Finite Group Databasebeta

Subgroups — #65964

Order 320.

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

normal subgroups only

This group has 334 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×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#4093	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#4041	C₂	1	1	yes	maximal, characteristic, complemented
7	#4041	C₂	1	2	yes	maximal, complemented
8	#4040	C₂	1	2	yes	maximal, complemented
9	#4039	C₂	1	1	yes	maximal, characteristic, complemented
10	#4028	C₂	1	2	yes	maximal, complemented
11	D₅×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₂³×D₅	C₂²	1	2	yes	—
13	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic
16	#713	—	2	4	no	—
17	C₂×D₂₀	—	2	4	no	—
18	C₂×D₂₀	C₂²	1	2	yes	—
19	C₂×D₂₀	C₂²	1	1	yes	characteristic
20	C₂×D₂₀	C₂²	1	1	yes	characteristic
21	C₂×D₂₀	C₂²	1	2	yes	complemented
22	C₂×D₂₀	C₂²	1	2	yes	complemented
23	C₂×C₄×D₅	C₂²	1	2	yes	complemented
24	C₂×C₄×D₅	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₄×D₅	C₂²	1	2	yes	complemented
26	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
27	C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
28	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
30	#685	C₂²	1	2	yes	complemented
31	#685	C₂²	1	2	yes	complemented
32	#685	C₂²	1	2	yes	complemented
33	#685	C₂²	1	2	yes	complemented
34	#685	C₂²	1	1	yes	characteristic, complemented
35	#683	C₂²	1	1	yes	characteristic, complemented
36	C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
37	#546	—	5	1	no	maximal, Sylow
38	C₂³×C₅	C₂³	1	1	yes	characteristic
39	C₂²×D₅	—	2	4	no	—
40	C₂²×D₅	—	2	4	no	—
41	C₂²×D₅	—	4	2	no	—
42	C₂²×D₅	C₂³	1	2	yes	—
43	C₂²×D₅	C₂³	1	1	yes	characteristic
44	C₂²×D₅	C₂³	1	2	yes	—
45	C₅×Q₈	—	2	2	no	—
46	C₂×C₄×C₅	—	2	2	no	—
47	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂²⋊D₅	—	2	2	no	—
52	C₂×Dic₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₅	C₂³	1	1	yes	characteristic
55	D₂₀	—	2	4	no	—
56	D₂₀	—	2	4	no	—
57	D₂₀	—	2	2	no	—
58	D₂₀	—	4	2	no	—
59	D₂₀	D₄	1	4	yes	complemented
60	C₄×D₅	—	2	4	no	—
61	C₄×D₅	—	2	2	no	—
62	C₄×D₅	—	4	2	no	—
63	C₂×#41	—	5	1	no	—
64	C₂²×D₄	—	5	1	no	—
65	#124	—	5	1	no	—
66	#123	—	5	2	no	—
67	C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
68	#121	—	5	1	no	—
69	#121	—	5	2	no	—
70	#120	—	5	2	no	—
71	C₄×D₄	—	5	1	no	—
72	C₄×D₄	—	5	1	no	—
73	C₂×C₂²⋊C₄	—	5	2	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	1	no	—
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	2	no	—
86	C₂×D₅	—	4	8	no	—
87	C₂×D₅	C₂×D₄	1	4	yes	—
88	C₄×C₅	—	2	2	no	—
89	C₄×C₅	—	2	1	no	—
90	C₄×C₅	—	2	2	no	—
91	C₄×C₅	C₂×D₄	1	2	yes	—
92	Dic₅	—	2	1	no	—
93	Dic₅	—	2	1	no	—
94	Dic₅	—	2	1	no	—
95	C₂⁴	—	5	2	no	—
96	#41	—	10	4	no	—
97	C₂×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
98	C₂×D₄	—	5	1	no	—
99	C₂×D₄	—	5	1	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	—	10	4	no	—
105	C₂²×C₄	—	5	2	no	—
106	C₂²×C₄	—	5	1	no	—
