Finite Group Databasebeta

Subgroups — #10601439

Order 640.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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×#4047	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#3951	C₂	1	1	yes	maximal, characteristic, complemented
4	#65255	C₂	1	2	yes	maximal
5	#64750	C₂	1	2	yes	maximal, complemented
6	C₂×#709	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×#681	C₂²	1	2	yes	—
10	C₂×#681	C₂²	1	1	yes	characteristic
11	#4047	—	2	2	no	—
12	#4047	C₄	1	4	yes	complemented
13	C₂×#676	—	2	1	no	—
14	C₂×C₈×D₅	—	2	1	no	—
15	C₅×#107	—	2	1	no	—
16	C₅×#107	—	2	1	no	—
17	#3951	—	2	2	no	—
18	#3942	—	2	2	no	—
19	#3941	—	2	2	no	—
20	#1816	—	5	1	no	maximal, Sylow
21	C₂²×C₄×C₅	—	2	2	no	—
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	C₂×C₄	1	2	yes	—
27	#709	Q₈	1	2	yes	—
28	C₂×D₂₀	D₄	1	1	yes	characteristic
29	C₂×C₄×D₅	—	2	1	no	—
30	C₂×Dic₁₀	D₄	1	1	yes	characteristic
31	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₈	D₄	1	2	yes	—
33	C₂×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₅×C₈	D₄	1	1	yes	characteristic
35	C₅×C₄⋊C₄	—	2	2	no	—
36	C₅×C₄⋊C₄	—	2	2	no	—
37	#681	—	2	1	no	—
38	#681	—	2	2	no	—
39	#681	—	2	2	no	—
40	#681	C₂×C₄	1	2	yes	—
41	C₂×#168	—	2	1	no	—
42	C₂×#168	—	2	2	no	—
43	#676	—	2	4	no	—
44	C₈×D₅	—	2	4	no	—
45	C₂×#131	—	5	1	no	—
46	C₂×#108	—	5	1	no	—
47	C₂×#107	D₅	1	1	yes	characteristic, complemented
48	#417	—	5	2	no	—
49	#343	—	5	2	no	—
50	C₂³×C₅	#41	1	1	yes	characteristic
51	C₂²×D₅	—	2	1	no	—
52	C₂×C₄×C₅	—	4	2	no	—
53	C₂×C₄×C₅	—	8	2	no	—
54	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅	#41	1	1	yes	characteristic
60	C₂²⋊D₅	—	2	2	no	—
61	C₂²⋊D₅	—	2	2	no	—
62	C₂×Dic₅	—	2	1	no	—
63	D₂₀	—	2	1	no	—
64	D₂₀	C₄⋊C₄	1	2	yes	—
65	C₄×D₅	—	2	2	no	—
66	C₄×D₅	—	2	2	no	—
67	Dic₁₀	—	2	1	no	—
68	Dic₁₀	C₄⋊C₄	1	2	yes	—
69	C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	#168	—	2	2	no	—
72	#168	—	4	2	no	—
73	C₂×#41	—	5	1	no	—
74	#131	—	5	4	no	—
75	#131	—	10	2	no	—
76	C₂×M₁₆	—	5	1	no	—
77	C₂×M₁₆	—	5	2	no	—
78	C₂×M₁₆	—	10	1	no	—
79	C₂²×C₈	—	10	1	no	—
80	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
82	#108	—	10	2	no	—
83	#107	—	2	1	no	—
84	#107	—	2	1	no	—
85	#103	—	10	2	no	—
86	#102	—	10	2	no	—
87	C₂²×C₅	—	2	1	no	—
88	C₂²×C₅	—	2	1	no	—
89	C₂²×C₅	#117	1	1	yes	characteristic
90	C₂²×C₅	#121	1	1	yes	characteristic
91	C₂²×C₅	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×D₅	—	2	2	no	—
93	C₂×D₅	—	4	1	no	—
94	C₂×D₅	—	4	1	no	—
95	C₄×C₅	—	8	2	no	—
96	C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₅	#121	1	1	yes	characteristic
99	C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
100	Dic₅	—	2	2	no	—
101	#41	—	5	2	no	—
102	#41	—	5	2	no	—
103	#41	—	10	1	no	—
104	#41	—	10	1	no	—
105	C₂×Q₈	—	5	1	no	—
106	C₂×D₄	—	5	1	no	—
107	C₂×D₄	—	10	1	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	10	1	no	—
110	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
111	M₁₆	—	5	2	no	—
112	M₁₆	—	10	1	no	—
113	M₁₆	—	10	4	no	—
114	M₁₆	—	10	2	no	—
115	M₁₆	—	10	2	no	—
116	C₂×C₈	—	10	1	no	—
117	C₂×C₈	—	10	4	no	—
118	C₂×C₈	—	10	2	no	—
119	C₂×C₈	C₄×D₅	1	2	yes	—
120	C₂×C₈	D₂₀	1	2	yes	—
121	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₂×C₅	—	2	1	no	—
126	C₂×C₅	—	2	1	no	—
127	C₂×C₅	#380	1	1	yes	characteristic
128	C₂×C₅	#472	1	1	yes	characteristic
129	C₂×C₅	#473	1	1	yes	characteristic
130	D₅	—	4	2	no	—
131	C₂³	—	10	1	no	—
132	C₂³	#711	1	1	yes	characteristic
133	Q₈	—	5	2	no	—
134	Q₈	—	10	1	no	—
135	D₄	—	5	2	no	—
136	D₄	—	10	2	no	—
137	D₄	—	10	2	no	—
138	D₄	—	10	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	10	1	no	—
142	C₂×C₄	—	10	2	no	—
143	C₂×C₄	—	10	2	no	—
144	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
145	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
146	C₂×C₄	D₄×D₅	1	1	yes	characteristic
147	C₂×C₄	Q₈×D₅	1	1	yes	characteristic
148	C₂×C₄	#713	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
150	C₈	—	10	2	no	—
151	C₈	—	20	2	no	—
152	C₈	#685	1	2	yes	—
153	C₈	#685	1	2	yes	—
154	C₅	#1816	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	10	2	no	—
158	C₂²	—	20	1	no	—
159	C₂²	—	20	1	no	—
160	C₂²	#4038	1	1	yes	characteristic
161	C₂²	#4043	1	1	yes	characteristic
162	C₂²	#4084	1	1	yes	characteristic
163	C₄	—	8	2	no	—
164	C₄	—	10	2	no	—
165	C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
166	C₄	#4041	1	1	yes	characteristic
167	C₄	C₂×#685	1	1	yes	characteristic
168	C₄	#4092	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	20	2	no	—
172	C₂	#65276	1	1	yes	characteristic
173	C₂	#65449	1	1	yes	characteristic
174	C₂	#65481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4084
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₂×C₅×#107	C₂
socle	C₂²×C₅	#121
prime radical	C₂×#709	C₂²
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	C₂×#107	D₅
hypocenter	C₅	#1816
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#107	C₂	semidirect
3. C₂×#3951	C₂	semidirect
5. #64750	C₂	semidirect
12. #4047	C₄	semidirect
47. C₂×#107	D₅	semidirect
154. C₅	#1816	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4047	2
C₂×C₅×#107	2
C₂×#3951	2
#65255	2
#64750	2
#1816	5