Finite Group Databasebeta

Subgroups — #65933

Order 320.

312 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 105 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×#711	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	#4086	C₂	1	2	yes	maximal, complemented
5	#4086	C₂	1	1	yes	maximal, characteristic, complemented
6	#4085	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#4041	C₂	1	1	yes	maximal, characteristic, complemented
9	#4033	C₂	1	1	yes	maximal, characteristic, complemented
10	#4030	C₂	1	2	yes	maximal, complemented
11	#4027	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₄	C₂²	1	2	yes	—
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	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
20	C₂²×Dic₅	C₂²	1	2	yes	—
21	#711	—	2	4	no	—
22	#709	—	2	4	no	—
23	C₂×D₂₀	C₂²	1	1	yes	characteristic
24	C₂×D₂₀	C₂²	1	2	yes	—
25	C₂×C₄×D₅	C₂²	1	2	yes	—
26	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
31	#685	C₂²	1	2	yes	—
32	#685	C₂²	1	1	yes	characteristic
33	#683	C₂²	1	1	yes	characteristic
34	#683	C₂²	1	2	yes	—
35	C₄×Dic₅	C₂²	1	2	yes	—
36	C₄×Dic₅	C₂²	1	1	yes	characteristic
37	#550	—	5	1	no	maximal, Sylow
38	C₂³×C₅	C₂³	1	1	yes	characteristic
39	C₂³×C₅	C₂³	1	2	yes	—
40	C₂²×D₅	C₂³	1	2	yes	—
41	C₂²×D₅	C₂³	1	1	yes	characteristic
42	C₅×D₄	—	2	2	no	—
43	C₅×D₄	—	4	2	no	—
44	C₂×C₄×C₅	—	2	2	no	—
45	C₂×C₄×C₅	C₂³	1	2	yes	—
46	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂²⋊D₅	—	2	4	no	—
49	C₂²⋊D₅	—	2	4	no	—
50	C₂²⋊D₅	—	2	4	no	—
51	C₂²⋊D₅	—	2	2	no	—
52	C₂²⋊D₅	—	4	2	no	—
53	C₂×Dic₅	—	2	4	no	—
54	C₂×Dic₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₅	C₂³	1	2	yes	—
57	C₂×Dic₅	C₂³	1	1	yes	characteristic
58	D₂₀	—	2	2	no	—
59	D₂₀	—	4	2	no	—
60	C₄×D₅	—	2	2	no	—
61	C₄×D₅	—	2	4	no	—
62	Dic₁₀	D₄	1	4	yes	complemented
63	C₂×#41	—	5	1	no	—
64	C₂×#41	—	5	1	no	—
65	#127	—	5	2	no	—
66	#127	—	5	1	no	—
67	#121	—	5	2	no	—
68	#121	—	5	1	no	—
69	#121	—	5	2	no	—
70	#121	D₅	1	1	yes	characteristic, complemented
71	C₄×Q₈	—	5	1	no	—
72	C₄×D₄	—	5	1	no	—
73	C₄×D₄	—	5	2	no	—
74	C₂²×C₅	—	2	2	no	—
75	C₂²×C₅	—	2	1	no	—
76	C₂²×C₅	—	2	1	no	—
77	C₂²×C₅	—	2	2	no	—
78	C₂²×C₅	—	2	2	no	—
79	C₂²×C₅	—	2	1	no	—
80	C₂²×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×D₅	—	2	2	no	—
82	C₂×D₅	—	2	1	no	—
83	C₂×D₅	—	2	2	no	—
84	C₂×D₅	—	2	1	no	—
85	C₂×D₅	—	2	1	no	—
86	C₂×D₅	—	2	2	no	—
87	C₄×C₅	—	2	1	no	—
88	C₄×C₅	—	2	2	no	—
89	C₄×C₅	C₂×D₄	1	2	yes	—
90	Dic₅	—	2	1	no	—
91	Dic₅	—	2	1	no	—
92	Dic₅	C₂×D₄	1	4	yes	—
93	Dic₅	#41	1	2	yes	complemented
94	#41	—	10	4	no	—
95	#41	—	10	4	no	—
96	C₂×Q₈	—	5	1	no	—
97	C₂×D₄	—	5	1	no	—
98	C₂×D₄	—	5	2	no	—