107	C₂²×C₄	—	5	2	no	—
108	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
109	C₄⋊C₄	—	5	1	no	—
110	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
111	C₄⋊C₄	C₂×D₅	1	2	yes	complemented
112	C₂²⋊C₄	—	5	1	no	—
113	C₂²⋊C₄	—	5	2	no	—
114	C₂²⋊C₄	—	5	2	no	—
115	C₂²⋊C₄	—	5	2	no	—
116	C₂²⋊C₄	—	5	1	no	—
117	C₂²⋊C₄	—	5	2	no	—
118	C₂²⋊C₄	C₂×D₅	1	2	yes	—
119	C₄²	—	5	1	no	—
120	C₂×C₅	—	2	2	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	—	2	4	no	—
125	D₅	—	4	1	no	—
126	D₅	—	4	2	no	—
127	C₂³	—	5	2	no	—
128	C₂³	—	5	2	no	—
129	C₂³	—	5	1	no	—
130	C₂³	—	10	4	no	—
131	C₂³	—	10	4	no	—
132	C₂³	—	20	2	no	—
133	C₂³	C₂²×D₅	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	5	4	no	—
136	D₄	—	10	4	no	—
137	D₄	—	10	4	no	—
138	D₄	—	10	2	no	—
139	D₄	—	10	2	no	—
140	D₄	—	20	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	5	1	no	—
143	C₂×C₄	—	5	1	no	—
144	C₂×C₄	—	5	1	no	—
145	C₂×C₄	—	10	4	no	—
146	C₂×C₄	—	10	2	no	—
147	C₂×C₄	—	20	2	no	—
148	C₂×C₄	C₂²×D₅	1	2	yes	—
149	C₂×C₄	C₂²×D₅	1	2	yes	—
150	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
152	C₅	#546	1	1	yes	characteristic, complemented, Sylow
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	5	4	no	—
156	C₂²	—	10	2	no	—
157	C₂²	—	10	2	no	—
158	C₂²	—	10	1	no	—
159	C₂²	—	10	2	no	—
160	C₂²	—	10	1	no	—
161	C₂²	—	10	2	no	—
162	C₂²	—	10	1	no	—
163	C₂²	—	10	2	no	—
164	C₂²	—	20	8	no	—
165	C₂²	#713	1	2	yes	complemented
166	C₂²	C₂³×D₅	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	10	1	no	—
171	C₄	—	10	1	no	—
172	C₄	—	10	1	no	—
173	C₄	D₄×D₅	1	2	yes	—
174	C₂	—	2	2	no	—
175	C₂	—	10	4	no	—
176	C₂	—	20	2	no	—
177	C₂	—	20	1	no	—
178	C₂	C₂×D₄×D₅	1	1	yes	characteristic
179	C₂	C₂×#713	1	1	yes	characteristic
180	C₂	#4149	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×C₂²⋊Q₈	C₂
hypercenter	C₂²⋊Q₈	D₅
hypocenter	C₅	#546
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×#713	C₂	semidirect
2. C₂²×D₂₀	C₂	semidirect
3. C₅×C₂²⋊Q₈	C₂	semidirect
4. #4093	C₂	semidirect
5. C₄×C₂²⋊D₅	C₂	semidirect
6. #4041	C₂	semidirect
7. #4041	C₂	semidirect
8. #4040	C₂	semidirect
9. #4039	C₂	semidirect
10. #4028	C₂	semidirect
11. D₅×C₂²⋊C₄	C₂	semidirect
13. C₂×C₅×Q₈	C₂²	semidirect
21. C₂×D₂₀	C₂²	semidirect
22. C₂×D₂₀	C₂²	semidirect
23. C₂×C₄×D₅	C₂²	semidirect
24. C₂×C₄×D₅	C₂²	semidirect
25. C₂×C₄×D₅	C₂²	semidirect
26. C₅×C₄⋊C₄	C₂²	semidirect
27. C₅×C₄⋊C₄	C₂²	semidirect
30. #685	C₂²	semidirect
31. #685	C₂²	semidirect
32. #685	C₂²	semidirect
33. #685	C₂²	semidirect
34. #685	C₂²	semidirect
35. #683	C₂²	semidirect
36. C₄×Dic₅	C₂²	semidirect
59. D₂₀	D₄	semidirect
67. C₂²⋊Q₈	D₅	semidirect
76. C₂²×C₅	#41	semidirect
97. C₂×Q₈	C₂×D₅	semidirect
110. C₄⋊C₄	C₂×D₅	semidirect
111. C₄⋊C₄	C₂×D₅	semidirect
152. C₅	#546	semidirect
165. C₂²	#713	semidirect

Maximal subgroups

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

subgroup	index
C₂×#713	2
C₂²×D₂₀	2
C₅×C₂²⋊Q₈	2
#4093	2
C₄×C₂²⋊D₅	2
#4041	2
#4040	2
#4039	2
#4028	2
D₅×C₂²⋊C₄	2
#546	5