Finite Group Databasebeta

Subgroups — #10599539

Order 640.

222 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 75 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	#65418	C₂	1	1	yes	maximal, characteristic
2	#65291	C₂	1	1	yes	maximal, characteristic
3	#65288	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#3937	C₂	1	1	yes	maximal, characteristic
5	C₂×#3935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#339	C₂	1	1	yes	maximal, characteristic, complemented
7	#64778	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	C₅×#117	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×#681	C₂²	1	1	yes	characteristic
13	C₂²×#168	C₂²	1	1	yes	characteristic
14	C₂²×#168	C₂²	1	1	yes	characteristic
15	#3952	—	2	2	no	—
16	#3942	C₄	1	2	yes	complemented
17	#3941	C₄	1	2	yes	complemented
18	#3937	—	2	2	no	—
19	C₄⋊#168	—	2	2	no	—
20	#3935	—	2	2	no	—
21	#1938	—	5	1	no	maximal, Sylow
22	C₂²×C₄×C₅	—	2	1	no	—
23	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₅	—	2	1	no	—
25	#709	—	2	1	no	—
26	#709	—	2	1	no	—
27	#709	—	2	1	no	—
28	#709	—	2	1	no	—
29	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅	—	2	1	no	—
31	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
32	C₅×M₁₆	—	2	1	no	—
33	C₅×M₁₆	—	2	1	no	—
34	C₂×C₅×C₈	—	2	1	no	—
35	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₅×C₂²⋊C₄	—	2	1	no	—
37	C₄²×C₅	—	2	1	no	—
38	C₄²×C₅	—	2	1	no	—
39	C₄²×C₅	D₄	1	2	yes	—
40	#681	—	2	1	no	—
41	#681	—	2	1	no	—
42	C₂×#168	—	2	1	no	—
43	C₂×#168	—	2	1	no	—
44	C₂×#168	—	2	1	no	—
45	C₂×#168	—	2	1	no	—
46	C₂×#168	—	2	1	no	—
47	C₂×#168	C₂×C₄	1	2	yes	—
48	C₂×#168	D₄	1	2	yes	—
49	#424	—	5	1	no	—
50	C₂×#104	—	5	1	no	—
51	#416	—	5	1	no	—
52	#408	—	5	1	no	—
53	C₂×#97	—	5	1	no	—
54	#341	—	5	1	no	—
55	#339	D₅	1	1	yes	characteristic, complemented
56	C₂³×C₅	#41	1	1	yes	characteristic
57	C₂²×D₅	—	2	1	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	—	4	2	no	—
61	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₄×C₅	#41	1	1	yes	characteristic
66	C₂×C₄×C₅	#41	1	1	yes	characteristic
67	C₂²⋊D₅	—	4	1	no	—
68	C₂²⋊D₅	—	4	1	no	—
69	C₂×Dic₅	—	2	1	no	—
70	D₂₀	—	2	1	no	—
71	D₂₀	—	2	1	no	—
72	C₄×D₅	—	4	1	no	—
73	C₄×D₅	—	4	1	no	—
74	Dic₁₀	—	2	1	no	—
75	Dic₁₀	—	2	1	no	—
76	C₅×C₈	—	4	1	no	—
77	#168	—	2	2	no	—
78	#168	—	2	2	no	—
79	#168	—	4	1	no	—
80	C₂×#41	—	5	1	no	—
81	C₂×M₁₆	—	5	1	no	—
82	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₈	—	5	1	no	—
84	C₂²×C₈	—	5	1	no	—
85	#117	C₂×D₅	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
87	C₄⋊C₈	—	10	2	no	—
88	#104	—	10	2	no	—
89	#103	—	5	2	no	—
90	#102	—	5	2	no	—
91	C₂²⋊C₈	—	10	2	no	—
92	#97	—	10	2	no	—
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
95	C₂²×C₅	#123	1	1	yes	characteristic
96	C₂²×C₅	#126	1	1	yes	characteristic
97	C₂×D₅	—	4	1	no	—
98	C₂×D₅	—	4	2	no	—
99	C₄×C₅	—	4	2	no	—
100	C₄×C₅	—	4	2	no	—
101	C₄×C₅	#117	1	1	yes	characteristic
102	C₄×C₅	C₄×D₄	1	1	yes	characteristic
103	C₄×C₅	#121	1	1	yes	characteristic
104	C₄×C₅	#124	1	1	yes	characteristic
105	Dic₅	—	4	1	no	—
106	#41	—	10	1	no	—
107	#41	—	10	1	no	—
108	#41	—	10	1	no	—
109	#41	—	10	1	no	—
110	C₂×Q₈	—	5	1	no	—
111	C₂×D₄	—	5	1	no	—
112	C₂×D₄	—	10	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	10	1	no	—
115	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
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	—	5	2	no	—
122	C₂×C₈	—	5	2	no	—
123	C₂×C₈	—	10	1	no	—
124	C₂×C₈	—	10	1	no	—
125	C₂×C₈	—	10	1	no	—
126	C₂×C₈	—	10	1	no	—
127	C₂×C₈	—	10	1	no	—
128	C₄⋊C₄	C₄×D₅	1	2	yes	—
129	C₂²⋊C₄	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	D₂₀	1	2	yes	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	—	2	1	no	—
135	C₂×C₅	#388	1	1	yes	characteristic
136	C₂×C₅	#444	1	2	yes	—
137	D₅	—	8	1	no	—
138	C₂³	—	10	1	no	—
139	C₂³	#711	1	1	yes	characteristic
140	Q₈	—	10	1	no	—
141	Q₈	—	10	1	no	—
142	D₄	—	10	1	no	—
143	D₄	—	10	1	no	—
144	D₄	—	20	1	no	—
145	D₄	—	20	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	10	1	no	—
150	C₂×C₄	—	20	1	no	—
151	C₂×C₄	—	20	1	no	—
152	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
153	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
154	C₂×C₄	#709	1	1	yes	characteristic
155	C₂×C₄	D₄×D₅	1	1	yes	characteristic
156	C₂×C₄	#711	1	1	yes	characteristic
157	C₂×C₄	#713	1	1	yes	characteristic
158	C₈	—	4	1	no	—
159	C₈	—	10	2	no	—
160	C₈	—	10	2	no	—
161	C₈	—	20	1	no	—
162	C₅	#1938	1	1	yes	characteristic, complemented, Sylow
163	C₂²	—	2	2	no	—
164	C₂²	—	20	2	no	—
165	C₂²	—	20	1	no	—
166	C₂²	#4027	1	1	yes	characteristic
167	C₂²	#4032	1	1	yes	characteristic
168	C₂²	#4044	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	20	1	no	—
172	C₄	C₄×D₂₀	1	1	yes	characteristic
173	C₄	#4031	1	1	yes	characteristic
174	C₄	#4038	1	1	yes	characteristic
175	C₄	#4041	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	40	1	no	—
179	C₂	#64959	1	1	yes	characteristic
180	C₂	#65112	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#339	C₂
socle	C₂²×C₅	C₄×D₄
prime radical	C₂×#709	C₂²
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	#339	D₅
hypocenter	C₅	#1938
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
3. #65288	C₂	semidirect
5. C₂×#3935	C₂	semidirect
6. C₅×#339	C₂	semidirect
7. #64778	C₂	semidirect
16. #3942	C₄	semidirect
17. #3941	C₄	semidirect
55. #339	D₅	semidirect
162. C₅	#1938	semidirect

Maximal subgroups

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

subgroup	index
#65418	2
#65291	2
#65288	2
C₂×#3937	2
C₂×#3935	2
C₅×#339	2
#64778	2
#1938	5