Finite Group Databasebeta

Subgroups — #66113

Order 320.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	#4150	C₂	1	1	yes	maximal, characteristic, complemented
2	#4147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#4096	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#689	C₂	1	1	yes	maximal, characteristic, complemented
6	#4064	C₂	1	2	yes	maximal, complemented
7	D₅×Q₁₆	C₂	1	2	yes	maximal, complemented
8	#4062	C₂	1	2	yes	maximal, complemented
9	#4061	C₂	1	2	yes	maximal, complemented
10	#4055	C₂	1	1	yes	maximal, characteristic, complemented
11	#4053	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
14	#713	—	2	1	no	—
15	#713	C₂²	1	2	yes	—
16	Q₈×D₅	—	2	1	no	—
17	Q₈×D₅	C₂²	1	2	yes	—
18	Q₈×D₅	C₂²	1	2	yes	—
19	#711	—	2	2	no	—
20	#711	C₂²	1	2	yes	—
21	#709	—	2	2	no	—
22	#709	—	2	1	no	—
23	#709	C₂²	1	1	yes	characteristic
24	#709	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₀	—	2	1	no	—
26	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
27	C₅×Q₁₆	C₂²	1	2	yes	—
28	C₅×SD₁₆	C₂²	1	2	yes	—
29	C₅×M₁₆	C₂²	1	1	yes	characteristic
30	#689	C₂²	1	1	yes	characteristic
31	#689	C₂²	1	2	yes	—
32	#689	C₂²	1	2	yes	—
33	#688	C₂²	1	1	yes	characteristic
34	#687	C₂²	1	1	yes	characteristic
35	#686	C₂²	1	1	yes	characteristic
36	C₂×#168	C₂²	1	1	yes	characteristic
37	Dic₂₀	C₂²	1	2	yes	—
38	#677	C₂²	1	2	yes	—
39	#676	C₂²	1	2	yes	—
40	C₈×D₅	C₂²	1	2	yes	—
41	#578	—	5	1	no	maximal, Sylow
42	C₅×Q₈	—	2	1	no	—
43	C₅×Q₈	C₂³	1	1	yes	characteristic
44	C₅×Q₈	C₂³	1	2	yes	—
45	C₅×D₄	—	2	1	no	—
46	C₅×D₄	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₅	—	2	1	no	—
48	C₂×C₄×C₅	—	2	1	no	—
49	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂²⋊D₅	—	2	1	no	—
51	C₂²⋊D₅	—	2	2	no	—
52	C₂²⋊D₅	D₄	1	2	yes	—
53	C₂×Dic₅	—	2	1	no	—
54	C₂×Dic₅	—	2	2	no	—
55	D₂₀	—	2	2	no	—
56	D₂₀	D₄	1	1	yes	characteristic
57	D₂₀	C₂³	1	1	yes	characteristic
58	C₄×D₅	—	2	2	no	—
59	C₄×D₅	—	2	2	no	—
60	C₄×D₅	—	2	1	no	—
61	C₄×D₅	—	2	2	no	—
62	C₄×D₅	C₂³	1	2	yes	—
63	Dic₁₀	—	2	1	no	—
64	Dic₁₀	—	2	2	no	—
65	Dic₁₀	—	2	2	no	—
66	Dic₁₀	D₄	1	1	yes	characteristic
67	Dic₁₀	C₂³	1	2	yes	—
68	Dic₁₀	C₂³	1	1	yes	characteristic
69	C₅×C₈	C₂³	1	2	yes	—
70	#168	C₂³	1	1	yes	characteristic
71	#168	C₂³	1	1	yes	characteristic
72	2₋¹⁺⁴	—	5	1	no	—
73	2₋¹⁺⁴	—	5	1	no	—
74	#137	—	5	2	no	—
75	#137	—	5	2	no	—
76	#137	—	5	1	no	—
77	#137	D₅	1	1	yes	characteristic, complemented
78	#135	—	5	1	no	—
79	#135	—	5	2	no	—
80	C₂×Q₁₆	—	5	1	no	—
81	C₂×Q₁₆	—	5	2	no	—
82	#131	—	5	1	no	—
83	C₂²×C₅	—	2	1	no	—
84	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×D₅	—	2	1	no	—
86	C₂×D₅	C₂×D₄	1	2	yes	—
87	C₄×C₅	—	2	1	no	—
88	C₄×C₅	—	2	2	no	—
