Finite Group Databasebeta

Subgroups — #66030

Order 320.

192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#4044	C₂	1	1	yes	maximal, characteristic
3	#4044	C₂	1	1	yes	maximal, characteristic
4	#4044	C₂	1	1	yes	maximal, characteristic
5	#4042	C₂	1	1	yes	maximal, characteristic
6	#4042	C₂	1	1	yes	maximal, characteristic
7	#4038	C₂	1	1	yes	maximal, characteristic
8	#4036	C₂	1	1	yes	maximal, characteristic, complemented
9	#4035	C₂	1	1	yes	maximal, characteristic, complemented
10	#4035	C₂	1	1	yes	maximal, characteristic, complemented
11	#4035	C₂	1	1	yes	maximal, characteristic, complemented
12	#4034	C₂	1	1	yes	maximal, characteristic, complemented
13	#4033	C₂	1	1	yes	maximal, characteristic, complemented
14	#4022	C₂	1	1	yes	maximal, characteristic
15	C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄²×C₅	C₂²	1	1	yes	characteristic
26	#685	C₂²	1	1	yes	characteristic
27	#685	C₂²	1	1	yes	characteristic
28	#685	C₂²	1	1	yes	characteristic
29	#685	C₂²	1	1	yes	characteristic
30	#685	C₂²	1	1	yes	characteristic
31	#685	C₂²	1	1	yes	characteristic
32	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
33	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
34	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
35	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
36	#683	C₂²	1	1	yes	characteristic
37	#683	C₂²	1	1	yes	characteristic
38	#683	C₂²	1	1	yes	characteristic
39	#683	C₂²	1	1	yes	characteristic
40	#683	C₂²	1	1	yes	characteristic
41	#683	C₂²	1	1	yes	characteristic
42	#683	C₂²	1	1	yes	characteristic
43	#683	C₂²	1	1	yes	characteristic
44	#683	C₂²	1	1	yes	characteristic
45	#683	C₂²	1	1	yes	characteristic
46	C₄×Dic₅	C₂²	1	1	yes	characteristic
47	C₄×Dic₅	C₂²	1	1	yes	characteristic
48	C₄×Dic₅	C₂²	1	1	yes	characteristic
49	C₄×Dic₅	C₂²	1	1	yes	characteristic
50	C₄×Dic₅	C₂²	1	1	yes	characteristic
51	#541	—	5	1	no	maximal, Sylow
52	C₂²×D₅	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×Dic₅	C₂³	1	1	yes	characteristic
63	C₂×Dic₅	C₂³	1	1	yes	characteristic
64	C₂×Dic₅	C₂³	1	1	yes	characteristic
65	C₂×Dic₅	C₂³	1	1	yes	characteristic
66	C₂×Dic₅	C₂³	1	1	yes	characteristic
67	C₄×D₅	—	2	2	no	—
68	Dic₁₀	—	2	2	no	—
69	Dic₁₀	—	2	2	no	—
70	C₄⋊Q₈	—	5	1	no	—
71	#126	—	5	1	no	—
72	#126	—	5	1	no	—
73	#126	—	5	1	no	—
74	#126	—	5	1	no	—
75	#125	—	5	1	no	—
76	#125	—	5	1	no	—
77	#125	—	5	1	no	—
78	#125	—	5	1	no	—
79	#125	D₅	1	1	yes	characteristic, complemented
80	C₂²⋊Q₈	—	5	1	no	—
81	C₂²⋊Q₈	—	5	1	no	—
82	C₄×Q₈	—	5	1	no	—
83	C₄×Q₈	—	5	1	no	—
84	#117	—	5	1	no	—
85	C₂²×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×D₅	—	2	1	no	—
87	C₂×D₅	—	2	1	no	—
88	C₂×D₅	—	2	1	no	—
89	C₄×C₅	—	2	1	no	—
90	C₄×C₅	—	2	1	no	—
