Finite Group Databasebeta

Subgroups — #13103068

Order 1600.

568 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 111 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	#11717523	C₂	1	1	yes	maximal, characteristic
2	C₅²×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717264	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717262	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716704	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716692	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716691	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅²×#41	C₂²	1	1	yes	characteristic
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	Q₈×C₅⋊D₅	—	2	1	no	—
11	#87902	—	2	1	no	—
12	#87902	—	2	2	no	—
13	#87900	—	2	1	no	—
14	#87900	C₂²	1	1	yes	characteristic
15	C₂×#6045	—	2	1	no	—
16	C₂×#6045	C₂²	1	1	yes	characteristic
17	C₅²×Q₁₆	—	2	1	no	—
18	C₅²×SD₁₆	—	2	1	no	—
19	C₅²×M₁₆	C₂²	1	1	yes	characteristic
20	#87812	—	2	1	no	—
21	#87811	—	2	1	no	—
22	#87806	—	2	2	no	—
23	C₄×#1039	C₂²	1	1	yes	characteristic
24	#87804	C₂²	1	1	yes	characteristic
25	#65494	—	5	6	no	maximal
26	C₅²×Q₈	—	2	1	no	—
27	C₅²×Q₈	—	2	1	no	—
28	C₅²×Q₈	D₄	1	1	yes	characteristic
29	C₅²×D₄	—	2	1	no	—
30	C₅²×D₄	D₄	1	1	yes	characteristic
31	C₂×C₄×C₅²	—	2	1	no	—
32	C₂×C₄×C₅²	—	2	1	no	—
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
35	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
36	C₂×#1039	—	2	2	no	—
37	C₂×#1039	—	2	1	no	—
38	C₂×#1039	D₄	1	2	yes	—
39	C₄⋊(C₅⋊D₅)	D₄	1	1	yes	characteristic
40	C₄×C₅⋊D₅	—	2	2	no	—
41	C₄×C₅⋊D₅	—	2	1	no	—
42	#6045	—	2	2	no	—
43	#6045	—	2	1	no	—
44	#6045	—	2	1	no	—
45	#6045	D₄	1	1	yes	characteristic
46	C₅²×C₈	—	2	1	no	—
47	#6029	—	2	1	no	—
48	#4150	—	5	6	no	—
49	C₅×#137	D₅	1	6	yes	complemented
50	#4090	—	5	6	no	—
51	#4088	—	5	6	no	—
52	#3969	—	5	6	no	—
53	#3957	—	5	6	no	—
54	#3956	—	5	6	no	—
55	C₂²×C₅²	—	2	1	no	—
56	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅⋊D₅	—	2	1	no	—
58	C₄×C₅²	—	2	1	no	—
59	C₄×C₅²	—	4	1	no	—
60	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
61	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
62	#1039	—	2	2	no	—
63	#1039	—	2	2	no	—
64	#1039	—	2	1	no	—
65	C₅×#41	C₂×D₅	1	6	yes	—
66	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
67	Q₈×D₅	—	10	6	no	—
68	#711	—	10	12	no	—
69	#711	—	10	6	no	—
70	#709	—	5	6	no	—
71	#709	—	10	6	no	—
72	C₂×Dic₁₀	—	5	6	no	—
73	C₂×Dic₁₀	—	10	6	no	—
74	C₅×Q₁₆	—	2	6	no	—
75	C₅×SD₁₆	—	2	6	no	—
76	C₅×M₁₆	C₂×D₅	1	6	yes	—
77	#689	—	10	6	no	—
78	#688	—	10	6	no	—
79	#683	—	10	12	no	—
80	C₄×Dic₅	—	5	6	no	—
81	#681	—	5	6	no	—
82	#456	—	25	1	no	Sylow
83	C₂×C₅²	—	2	1	no	—
84	C₂×C₅²	—	4	1	no	—
85	C₂×C₅²	#120	1	1	yes	characteristic
86	C₅⋊D₅	—	4	1	no	—
87	C₅×Q₈	—	2	6	no	—
88	C₅×Q₈	—	2	6	no	—
89	C₅×Q₈	C₂²⋊D₅	1	6	yes	—
90	C₅×D₄	—	2	6	no	—
91	C₅×D₄	C₂²⋊D₅	1	6	yes	—
92	C₂×C₄×C₅	—	2	6	no	—
93	C₂×C₄×C₅	—	2	6	no	—
94	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