89	C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄×C₅	C₂⁴	1	1	yes	characteristic
91	Dic₅	—	2	1	no	—
92	Dic₅	—	2	2	no	—
93	Dic₅	C₂×D₄	1	2	yes	—
94	#41	—	5	1	no	—
95	#41	—	5	2	no	—
96	#41	—	5	2	no	—
97	#41	—	5	1	no	—
98	#41	—	10	2	no	—
99	#41	—	10	1	no	—
100	#41	—	10	2	no	—
101	#41	—	10	1	no	—
102	#41	C₂×D₅	1	1	yes	characteristic
103	C₂×Q₈	—	5	2	no	—
104	C₂×Q₈	—	5	2	no	—
105	C₂×Q₈	—	5	1	no	—
106	C₂×Q₈	—	10	1	no	—
107	C₂×Q₈	—	10	1	no	—
108	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
109	Q₁₆	—	5	1	no	—
110	Q₁₆	—	5	2	no	—
111	Q₁₆	—	5	2	no	—
112	Q₁₆	—	5	2	no	—
113	Q₁₆	C₂×D₅	1	2	yes	—
114	SD₁₆	—	5	1	no	—
115	SD₁₆	—	5	1	no	—
116	SD₁₆	—	5	2	no	—
117	SD₁₆	C₂×D₅	1	2	yes	—
118	D₈	—	5	1	no	—
119	M₁₆	—	5	2	no	—
120	M₁₆	C₂×D₅	1	1	yes	characteristic
121	C₂×C₈	—	5	1	no	—
122	C₂×C₈	—	5	2	no	—
123	C₂×C₅	—	2	1	no	—
124	C₂×C₅	—	4	1	no	—
125	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
126	D₅	—	2	2	no	—
127	D₅	—	4	1	no	—
128	Q₈	—	2	1	no	—
129	Q₈	—	5	1	no	—
130	Q₈	—	5	1	no	—
131	Q₈	—	5	2	no	—
132	Q₈	—	10	2	no	—
133	Q₈	—	10	2	no	—
134	Q₈	—	10	1	no	—
135	Q₈	C₂²×D₅	1	1	yes	characteristic
136	Q₈	C₂²×D₅	1	2	yes	—
137	D₄	—	2	1	no	—
138	D₄	—	5	2	no	—
139	D₄	—	5	1	no	—
140	D₄	—	5	1	no	—
141	D₄	—	10	2	no	—
142	D₄	—	10	2	no	—
143	D₄	—	10	1	no	—
144	D₄	C₂²×D₅	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	5	2	no	—
148	C₂×C₄	—	10	2	no	—
149	C₂×C₄	—	10	2	no	—
150	C₂×C₄	—	10	2	no	—
151	C₂×C₄	—	10	2	no	—
152	C₂×C₄	—	10	1	no	—
153	C₂×C₄	—	10	1	no	—
154	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₈	—	5	1	no	—
156	C₈	—	5	1	no	—
157	C₈	C₂²×D₅	1	2	yes	—
158	C₅	#578	1	1	yes	characteristic, complemented, Sylow
159	C₂²	—	2	1	no	—
160	C₂²	—	5	2	no	—
161	C₂²	—	10	1	no	—
162	C₂²	D₄×D₅	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	2	no	—
165	C₄	—	5	2	no	—
166	C₄	—	10	1	no	—
167	C₄	—	10	2	no	—
168	C₄	D₄×D₅	1	1	yes	characteristic
169	C₄	C₂³×D₅	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	4	1	no	—
172	C₂	—	10	2	no	—
173	C₂	—	20	1	no	—
174	C₂	C₂×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴
Frattini	C₄	C₂³×D₅
Fitting	C₅×#137	C₂
socle	C₂×C₅	C₂²×D₄
hypercenter	#137	D₅
hypocenter	C₅	#578
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #4150	C₂	semidirect
2. #4147	C₂	semidirect
3. C₅×#137	C₂	semidirect
4. #4096	C₂	semidirect
5. C₂×#689	C₂	semidirect
6. #4064	C₂	semidirect
7. D₅×Q₁₆	C₂	semidirect
8. #4062	C₂	semidirect
9. #4061	C₂	semidirect
10. #4055	C₂	semidirect
11. #4053	C₂	semidirect
77. #137	D₅	semidirect
158. C₅	#578	semidirect

Maximal subgroups

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

subgroup	index
#4150	2
#4147	2
C₅×#137	2
#4096	2
C₂×#689	2
#4064	2
D₅×Q₁₆	2
#4062	2
#4061	2
#4055	2
#4053	2
#578	5