91	C₄×C₅	—	2	1	no	—
92	C₄×C₅	—	2	1	no	—
93	C₄×C₅	—	2	1	no	—
94	C₄×C₅	—	2	1	no	—
95	C₄×C₅	—	2	1	no	—
96	Dic₅	—	2	1	no	—
97	Dic₅	—	2	1	no	—
98	Dic₅	—	2	1	no	—
99	Dic₅	—	2	1	no	—
100	Dic₅	—	2	1	no	—
101	Dic₅	—	2	1	no	—
102	Dic₅	#41	1	2	yes	—
103	C₂×Q₈	—	5	1	no	—
104	C₂×Q₈	—	5	1	no	—
105	C₂²×C₄	—	5	1	no	—
106	C₄⋊C₄	—	5	1	no	—
107	C₄⋊C₄	—	5	1	no	—
108	C₄⋊C₄	—	5	1	no	—
109	C₄⋊C₄	—	5	1	no	—
110	C₄⋊C₄	—	5	1	no	—
111	C₄⋊C₄	—	5	1	no	—
112	C₄⋊C₄	—	5	1	no	—
113	C₄⋊C₄	—	5	1	no	—
114	C₄⋊C₄	—	5	1	no	—
115	C₄⋊C₄	—	5	1	no	—
116	C₄⋊C₄	—	5	1	no	—
117	C₄⋊C₄	—	5	1	no	—
118	C₄⋊C₄	—	5	1	no	—
119	C₄⋊C₄	—	5	1	no	—
120	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
121	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
122	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
123	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
124	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
125	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
126	C₂²⋊C₄	—	5	1	no	—
127	C₂²⋊C₄	—	5	1	no	—
128	C₂²⋊C₄	—	5	1	no	—
129	C₂²⋊C₄	—	5	1	no	—
130	C₂²⋊C₄	—	5	1	no	—
131	C₂²⋊C₄	—	5	1	no	—
132	C₄²	—	5	1	no	—
133	C₄²	—	5	1	no	—
134	C₄²	—	5	1	no	—
135	C₄²	—	5	1	no	—
136	C₄²	—	5	1	no	—
137	C₄²	C₂×D₅	1	1	yes	characteristic
138	C₂×C₅	C₂×#41	1	1	yes	characteristic
139	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
141	D₅	—	4	1	no	—
142	C₂³	—	5	1	no	—
143	Q₈	—	10	2	no	—
144	Q₈	—	10	2	no	—
145	C₂×C₄	—	5	1	no	—
146	C₂×C₄	—	5	1	no	—
147	C₂×C₄	—	5	1	no	—
148	C₂×C₄	—	5	1	no	—
149	C₂×C₄	—	5	1	no	—
150	C₂×C₄	—	5	1	no	—
151	C₂×C₄	—	5	1	no	—
152	C₂×C₄	—	10	2	no	—
153	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
154	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₅	#541	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	10	1	no	—
162	C₂²	—	10	1	no	—
163	C₂²	—	10	1	no	—
164	C₂²	C₂³×D₅	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	5	2	no	—
173	C₄	—	10	1	no	—
174	C₄	—	10	1	no	—
175	C₄	—	10	1	no	—
176	C₄	—	10	1	no	—
177	C₄	—	10	1	no	—
178	C₄	—	10	1	no	—
179	C₂	—	20	1	no	—
180	C₂	#4147	1	1	yes	characteristic
181	C₂	D₅×#41	1	1	yes	characteristic
182	C₂	#4150	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#125	C₂
prime radical	C₂²×D₅	C₂³
hypercenter	#125	D₅
hypocenter	C₅	#541
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
8. #4036	C₂	semidirect
9. #4035	C₂	semidirect
10. #4035	C₂	semidirect
11. #4035	C₂	semidirect
12. #4034	C₂	semidirect
13. #4033	C₂	semidirect
15. C₄×Dic₁₀	C₂	semidirect
79. #125	D₅	semidirect
160. C₅	#541	semidirect

Maximal subgroups

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

subgroup	index
C₅×#125	2
#4044	2
#4042	2
#4038	2
#4036	2
#4035	2
#4034	2
#4033	2
#4022	2
C₄×Dic₁₀	2
#541	5