Finite Group Databasebeta

Subgroups — #10599836

Order 640.

190 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 57 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	#65428	C₂	1	1	yes	maximal, characteristic
2	#65421	C₂	1	1	yes	maximal, characteristic
3	#65408	C₂	1	1	yes	maximal, characteristic, complemented
4	#65291	C₂	1	1	yes	maximal, characteristic
5	C₅×#344	C₂	1	1	yes	maximal, characteristic, complemented
6	#64778	C₂	1	1	yes	maximal, characteristic, complemented
7	#64751	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#709	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₅×#117	C₂²	1	1	yes	characteristic
12	#4072	C₂²	1	1	yes	characteristic
13	C₂×#681	C₂²	1	1	yes	characteristic
14	C₂²×#168	C₂²	1	1	yes	characteristic
15	#3957	—	2	2	no	—
16	#3956	—	2	1	no	—
17	#3955	—	2	1	no	—
18	#3946	—	2	2	no	—
19	#3942	—	2	1	no	—
20	#3941	—	2	1	no	—
21	#2080	—	5	1	no	maximal, Sylow
22	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₅	—	2	1	no	—
24	#709	—	2	1	no	—
25	#709	—	2	1	no	—
26	#709	—	2	1	no	—
27	#709	—	2	1	no	—
28	C₂×D₂₀	D₄	1	1	yes	characteristic
29	C₂×C₄×D₅	—	2	1	no	—
30	C₂×Dic₁₀	D₄	1	1	yes	characteristic
31	C₅×M₁₆	—	2	1	no	—
32	C₅×M₁₆	—	2	1	no	—
33	C₅×M₁₆	—	2	1	no	—
34	C₅×M₁₆	—	2	1	no	—
35	C₂×C₅×C₈	—	2	1	no	—
36	C₂×C₅×C₈	—	2	1	no	—
37	C₅×C₄⋊C₄	—	2	1	no	—
38	C₅×C₂²⋊C₄	—	2	1	no	—
39	C₄²×C₅	—	2	1	no	—
40	C₂²⋊Dic₅	—	2	1	no	—
41	C₄⋊Dic₅	—	2	1	no	—
42	C₄×Dic₅	D₄	1	2	yes	—
43	#681	—	2	1	no	—
44	#681	—	2	1	no	—
45	C₂×#168	—	2	1	no	—
46	C₂×#168	—	2	1	no	—
47	C₂×#168	—	2	1	no	—
48	C₂×#168	—	2	1	no	—
49	#424	—	5	1	no	—
50	#421	—	5	1	no	—
51	#416	—	5	1	no	—
52	#413	—	5	1	no	—
53	#344	D₅	1	1	yes	characteristic, complemented
54	#341	—	5	1	no	—
55	H(3,ℤ/4)	—	5	1	no	—
56	C₂³×C₅	#41	1	1	yes	characteristic
57	C₂²×D₅	—	2	1	no	—
58	C₂×C₄×C₅	—	4	1	no	—
59	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅	#41	1	1	yes	characteristic
62	C₂×C₄×C₅	#41	1	1	yes	characteristic
63	C₂×C₄×C₅	#41	1	1	yes	characteristic
64	C₂×C₄×C₅	#41	1	1	yes	characteristic
65	C₂²⋊D₅	—	4	1	no	—
66	C₂²⋊D₅	—	4	1	no	—
67	C₂×Dic₅	—	2	2	no	—
68	C₂×Dic₅	—	2	1	no	—
69	D₂₀	—	2	1	no	—
70	D₂₀	—	2	1	no	—
71	C₄×D₅	—	4	1	no	—
72	C₄×D₅	—	4	1	no	—
73	Dic₁₀	—	2	1	no	—
74	Dic₁₀	—	2	1	no	—
75	C₅×C₈	—	4	1	no	—
76	C₅×C₈	—	4	1	no	—
77	#168	—	4	1	no	—
78	#168	—	4	1	no	—
79	C₂×#41	—	5	1	no	—
80	C₂×M₁₆	—	5	1	no	—
81	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
82	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₈	—	5	1	no	—
84	#117	—	5	1	no	—
85	#117	C₂×D₅	1	1	yes	characteristic
86	C₄⋊C₈	—	10	2	no	—
87	#104	—	10	2	no	—
88	#103	—	10	1	no	—
89	#102	—	10	1	no	—
90	#101	—	10	1	no	—
91	#100	—	10	1	no	—
92	C₂²×C₅	—	4	1	no	—
93	C₂²×C₅	#123	1	1	yes	characteristic