99	C₂×D₄	—	5	2	no	—
100	C₂×D₄	—	5	2	no	—
101	C₂×D₄	—	5	2	no	—
102	C₂×D₄	—	5	1	no	—
103	C₂×D₄	—	5	2	no	—
104	C₂×D₄	C₂×D₅	1	1	yes	characteristic
105	C₂×D₄	C₂×D₅	1	2	yes	—
106	C₂²×C₄	—	5	2	no	—
107	C₂²×C₄	—	5	2	no	—
108	C₂²×C₄	—	5	1	no	—
109	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
110	C₄⋊C₄	—	5	1	no	—
111	C₄⋊C₄	—	5	2	no	—
112	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
113	C₂²⋊C₄	—	5	1	no	—
114	C₂²⋊C₄	—	5	1	no	—
115	C₂²⋊C₄	—	5	2	no	—
116	C₂²⋊C₄	C₂×D₅	1	2	yes	—
117	C₄²	—	5	2	no	—
118	C₄²	—	5	1	no	—
119	C₂×C₅	—	4	2	no	—
120	C₂×C₅	—	4	1	no	—
121	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₅	C₂×#41	1	1	yes	characteristic
123	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
124	D₅	—	4	2	no	—
125	D₅	—	4	1	no	—
126	C₂³	—	5	1	no	—
127	C₂³	—	5	2	no	—
128	C₂³	C₂²×D₅	1	1	yes	characteristic
129	C₂³	C₂²×D₅	1	2	yes	—
130	Q₈	—	5	4	no	—
131	D₄	—	2	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	10	4	no	—
134	D₄	—	10	4	no	—
135	D₄	—	10	2	no	—
136	D₄	—	10	4	no	—
137	D₄	—	10	2	no	—
138	D₄	—	20	2	no	—
139	D₄	—	20	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	5	1	no	—
142	C₂×C₄	—	5	2	no	—
143	C₂×C₄	—	5	1	no	—
144	C₂×C₄	—	5	1	no	—
145	C₂×C₄	—	10	4	no	—
146	C₂×C₄	—	10	4	no	—
147	C₂×C₄	—	10	2	no	—
148	C₂×C₄	C₂²×D₅	1	2	yes	—
149	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
151	C₅	#550	1	1	yes	characteristic, complemented, Sylow
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	10	2	no	—
159	C₂²	—	10	1	no	—
160	C₂²	—	10	1	no	—
161	C₂²	—	10	2	no	—
162	C₂²	—	10	1	no	—
163	C₂²	—	10	2	no	—
164	C₂²	C₂³×D₅	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	2	no	—
167	C₄	—	5	4	no	—
168	C₄	—	5	2	no	—
169	C₄	—	10	1	no	—
170	C₄	—	10	1	no	—
171	C₄	D₄×D₅	1	2	yes	—
172	C₂	—	4	1	no	—
173	C₂	—	4	2	no	—
174	C₂	—	20	2	no	—
175	C₂	—	20	1	no	—
176	C₂	C₂×D₄×D₅	1	1	yes	characteristic
177	C₂	#4144	1	1	yes	characteristic
178	C₂	D₅×#41	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#121	C₂
hypercenter	#121	D₅
hypocenter	C₅	#550
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#711	C₂	semidirect
2. C₂×#709	C₂	semidirect
3. C₅×#121	C₂	semidirect
4. #4086	C₂	semidirect
5. #4086	C₂	semidirect
6. #4085	C₂	semidirect
7. C₄×C₂²⋊D₅	C₂	semidirect
8. #4041	C₂	semidirect
9. #4033	C₂	semidirect
10. #4030	C₂	semidirect
11. #4027	C₂	semidirect
62. Dic₁₀	D₄	semidirect
70. #121	D₅	semidirect
93. Dic₅	#41	semidirect
151. C₅	#550	semidirect

Maximal subgroups

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

subgroup	index
C₂×#711	2
C₂×#709	2
C₅×#121	2
#4086	2
#4085	2
C₄×C₂²⋊D₅	2
#4041	2
#4033	2
#4030	2
#4027	2
#550	5