Finite Group Databasebeta

Subgroups — #10600925

Order 640.

190 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₅×#413	C₂	1	1	yes	maximal, characteristic
2	#65525	C₂	1	1	yes	maximal, characteristic
3	#65522	C₂	1	1	yes	maximal, characteristic
4	#65408	C₂	1	1	yes	maximal, characteristic, complemented
5	#64779	C₂	1	1	yes	maximal, characteristic, complemented
6	#64778	C₂	1	1	yes	maximal, characteristic, complemented
7	#64777	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#41	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	#4072	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	C₅×#101	—	2	1	no	—
16	C₅×#100	—	2	1	no	—
17	#3969	—	2	2	no	—
18	#3967	—	2	1	no	—
19	#3964	—	2	1	no	—
20	#3946	—	2	2	no	—
21	#2080	—	5	1	no	maximal, Sylow
22	C₅×#41	—	2	1	no	—
23	C₅×#41	—	2	1	no	—
24	C₅×#41	—	2	1	no	—
25	C₅×#41	—	2	1	no	—
26	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
27	C₂×C₅×D₄	—	2	1	no	—
28	C₂×C₅×D₄	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅	—	2	1	no	—
30	C₂²×C₄×C₅	C₂³	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	C₂²⋊Dic₅	—	2	1	no	—
38	C₂²⋊Dic₅	—	2	1	no	—
39	C₄⋊Dic₅	—	2	1	no	—
40	C₄⋊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	D₅	1	1	yes	characteristic
53	#344	—	5	1	no	—
54	#341	—	5	1	no	—
55	H(3,ℤ/4)	—	5	1	no	—
56	C₂³×C₅	—	2	1	no	—
57	C₂³×C₅	#41	1	1	yes	characteristic
58	C₅×Q₈	—	2	1	no	—
59	C₅×Q₈	—	2	1	no	—
60	C₅×D₄	—	2	1	no	—
61	C₅×D₄	—	2	1	no	—
62	C₅×D₄	—	4	1	no	—
63	C₅×D₄	—	4	1	no	—
64	C₂×C₄×C₅	—	2	1	no	—
65	C₂×C₄×C₅	—	4	1	no	—
66	C₂×C₄×C₅	—	4	1	no	—
67	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅	#41	1	1	yes	characteristic
70	C₂×C₄×C₅	#41	1	1	yes	characteristic
71	C₂×C₄×C₅	#41	1	1	yes	characteristic
72	C₂×C₄×C₅	#41	1	1	yes	characteristic
73	C₂×Dic₅	—	2	2	no	—
74	C₂×Dic₅	—	4	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	C₂×D₅	1	1	yes	characteristic
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	—	5	1	no	—
86	C₄⋊C₈	—	10	2	no	—
87	#104	—	10	2	no	—
88	#103	—	10	1	no	—
89	#102	—	10	1	no	—
90	#101	—	2	1	no	—
91	#100	—	2	1	no	—
92	C₂²×C₅	—	4	1	no	—
93	C₂²×C₅	—	4	1	no	—
94	C₂²×C₅	—	8	1	no	—
95	C₂²×C₅	#123	1	1	yes	characteristic
96	C₂²×C₅	#123	1	1	yes	characteristic
97	C₂²×C₅	#126	1	1	yes	characteristic
98	C₄×C₅	—	4	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	—	8	1	no	—
105	#41	—	2	1	no	—
106	#41	—	2	1	no	—
107	#41	—	2	1	no	—
108	#41	—	2	1	no	—
109	C₂×Q₈	C₂²⋊D₅	1	1	yes	characteristic
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	C₂²⋊D₅	1	1	yes	characteristic
112	C₂²×C₄	—	2	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	—	10	1	no	—
127	C₄⋊C₄	—	10	1	no	—
128	C₂²⋊C₄	—	10	1	no	—
129	C₂²⋊C₄	—	10	1	no	—
130	C₄²	—	5	2	no	—
131	C₄²	—	10	1	no	—
132	C₂×C₅	—	2	1	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	—	2	1	no	—
135	C₂×C₅	—	8	1	no	—
136	C₂×C₅	#397	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	C₂³	#709	1	1	yes	characteristic
139	Q₈	—	2	1	no	—
140	Q₈	—	2	1	no	—
141	D₄	—	2	1	no	—
142	D₄	—	2	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	4	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	10	2	no	—
149	C₂×C₄	—	20	1	no	—
150	C₂×C₄	#709	1	1	yes	characteristic
151	C₂×C₄	D₄×D₅	1	1	yes	characteristic
152	C₂×C₄	#711	1	1	yes	characteristic
153	C₂×C₄	#711	1	1	yes	characteristic
154	C₂×C₄	#711	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₅	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	—	4	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	#4025	1	1	yes	characteristic
165	C₂²	#4029	1	1	yes	characteristic
166	C₂²	#4075	1	1	yes	characteristic
167	C₄	—	4	1	no	—
168	C₄	—	20	2	no	—
169	C₄	—	40	1	no	—
170	C₄	#4025	1	1	yes	characteristic
171	C₄	#4031	1	1	yes	characteristic
172	C₄	#4081	1	1	yes	characteristic
173	C₄	#4085	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#65241	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4031
commutator, cosocle	C₂²×C₄×C₅	C₂³
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#413	C₂
socle	C₂×C₅	#397
prime radical	C₂×C₅×#41	C₂²
hypercenter	#413	D₅
hypocenter	C₅	#2080
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

Complements

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

N	H ≅ G/N	type
4. #65408	C₂	semidirect
5. #64779	C₂	semidirect
6. #64778	C₂	semidirect
7. #64777	C₂	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
C₅×#413	2
#65525	2
#65522	2
#65408	2
#64779	2
#64778	2
#64777	2
#2080	5