Finite Group Databasebeta

Subgroups — #65966

Order 320.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
6	#4043	C₂	1	1	yes	maximal, characteristic, complemented
7	#4042	C₂	1	2	yes	maximal, complemented
8	#4034	C₂	1	2	yes	maximal, complemented
9	#4033	C₂	1	1	yes	maximal, characteristic, complemented
10	#4027	C₂	1	2	yes	maximal
11	#4024	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic
15	C₂²×Dic₅	C₂²	1	2	yes	—
16	Q₈×D₅	—	2	4	no	—
17	C₂×C₄×D₅	C₂²	1	2	yes	—
18	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀	—	2	4	no	—
20	C₂×Dic₁₀	C₂²	1	2	yes	—
21	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₀	C₂²	1	2	yes	—
23	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₄	C₂²	1	2	yes	—
26	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
28	#685	C₂²	1	2	yes	—
29	#685	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₅	C₂²	1	2	yes	—
31	#683	C₂²	1	2	yes	—
32	#683	C₂²	1	2	yes	—
33	#683	C₂²	1	2	yes	—
34	#683	C₂²	1	1	yes	characteristic
35	C₄×Dic₅	C₂²	1	2	yes	—
36	C₄×Dic₅	C₂²	1	1	yes	characteristic
37	D₄×Q₈	—	5	1	no	maximal, Sylow
38	C₂³×C₅	C₂³	1	1	yes	characteristic
39	C₂²×D₅	C₂³	1	1	yes	characteristic
40	C₅×Q₈	—	2	2	no	—
41	C₂×C₄×C₅	—	2	2	no	—
42	C₂×C₄×C₅	C₂³	1	2	yes	—
43	C₂×C₄×C₅	C₂³	1	2	yes	—
44	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂²⋊D₅	Q₈	1	4	yes	complemented
47	C₂×Dic₅	—	2	4	no	—
48	C₂×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×Dic₅	C₂³	1	2	yes	—
50	C₂×Dic₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₅	C₂³	1	2	yes	—
52	C₂×Dic₅	C₂³	1	1	yes	characteristic
53	C₄×D₅	—	2	2	no	—
54	C₄×D₅	—	2	4	no	—
55	Dic₁₀	—	2	2	no	—
56	Dic₁₀	—	2	4	no	—
57	Dic₁₀	—	2	4	no	—
58	Dic₁₀	D₄	1	4	yes	complemented
59	C₂²×Q₈	—	5	1	no	—
60	C₂²×Q₈	—	5	1	no	—
61	C₄⋊Q₈	—	5	1	no	—
62	C₄⋊Q₈	—	5	2	no	—
63	C₂²⋊Q₈	—	5	2	no	—
64	C₂²⋊Q₈	—	5	1	no	—
65	C₂²⋊Q₈	—	5	2	no	—
66	C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
67	C₄×Q₈	—	5	1	no	—
68	C₄×D₄	—	5	1	no	—
69	C₄×D₄	—	5	2	no	—
70	C₂²×C₅	—	2	1	no	—
71	C₂²×C₅	—	2	1	no	—
72	C₂²×C₅	C₂×Q₈	1	2	yes	complemented
73	C₂²×C₅	C₂⁴	1	1	yes	characteristic
74	C₂×D₅	—	2	1	no	—
75	C₂×D₅	—	2	1	no	—
76	C₂×D₅	C₂×Q₈	1	2	yes	complemented
77	C₄×C₅	—	2	1	no	—
78	C₄×C₅	—	2	2	no	—
79	C₄×C₅	—	2	2	no	—
80	C₄×C₅	C₂×D₄	1	2	yes	—
81	Dic₅	—	2	1	no	—
82	Dic₅	—	2	2	no	—
83	Dic₅	—	2	1	no	—
84	Dic₅	C₂×D₄	1	4	yes	—
85	Dic₅	C₂×Q₈	1	2	yes	complemented
86	C₂×Q₈	—	5	1	no	—
87	C₂×Q₈	—	5	2	no	—
88	C₂×Q₈	—	5	2	no	—
89	C₂×Q₈	—	5	1	no	—
