Finite Group Databasebeta

Subgroups — #65886

Order 320.

290 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 107 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×#709	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#4085	C₂	1	2	yes	maximal, complemented
5	#4076	C₂	1	1	yes	maximal, characteristic, complemented
6	#4070	C₂	1	1	yes	maximal, characteristic, complemented
7	#4042	C₂	1	2	yes	maximal, complemented
8	#4031	C₂	1	2	yes	maximal, complemented
9	#4027	C₂	1	2	yes	maximal, complemented
10	#4021	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×Dic₅	C₂²	1	2	yes	—
18	#709	—	2	4	no	—
19	C₂×D₂₀	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₅	C₂²	1	2	yes	—
21	C₂×Dic₁₀	—	2	4	no	—
22	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₀	C₂²	1	2	yes	—
24	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₄²×C₅	C₂²	1	1	yes	characteristic
29	C₂²⋊Dic₅	C₂²	1	2	yes	—
30	#685	C₂²	1	2	yes	—
31	#685	C₂²	1	2	yes	—
32	#685	C₂²	1	2	yes	—
33	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
34	#683	C₂²	1	2	yes	—
35	#683	C₂²	1	2	yes	—
36	C₄×Dic₅	C₂²	1	2	yes	—
37	#548	—	5	1	no	maximal, Sylow
38	C₂³×C₅	C₂³	1	1	yes	characteristic
39	C₂³×C₅	C₂³	1	1	yes	characteristic
40	C₂²×D₅	C₂³	1	2	yes	—
41	C₅×D₄	—	2	2	no	—
42	C₂×C₄×C₅	—	2	2	no	—
43	C₂×C₄×C₅	—	2	2	no	—
44	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂²⋊D₅	—	2	4	no	—
50	C₂²⋊D₅	—	2	4	no	—
51	C₂×Dic₅	—	2	4	no	—
52	C₂×Dic₅	C₂³	1	2	yes	—
53	C₂×Dic₅	C₂³	1	2	yes	—
54	C₂×Dic₅	C₂³	1	2	yes	—
55	D₂₀	—	2	2	no	—
56	C₄×D₅	—	2	4	no	—
57	Dic₁₀	—	2	2	no	—
58	Dic₁₀	—	2	4	no	—
59	Dic₁₀	D₄	1	4	yes	complemented
60	C₂×#41	—	5	1	no	—
61	C₂²×Q₈	—	5	1	no	—
62	#124	—	5	2	no	—
63	#124	—	5	1	no	—
64	C₂²⋊Q₈	—	5	1	no	—
65	C₂²⋊Q₈	—	5	2	no	—
66	#121	—	5	2	no	—
67	#121	—	5	1	no	—
68	C₄×Q₈	—	5	1	no	—
69	C₄×D₄	—	5	2	no	—
70	C₄×D₄	D₅	1	1	yes	characteristic, complemented
71	C₂²×C₅	—	2	1	no	—
72	C₂²×C₅	—	2	1	no	—
73	C₂²×C₅	—	2	1	no	—
74	C₂²×C₅	—	2	1	no	—
75	C₂²×C₅	—	2	1	no	—
76	C₂²×C₅	#41	1	2	yes	complemented
77	C₂²×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×D₅	—	2	2	no	—
79	C₂×D₅	—	2	2	no	—
80	C₂×D₅	—	2	2	no	—
81	C₄×C₅	—	2	1	no	—
82	C₄×C₅	—	2	1	no	—
83	C₄×C₅	—	2	1	no	—
84	C₄×C₅	—	2	1	no	—
85	C₄×C₅	C₂×D₄	1	2	yes	—
86	Dic₅	—	2	2	no	—
87	Dic₅	—	2	2	no	—
88	Dic₅	C₂×D₄	1	4	yes	—
89	#41	—	10	4	no	—
90	C₂×Q₈	—	5	1	no	—
91	C₂×Q₈	—	5	2	no	—
92	C₂×Q₈	—	5	1	no	—
93	C₂×Q₈	—	10	4	no	—
94	C₂×D₄	—	5	2	no	—
95	C₂×D₄	—	5	2	no	—
96	C₂×D₄	—	5	1	no	—
97	C₂×D₄	C₂×D₅	1	1	yes	characteristic
98	C₂²×C₄	—	5	2	no	—
99	C₂²×C₄	—	5	2	no	—
100	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
101	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
102	C₄⋊C₄	—	5	2	no	—
103	C₄⋊C₄	—	5	2	no	—
104	C₄⋊C₄	—	5	1	no	—
105	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
106	C₂²⋊C₄	—	5	2	no	—
107	C₂²⋊C₄	—	5	2	no	—
108	C₂²⋊C₄	—	5	2	no	—
109	C₂²⋊C₄	—	5	2	no	—
110	C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
111	C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
112	C₄²	—	5	2	no	—
113	C₄²	C₂×D₅	1	1	yes	characteristic
114	C₂×C₅	—	2	2	no	—
115	C₂×C₅	—	4	1	no	—
116	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₅	C₂×#41	1	1	yes	characteristic
118	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
119	D₅	—	4	2	no	—
120	C₂³	—	5	2	no	—
121	C₂³	C₂²×D₅	1	1	yes	characteristic
122	C₂³	C₂²×D₅	1	1	yes	characteristic
123	Q₈	—	5	4	no	—
124	Q₈	—	10	4	no	—
125	Q₈	—	10	2	no	—
126	D₄	—	2	2	no	—
127	D₄	—	10	4	no	—
128	D₄	—	10	4	no	—
129	D₄	—	10	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	5	2	no	—
133	C₂×C₄	—	5	2	no	—
134	C₂×C₄	—	5	2	no	—
135	C₂×C₄	—	10	4	no	—
136	C₂×C₄	—	10	4	no	—
137	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
139	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
140	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
141	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
142	C₅	#548	1	1	yes	characteristic, complemented, Sylow
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	10	2	no	—
149	C₂²	—	10	2	no	—
150	C₂²	—	10	2	no	—
151	C₂²	#709	1	2	yes	complemented
152	C₂²	C₂³×D₅	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	5	4	no	—
158	C₄	—	10	2	no	—
159	C₄	—	10	2	no	—
160	C₄	D₄×D₅	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	—	4	1	no	—
163	C₂	—	20	2	no	—
164	C₂	C₂×#709	1	1	yes	characteristic
165	C₂	C₂×D₄×D₅	1	1	yes	characteristic
166	C₂	#4150	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₄×C₅×D₄	C₂
hypercenter	C₄×D₄	D₅
hypocenter	C₅	#548
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#709	C₂	semidirect
2. C₂²×Dic₁₀	C₂	semidirect
3. C₄×C₅×D₄	C₂	semidirect
4. #4085	C₂	semidirect
5. #4076	C₂	semidirect
6. #4070	C₂	semidirect
7. #4042	C₂	semidirect
8. #4031	C₂	semidirect
9. #4027	C₂	semidirect
10. #4021	C₂	semidirect
11. C₄×Dic₁₀	C₂	semidirect
59. Dic₁₀	D₄	semidirect
70. C₄×D₄	D₅	semidirect
76. C₂²×C₅	#41	semidirect
142. C₅	#548	semidirect
151. C₂²	#709	semidirect

Maximal subgroups

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

subgroup	index
C₂×#709	2
C₂²×Dic₁₀	2
C₄×C₅×D₄	2
#4085	2
#4076	2
#4070	2
#4042	2
#4031	2
#4027	2
#4021	2
C₄×Dic₁₀	2
#548	5