Finite Group Databasebeta

Subgroups — #10617383

Order 640.

330 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 152 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	#66062	C₂	1	1	yes	maximal, characteristic
2	#65788	C₂	1	2	yes	maximal, complemented
3	Q₈×#170	C₂	1	2	yes	maximal, complemented
4	#65709	C₂	1	2	yes	maximal, complemented
5	#65706	C₂	1	4	yes	maximal
6	#65703	C₂	1	2	yes	maximal, complemented
7	#65680	C₂	1	1	yes	maximal, characteristic
8	#65680	C₂	1	1	yes	maximal, characteristic
9	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	#4092	C₄	1	2	yes	—
11	Q₈×Dic₅	C₂²	1	2	yes	—
12	#4043	C₄	1	2	yes	—
13	#4038	C₂²	1	2	yes	—
14	#4036	C₄	1	2	yes	—
15	#4033	C₂²	1	2	yes	—
16	C₄²×D₅	C₂²	1	1	yes	characteristic
17	#4015	C₄	1	1	yes	characteristic
18	#4004	C₂²	1	4	yes	—
19	#4003	C₂²	1	2	yes	—
20	#4003	C₂²	1	2	yes	—
21	#4002	C₂²	1	4	yes	—
22	C₄⋊#170	C₂²	1	1	yes	characteristic
23	C₄⋊#170	C₂²	1	1	yes	characteristic
24	C₄⋊#170	C₂²	1	1	yes	characteristic
25	C₄⋊#170	C₂²	1	1	yes	characteristic
26	C₄⋊#170	C₂²	1	4	yes	—
27	C₄×#170	C₂²	1	2	yes	—
28	C₄×#170	C₂²	1	2	yes	—
29	C₄×#170	C₂²	1	4	yes	—
30	#2986	—	5	1	no	maximal, Sylow
31	C₂×C₅×Q₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
35	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
36	C₂×#170	C₂³	1	2	yes	—
37	C₂×#170	C₂³	1	2	yes	—
38	C₂×#170	C₂³	1	2	yes	—
39	C₂×#170	C₂³	1	2	yes	—
40	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	#685	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
47	#683	C₂×C₄	1	4	yes	—
48	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₅	C₂³	1	1	yes	characteristic
50	C₄×Dic₅	C₂³	1	1	yes	characteristic
51	C₄×Dic₅	C₂³	1	4	yes	—
52	#533	—	5	1	no	—
53	#446	—	5	2	no	—
54	#446	—	5	2	no	—
55	C₈×Q₈	—	5	2	no	—
56	C₈×Q₈	—	5	2	no	—
57	#433	—	5	4	no	—
58	#431	—	5	1	no	—
59	#431	—	5	1	no	—
60	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
61	C₅×Q₈	—	2	4	no	—
62	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₅	C₂²×C₄	1	4	yes	—
69	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
70	C₄×D₅	—	2	2	no	—
71	C₄×D₅	—	2	2	no	—
72	C₄×D₅	—	2	1	no	—
73	C₄×D₅	—	2	1	no	—
74	Dic₁₀	—	2	4	no	—
75	#170	—	2	2	no	—
76	#170	—	2	2	no	—
77	#170	—	2	2	no	—
78	#170	—	2	2	no	—
79	C₄⋊Q₈	—	5	1	no	—
80	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
81	#125	—	5	2	no	—
82	C₂²⋊Q₈	—	5	2	no	—
83	C₂²⋊Q₈	—	5	2	no	—
84	C₄×Q₈	—	5	2	no	—
85	C₄×Q₈	—	5	2	no	—
86	#117	—	5	2	no	—
87	C₂×C₄²	—	5	1	no	—
88	C₄⋊C₈	—	5	1	no	—
89	C₄⋊C₈	—	5	1	no	—
90	C₄⋊C₈	—	5	1	no	—
91	C₄⋊C₈	—	5	1	no	—
92	C₄⋊C₈	—	5	4	no	—
93	C₄⋊C₈	—	5	4	no	—
94	C₂²⋊C₈	—	5	2	no	—
95	C₂²⋊C₈	—	5	2	no	—
96	#97	—	5	4	no	—
97	C₄×C₈	—	5	2	no	—
98	C₄×C₈	—	5	2	no	—
99	C₄×C₈	—	5	4	no	—
100	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₅	—	2	1	no	—
102	C₂×D₅	—	2	1	no	—
103	C₂×D₅	—	2	1	no	—
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	—	2	4	no	—
106	C₄×C₅	#131	1	2	yes	—
107	C₄×C₅	#131	1	2	yes	—
108	Dic₅	—	2	4	no	—
109	Dic₅	—	2	1	no	—
110	Dic₅	—	2	1	no	—
111	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
112	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂×Q₈	—	5	2	no	—
114	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
115	C₂²×C₄	—	5	1	no	—
116	C₂²×C₄	—	5	1	no	—
117	C₂²×C₄	—	5	1	no	—
118	C₂×C₈	—	5	2	no	—
119	C₂×C₈	—	5	2	no	—
120	C₂×C₈	—	5	2	no	—
121	C₂×C₈	—	5	2	no	—
122	C₄⋊C₄	—	5	2	no	—
123	C₄⋊C₄	—	5	4	no	—
124	C₄⋊C₄	—	5	4	no	—
125	C₄⋊C₄	—	5	2	no	—
126	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
127	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
128	C₂²⋊C₄	—	5	4	no	—
129	C₄²	—	5	1	no	—
130	C₄²	—	5	1	no	—
131	C₄²	—	5	4	no	—
132	C₄²	—	5	1	no	—
133	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂×C₅	#519	1	1	yes	characteristic
135	C₂×C₅	C₂×#131	1	1	yes	characteristic
136	C₂×C₅	C₂×#131	1	1	yes	characteristic
137	D₅	—	4	1	no	—
138	C₂³	—	5	1	no	—
139	Q₈	—	2	4	no	—
140	Q₈	—	10	4	no	—
141	C₂×C₄	—	5	1	no	—
142	C₂×C₄	—	5	1	no	—
143	C₂×C₄	—	5	1	no	—
144	C₂×C₄	—	5	4	no	—
145	C₂×C₄	—	10	1	no	—
146	C₂×C₄	—	10	1	no	—
147	C₂×C₄	—	10	2	no	—
148	C₂×C₄	—	10	2	no	—
149	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
151	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₈	—	10	2	no	—
154	C₈	—	10	2	no	—
155	C₈	—	10	2	no	—
156	C₈	—	10	2	no	—
157	C₅	#2986	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	10	1	no	—
159	C₂²	—	10	1	no	—
160	C₂²	—	10	1	no	—
161	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	1	no	—
164	C₄	—	5	1	no	—
165	C₄	—	5	1	no	—
166	C₄	—	10	1	no	—
167	C₄	—	10	1	no	—
168	C₄	—	10	4	no	—
169	C₄	#4131	1	2	yes	—
170	C₄	#4133	1	2	yes	—
171	C₂	—	20	1	no	—
172	C₂	C₂×#4131	1	1	yes	characteristic
173	C₂	C₂×#4133	1	1	yes	characteristic
174	C₂	#66262	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×C₄⋊Q₈	C₄
prime radical	C₂²×D₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2986
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #65788	C₂	semidirect
3. Q₈×#170	C₂	semidirect
4. #65709	C₂	semidirect
6. #65703	C₂	semidirect
157. C₅	#2986	semidirect

Maximal subgroups

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

subgroup	index
#66062	2
#65788	2
Q₈×#170	2
#65709	2
#65706	2
#65703	2
#65680	2
#2986	5