90	C₂×Q₈	—	10	4	no	—
91	C₂×Q₈	—	10	4	no	—
92	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
93	C₂×D₄	—	5	1	no	—
94	C₂²×C₄	—	5	2	no	—
95	C₂²×C₄	—	5	2	no	—
96	C₂²×C₄	—	5	1	no	—
97	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
98	C₄⋊C₄	—	5	1	no	—
99	C₄⋊C₄	—	5	2	no	—
100	C₄⋊C₄	—	5	2	no	—
101	C₄⋊C₄	—	5	2	no	—
102	C₄⋊C₄	—	5	2	no	—
103	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
104	C₄⋊C₄	C₂×D₅	1	2	yes	—
105	C₂²⋊C₄	—	5	1	no	—
106	C₂²⋊C₄	—	5	1	no	—
107	C₂²⋊C₄	—	5	2	no	—
108	C₂²⋊C₄	C₂×D₅	1	2	yes	—
109	C₄²	—	5	2	no	—
110	C₄²	—	5	1	no	—
111	C₂×C₅	—	2	2	no	—
112	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
115	D₅	—	2	2	no	—
116	C₂³	—	5	1	no	—
117	C₂³	C₂²×D₅	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	Q₈	—	5	4	no	—
120	Q₈	—	10	4	no	—
121	Q₈	—	10	4	no	—
122	Q₈	—	10	2	no	—
123	D₄	—	5	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	5	1	no	—
126	C₂×C₄	—	5	2	no	—
127	C₂×C₄	—	5	1	no	—
128	C₂×C₄	—	5	1	no	—
129	C₂×C₄	—	5	2	no	—
130	C₂×C₄	—	10	4	no	—
131	C₂×C₄	—	10	4	no	—
132	C₂×C₄	—	10	2	no	—
133	C₂×C₄	C₂²×D₅	1	2	yes	—
134	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₂×C₄	C₂²×D₅	1	2	yes	—
136	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
137	C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	5	2	no	—
141	C₂²	—	10	1	no	—
142	C₂²	—	10	1	no	—
143	C₂²	Q₈×D₅	1	2	yes	complemented
144	C₂²	C₂³×D₅	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	5	4	no	—
149	C₄	—	5	2	no	—
150	C₄	—	10	2	no	—
151	C₄	—	10	1	no	—
152	C₄	—	10	1	no	—
153	C₄	D₄×D₅	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	—	10	2	no	—
156	C₂	C₂×D₄×D₅	1	1	yes	characteristic
157	C₂	C₂×Q₈×D₅	1	1	yes	characteristic
158	C₂	#4150	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×C₂²⋊Q₈	C₂
prime radical	C₂×C₂²⋊D₅	C₂²
hypercenter	C₂²⋊Q₈	D₅
hypocenter	C₅	D₄×Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₅	C₂	semidirect
2. C₂²×Dic₁₀	C₂	semidirect
3. C₅×C₂²⋊Q₈	C₂	semidirect
4. #4090	C₂	semidirect
6. #4043	C₂	semidirect
7. #4042	C₂	semidirect
8. #4034	C₂	semidirect
9. #4033	C₂	semidirect
11. #4024	C₂	semidirect
46. C₂²⋊D₅	Q₈	semidirect
58. Dic₁₀	D₄	semidirect
66. C₂²⋊Q₈	D₅	semidirect
72. C₂²×C₅	C₂×Q₈	semidirect
76. C₂×D₅	C₂×Q₈	semidirect
85. Dic₅	C₂×Q₈	semidirect
137. C₅	D₄×Q₈	semidirect
143. C₂²	Q₈×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₅	2
C₂²×Dic₁₀	2
C₅×C₂²⋊Q₈	2
#4090	2
C₄×C₂²⋊D₅	2
#4043	2
#4042	2
#4034	2
#4033	2
#4027	2
#4024	2
D₄×Q₈	5