95	C₂²⋊D₅	—	10	12	no	—
96	C₂²⋊D₅	—	10	6	no	—
97	C₂×Dic₅	—	5	12	no	—
98	C₂×Dic₅	—	10	12	no	—
99	C₂×Dic₅	—	10	6	no	—
100	D₂₀	—	5	6	no	—
101	C₄×D₅	—	10	12	no	—
102	C₄×D₅	—	10	6	no	—
103	Dic₁₀	—	5	6	no	—
104	Dic₁₀	—	10	12	no	—
105	Dic₁₀	—	10	6	no	—
106	Dic₁₀	—	10	6	no	—
107	C₅×C₈	—	2	6	no	—
108	#168	—	10	6	no	—
109	2₋¹⁺⁴	—	25	1	no	—
110	#137	—	25	1	no	—
111	#137	C₅⋊D₅	1	1	yes	characteristic, complemented
112	C₄⋊Q₈	—	25	1	no	—
113	#104	—	25	1	no	—
114	#104	—	25	1	no	—
115	#101	—	25	1	no	—
116	C₅²	#456	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₅	—	2	6	no	—
118	C₂²×C₅	D₄×D₅	1	6	yes	—
119	C₂×D₅	—	10	6	no	—
120	C₄×C₅	—	2	6	no	—
121	C₄×C₅	—	4	6	no	—
122	C₄×C₅	D₄×D₅	1	6	yes	—
123	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
124	Dic₅	—	10	12	no	—
125	Dic₅	—	10	6	no	—
126	Dic₅	—	10	12	no	—
127	#41	—	25	1	no	—
128	#41	—	50	1	no	—
129	#41	—	50	2	no	—
130	#41	—	50	1	no	—
131	#41	C₂×C₅⋊D₅	1	1	yes	characteristic
132	C₂×Q₈	—	25	1	no	—
133	C₂×Q₈	—	50	1	no	—
134	C₂×Q₈	—	50	1	no	—
135	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
136	Q₁₆	—	2	1	no	—
137	Q₁₆	—	50	1	no	—
138	SD₁₆	—	2	1	no	—
139	SD₁₆	—	50	1	no	—
140	M₁₆	—	25	1	no	—
141	M₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
142	C₄⋊C₄	—	50	2	no	—
143	C₄²	—	25	1	no	—
144	C₂×C₅	—	2	6	no	—
145	C₂×C₅	—	4	6	no	—
146	C₂×C₅	#4083	1	6	yes	—
147	D₅	—	20	6	no	—
148	Q₈	—	2	1	no	—
149	Q₈	—	2	1	no	—
150	Q₈	—	25	1	no	—
151	Q₈	—	50	2	no	—
152	Q₈	—	50	1	no	—
153	Q₈	—	50	1	no	—
154	Q₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
155	D₄	—	2	1	no	—
156	D₄	—	25	1	no	—
157	D₄	—	50	2	no	—
158	D₄	—	50	1	no	—
159	D₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	25	2	no	—
163	C₂×C₄	—	50	2	no	—
164	C₂×C₄	—	50	2	no	—
165	C₂×C₄	—	50	1	no	—
166	C₂×C₄	—	50	1	no	—
167	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	50	1	no	—
170	C₅	#65494	1	6	yes	complemented
171	C₂²	—	2	1	no	—
172	C₂²	—	50	1	no	—
173	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	50	2	no	—
177	C₄	—	50	2	no	—
178	C₄	—	50	1	no	—
179	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
180	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	100	1	no	—
184	C₂	#11717257	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11717257
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#137	C₂
socle	C₂×C₅²	#120
prime radical	#11717523	C₂
hypercenter	#137	C₅⋊D₅
hypocenter	C₅²	#456
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. C₅²×#137	C₂	semidirect
3. #11717264	C₂	semidirect
4. #11717262	C₂	semidirect
5. #11716704	C₂	semidirect
6. #11716692	C₂	semidirect
7. #11716691	C₂	semidirect
49. C₅×#137	D₅	semidirect
111. #137	C₅⋊D₅	semidirect
116. C₅²	#456	semidirect
170. C₅	#65494	semidirect

Maximal subgroups

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

subgroup	index
#11717523	2
C₅²×#137	2
#11717264	2
#11717262	2
#11716704	2
#11716692	2
#11716691	2
#65494	5