Finite Group Databasebeta

Subgroups — #65990

Order 320.

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

normal subgroups only

This group has 330 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×#711	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	#4083	C₂	1	1	yes	maximal, characteristic, complemented
5	#4075	C₂	1	1	yes	maximal, characteristic, complemented
6	#4042	C₂	1	2	yes	maximal, complemented
7	#4039	C₂	1	2	yes	maximal, complemented
8	#4030	C₂	1	2	yes	maximal, complemented
9	#4028	C₂	1	1	yes	maximal, characteristic, complemented
10	#4027	C₂	1	2	yes	maximal, complemented
11	#4023	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic, complemented
18	C₂²×Dic₅	C₂²	1	1	yes	characteristic
19	C₂²×Dic₅	C₂²	1	1	yes	characteristic, complemented
20	#711	—	2	4	no	—
21	C₂×D₂₀	C₂²	1	2	yes	—
22	C₂×C₄×D₅	—	2	4	no	—
23	C₂×C₄×D₅	C₂²	1	2	yes	—
24	C₂×C₄×D₅	C₂²	1	2	yes	—
25	C₂×C₄×D₅	C₂²	1	1	yes	characteristic, complemented
26	C₂×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
27	C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
28	C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
30	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
31	#685	C₂²	1	2	yes	—
32	#685	C₂²	1	2	yes	—
33	#685	C₂²	1	2	yes	complemented
34	#683	C₂²	1	2	yes	complemented
35	#683	C₂²	1	2	yes	complemented
36	C₄×Dic₅	C₂²	1	2	yes	complemented
37	#525	—	5	1	no	maximal, Sylow
38	C₂³×C₅	C₂³	1	1	yes	characteristic
39	C₂³×C₅	C₂³	1	1	yes	characteristic
40	C₂²×D₅	—	2	2	no	—
41	C₂²×D₅	—	2	4	no	—
42	C₂²×D₅	C₂³	1	1	yes	characteristic
43	C₂²×D₅	C₂³	1	2	yes	—
44	C₅×D₄	—	2	2	no	—
45	C₂×C₄×C₅	—	2	2	no	—
46	C₂×C₄×C₅	C₂³	1	2	yes	—
47	C₂×C₄×C₅	C₂³	1	2	yes	—
48	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂²⋊D₅	—	2	2	no	—
50	C₂²⋊D₅	—	2	4	no	—
51	C₂²⋊D₅	—	2	2	no	—
52	C₂×Dic₅	—	2	2	no	—
53	C₂×Dic₅	D₄	1	4	yes	complemented
54	C₂×Dic₅	C₂³	1	2	yes	—
55	C₂×Dic₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₅	C₂³	1	1	yes	characteristic
58	D₂₀	—	2	4	no	—
59	C₄×D₅	—	2	4	no	—
60	C₄×D₅	—	2	4	no	—
61	C₄×D₅	—	2	2	no	—
62	Dic₁₀	—	2	2	no	—
63	C₂×#41	—	5	1	no	—
64	C₂³×C₄	—	5	1	no	—
65	#123	—	5	1	no	—
66	#123	D₅	1	1	yes	characteristic, complemented
67	C₂²⋊Q₈	—	5	2	no	—
68	#121	—	5	2	no	—
69	#120	—	5	1	no	—
70	#120	—	5	1	no	—
71	C₄×D₄	—	5	2	no	—
72	C₄×D₄	—	5	2	no	—
73	#117	—	5	1	no	—
74	C₂²×C₅	—	2	2	no	—
75	C₂²×C₅	—	2	1	no	—
76	C₂²×C₅	—	2	2	no	—
77	C₂²×C₅	C₂×D₄	1	2	yes	—
78	C₂²×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×D₅	—	2	8	no	—
80	C₂×D₅	—	2	2	no	—
81	C₂×D₅	—	2	2	no	—
82	C₂×D₅	—	2	1	no	—
83	C₂×D₅	—	2	2	no	—
84	C₂×D₅	#41	1	4	yes	complemented
85	C₄×C₅	—	2	1	no	—
86	C₄×C₅	—	2	2	no	—
87	C₄×C₅	—	2	2	no	—
88	Dic₅	—	2	1	no	—
89	Dic₅	—	2	2	no	—
90	Dic₅	C₂×D₄	1	2	yes	—
91	Dic₅	C₂×D₄	1	2	yes	—
92	C₂⁴	—	5	1	no	—
93	#41	—	10	4	no	—
94	C₂×Q₈	—	5	1	no	—
95	C₂×D₄	—	5	2	no	—
96	C₂×D₄	—	5	1	no	—
97	C₂×D₄	—	5	1	no	—