94	C₂²×C₅	#123	1	1	yes	characteristic
95	C₂²×C₅	#126	1	1	yes	characteristic
96	C₂×D₅	—	4	1	no	—
97	C₂×D₅	—	8	1	no	—
98	C₄×C₅	—	8	1	no	—
99	C₄×C₅	#121	1	1	yes	characteristic
100	C₄×C₅	#123	1	1	yes	characteristic
101	C₄×C₅	#124	1	1	yes	characteristic
102	C₄×C₅	#126	1	1	yes	characteristic
103	Dic₅	—	4	2	no	—
104	Dic₅	—	4	1	no	—
105	#41	—	10	1	no	—
106	#41	—	10	1	no	—
107	#41	—	10	1	no	—
108	#41	—	10	1	no	—
109	C₂×Q₈	—	5	1	no	—
110	C₂×D₄	—	5	1	no	—
111	C₂×D₄	—	10	1	no	—
112	C₂²×C₄	—	10	1	no	—
113	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	10	1	no	—
119	M₁₆	—	10	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	10	1	no	—
123	C₂×C₈	—	10	1	no	—
124	C₂×C₈	—	10	1	no	—
125	C₂×C₈	—	10	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	10	1	no	—
128	C₂²⋊C₄	—	2	1	no	—
129	C₂²⋊C₄	—	10	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	5	2	no	—
132	C₂×C₅	—	2	1	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	—	2	1	no	—
135	C₂×C₅	#397	1	1	yes	characteristic
136	D₅	—	8	1	no	—
137	C₂³	—	10	1	no	—
138	C₂³	#709	1	1	yes	characteristic
139	Q₈	—	10	1	no	—
140	Q₈	—	10	1	no	—
141	D₄	—	10	1	no	—
142	D₄	—	10	1	no	—
143	D₄	—	20	1	no	—
144	D₄	—	20	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	10	1	no	—
147	C₂×C₄	—	10	2	no	—
148	C₂×C₄	—	20	1	no	—
149	C₂×C₄	—	20	1	no	—
150	C₂×C₄	#709	1	1	yes	characteristic
151	C₂×C₄	#709	1	1	yes	characteristic
152	C₂×C₄	D₄×D₅	1	1	yes	characteristic
153	C₂×C₄	D₄×D₅	1	1	yes	characteristic
154	C₂×C₄	#711	1	1	yes	characteristic
155	C₂×C₄	#713	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	20	1	no	—
159	C₈	—	20	1	no	—
160	C₅	#2080	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	4	1	no	—
162	C₂²	—	20	1	no	—
163	C₂²	—	40	1	no	—
164	C₂²	#4022	1	1	yes	characteristic
165	C₂²	#4029	1	1	yes	characteristic
166	C₂²	#4040	1	1	yes	characteristic
167	C₄	—	8	1	no	—
168	C₄	—	20	2	no	—
169	C₄	—	20	1	no	—
170	C₄	#4030	1	1	yes	characteristic
171	C₄	#4031	1	1	yes	characteristic
172	C₄	#4040	1	1	yes	characteristic
173	C₄	#4044	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	40	1	no	—
178	C₂	#64964	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4031
commutator, cosocle	C₂²×C₄×C₅	C₂³
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#344	C₂
socle	C₂×C₅	#397
prime radical	C₂×#709	C₂²
hypercenter	#344	D₅
hypocenter	C₅	#2080
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
3. #65408	C₂	semidirect
5. C₅×#344	C₂	semidirect
6. #64778	C₂	semidirect
7. #64751	C₂	semidirect
53. #344	D₅	semidirect
160. C₅	#2080	semidirect

Maximal subgroups

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

subgroup	index
#65428	2
#65421	2
#65408	2
#65291	2
C₅×#344	2
#64778	2
#64751	2
#2080	5