	—	10	2	no	—
83	C₂×C₄×C₅	—	10	2	no	—
84	C₂×Dic₅	—	2	2	no	—
85	C₂×Dic₅	—	2	2	no	—
86	C₂×Dic₅	—	5	1	no	—
87	C₂×Dic₅	—	5	1	no	—
88	C₂×Dic₅	—	5	1	no	—
89	C₂×Dic₅	—	5	1	no	—
90	C₂×Dic₅	—	10	2	no	—
91	C₂×Dic₅	—	10	2	no	—
92	C₂×Dic₅	—	10	2	no	—
93	C₂×Dic₅	—	10	2	no	—
94	C₂×Dic₅	—	20	4	no	—
95	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₂×Dic₅	C₂×AGL(1,𝔽₅)	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	#170	—	10	2	no	—
100	#170	—	10	2	no	—
101	#170	—	10	2	no	—
102	#170	—	10	2	no	—
103	#170	—	20	4	no	—
104	C₂×C₄²	—	25	1	no	—
105	C₄⋊C₈	—	25	2	no	—
106	C₂²⋊C₈	—	25	1	no	—
107	C₂²⋊C₈	—	25	1	no	—
108	#97	—	25	2	no	—
109	C₅²	#432	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₅	—	2	1	no	—
111	C₂²×C₅	—	2	1	no	—
112	C₂²×C₅	—	2	1	no	—
113	C₂²×C₅	—	2	1	no	—
114	C₂²×C₅	—	2	4	no	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	—	4	2	no	—
117	C₂²×C₅	—	4	2	no	—
118	C₂²×C₅	#700	1	2	yes	—
119	C₂²×C₅	#704	1	2	yes	—
120	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₄×C₅	—	5	2	no	—
123	C₄×C₅	—	10	1	no	—
124	C₄×C₅	—	10	1	no	—
125	C₄×C₅	—	10	1	no	—
126	Dic₅	—	2	1	no	—
127	Dic₅	—	2	1	no	—
128	Dic₅	—	2	1	no	—
129	Dic₅	—	5	2	no	—
130	Dic₅	—	5	2	no	—
131	Dic₅	—	10	4	no	—
132	Dic₅	—	10	1	no	—
133	Dic₅	—	10	1	no	—
134	Dic₅	—	20	2	no	—
135	Dic₅	#704	1	2	yes	—
136	C₂²×C₄	—	5	1	no	—
137	C₂²×C₄	—	5	1	no	—
138	C₂²×C₄	—	25	1	no	—
139	C₂×C₈	—	25	2	no	—
140	C₂×C₈	—	25	2	no	—
141	C₄²	—	25	1	no	—
142	C₄²	—	25	1	no	—
143	C₄²	—	25	1	no	—
144	C₄²	—	25	1	no	—
145	C₂×C₅	—	2	2	no	—
146	C₂×C₅	—	2	2	no	—
147	C₂×C₅	—	2	2	no	—
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	—	2	2	no	—
150	C₂×C₅	—	4	4	no	—
151	C₂×C₅	C₂×#700	1	1	yes	characteristic
152	C₂×C₅	D₅⋊M₁₆	1	1	yes	characteristic
153	C₂×C₅	#4130	1	1	yes	characteristic
154	C₂×C₅	#4130	1	1	yes	characteristic
155	C₂×C₅	C₂×#704	1	1	yes	characteristic
156	C₂×C₅	C₂×#704	1	1	yes	characteristic
157	C₂³	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×C₄	—	5	1	no	—
159	C₂×C₄	—	5	1	no	—
160	C₂×C₄	—	5	1	no	—
161	C₂×C₄	—	5	1	no	—
162	C₂×C₄	—	10	2	no	—
163	C₂×C₄	—	10	2	no	—
164	C₂×C₄	—	25	1	no	—
165	C₂×C₄	—	25	1	no	—
166	C₂×C₄	—	50	2	no	—
167	C₈	—	50	2	no	—
168	C₈	—	50	2	no	—
169	C₅	—	2	2	no	—
170	C₅	#65757	1	1	yes	characteristic, complemented
171	C₅	#65759	1	1	yes	characteristic, complemented
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#87834	1	2	yes	—
175	C₂²	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₄	—	5	2	no	—
177	C₄	—	10	1	no	—
178	C₄	—	10	1	no	—
179	C₄	—	10	1	no	—
180	C₄	—	25	2	no	—
181	C₄	—	50	1	no	—
182	C₂	—	2	2	no	—
183	C₂	#11717367	1	1	yes	characteristic
184	C₂	D₅⋊#704	1	1	yes	characteristic
185	C₂	C₂×#87834	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting, prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₂³	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#432
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
4. #11716783	C₂	semidirect
5. #11716778	C₂	semidirect
109. C₅²	#432	semidirect
170. C₅	#65757	semidirect
171. C₅	#65759	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₅²	2
C₂²⋊#6034	2
C₂²⋊#6033	2
#11716783	2
#11716778	2
#65759	5
#65757	5