98	C₂×D₄	—	5	2	no	—
99	C₂×D₄	C₂×D₅	1	1	yes	characteristic
100	C₂²×C₄	—	5	1	no	—
101	C₂²×C₄	—	5	1	no	—
102	C₂²×C₄	—	5	2	no	—
103	C₂²×C₄	—	5	1	no	—
104	C₂²×C₄	—	5	2	no	—
105	C₂²×C₄	—	10	4	no	—
106	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
107	C₄⋊C₄	—	5	2	no	—
108	C₄⋊C₄	—	5	2	no	—
109	C₄⋊C₄	C₂×D₅	1	2	yes	complemented
110	C₂²⋊C₄	—	5	1	no	—
111	C₂²⋊C₄	—	5	2	no	—
112	C₂²⋊C₄	—	5	2	no	—
113	C₂²⋊C₄	—	5	2	no	—
114	C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
115	C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
116	C₄²	—	5	2	no	—
117	C₂×C₅	—	2	2	no	—
118	C₂×C₅	—	4	1	no	—
119	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₅	C₂×#41	1	2	yes	—
121	D₅	—	2	4	no	—
122	D₅	—	4	1	no	—
123	C₂³	—	5	2	no	—
124	C₂³	—	5	1	no	—
125	C₂³	—	10	4	no	—
126	C₂³	—	10	2	no	—
127	C₂³	C₂²×D₅	1	1	yes	characteristic
128	C₂³	C₂²×D₅	1	1	yes	characteristic
129	Q₈	—	10	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	10	4	no	—
132	D₄	—	10	2	no	—
133	D₄	—	10	4	no	—
134	D₄	—	10	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	5	4	no	—
137	C₂×C₄	—	5	1	no	—
138	C₂×C₄	—	5	1	no	—
139	C₂×C₄	—	5	2	no	—
140	C₂×C₄	—	5	1	no	—
141	C₂×C₄	—	10	4	no	—
142	C₂×C₄	—	10	4	no	—
143	C₂×C₄	—	10	2	no	—
144	C₂×C₄	—	10	2	no	—
145	C₂×C₄	C₂²×D₅	1	2	yes	—
146	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₅	1	2	yes	—
148	C₅	#525	1	1	yes	characteristic, complemented, Sylow
149	C₂²	—	2	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	5	4	no	—
153	C₂²	—	10	8	no	—
154	C₂²	—	10	2	no	—
155	C₂²	—	10	2	no	—
156	C₂²	—	10	1	no	—
157	C₂²	—	10	2	no	—
158	C₂²	D₄×D₅	1	2	yes	—
159	C₂²	C₂³×D₅	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	2	no	—
163	C₄	—	5	2	no	—
164	C₄	—	5	2	no	—
165	C₄	—	10	1	no	—
166	C₄	—	10	2	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	1	no	—
169	C₂	—	10	4	no	—
170	C₂	—	20	1	no	—
171	C₂	C₂×D₄×D₅	1	1	yes	characteristic
172	C₂	D₅×#41	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#123	C₂
hypercenter	#123	D₅
hypocenter	C₅	#525
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. C₂×#711	C₂	semidirect
2. C₂²×C₄×D₅	C₂	semidirect
3. C₅×#123	C₂	semidirect
4. #4083	C₂	semidirect
5. #4075	C₂	semidirect
6. #4042	C₂	semidirect
7. #4039	C₂	semidirect
8. #4030	C₂	semidirect
9. #4028	C₂	semidirect
10. #4027	C₂	semidirect
11. #4023	C₂	semidirect
17. C₂×C₂²⋊D₅	C₂²	semidirect
19. C₂²×Dic₅	C₂²	semidirect
25. C₂×C₄×D₅	C₂²	semidirect
26. C₂×Dic₁₀	C₂²	semidirect
27. C₅×C₄⋊C₄	C₂²	semidirect
29. C₅×C₂²⋊C₄	C₂²	semidirect
33. #685	C₂²	semidirect
34. #683	C₂²	semidirect
35. #683	C₂²	semidirect
36. C₄×Dic₅	C₂²	semidirect
53. C₂×Dic₅	D₄	semidirect
66. #123	D₅	semidirect
84. C₂×D₅	#41	semidirect
109. C₄⋊C₄	C₂×D₅	semidirect
115. C₂²⋊C₄	C₂×D₅	semidirect
148. C₅	#525	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
C₂²×C₄×D₅	2
C₅×#123	2
#4083	2
#4075	2
#4042	2
#4039	2
#4030	2
#4028	2
#4027	2
#4023	2
#525	5