Finite Group Databasebeta

Subgroups — #13027554

Order 1280.

290 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 124 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×#65429	C₂	1	1	yes	maximal, characteristic
2	C₂×#64877	C₂	1	2	yes	maximal, complemented
3	#10598759	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×#131	C₄	1	1	yes	characteristic
5	#65429	—	2	2	no	—
6	#65429	—	2	2	no	—
7	C₂×#3944	C₄	1	1	yes	characteristic
8	C₂×#3944	C₂²	1	2	yes	—
9	C₂²×#672	C₄	1	2	yes	—
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	#64877	—	2	2	no	—
12	#64877	C₂²	1	4	yes	—
13	C₂×#3928	—	2	2	no	—
14	#13556	—	5	1	no	maximal, Sylow
15	C₂×C₅×#41	C₈	1	1	yes	characteristic
16	C₅×#131	—	2	2	no	—
17	C₅×#131	—	2	2	no	—
18	C₂×C₅×M₁₆	C₈	1	1	yes	characteristic
19	C₂×C₅×M₁₆	C₈	1	2	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	#3944	—	2	2	no	—
23	#3944	—	2	4	no	—
24	C₂×#672	—	2	2	no	—
25	C₂×#672	—	2	2	no	—
26	C₂×#672	C₂×C₄	1	2	yes	—
27	C₂×#672	C₂×C₄	1	1	yes	characteristic
28	C₂×#672	D₄	1	2	yes	—
29	C₂×#672	D₄	1	2	yes	—
30	C₂×#672	C₂³	1	1	yes	characteristic
31	#3928	—	2	4	no	—
32	C₂×#504	—	5	1	no	—
33	C₂×M₆₄	—	5	2	no	—
34	#1400	—	5	4	no	—
35	C₅×#41	—	2	2	no	—
36	C₅×#41	—	2	2	no	—
37	C₂×C₅×Q₈	C₁₆	1	1	yes	characteristic
38	C₂×C₅×D₄	C₁₆	1	2	yes	—
39	C₂×C₅×D₄	C₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
42	C₅×M₁₆	—	2	2	no	—
43	C₅×M₁₆	—	2	4	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
47	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
48	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₅×C₈	M₁₆	1	2	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
51	#672	—	2	2	no	—
52	#672	C₂²⋊C₄	1	1	yes	characteristic
53	#672	C₂²⋊C₄	1	1	yes	characteristic
54	#672	C₂×D₄	1	2	yes	—
55	C₂×#131	AGL(1,𝔽₅)	1	1	yes	characteristic
56	#504	—	10	2	no	—
57	#504	—	10	2	no	—
58	C₂×M₃₂	—	5	2	no	—
59	C₂×M₃₂	—	5	1	no	—
60	C₂²×C₁₆	—	5	1	no	—
61	C₂²×C₁₆	—	5	2	no	—
62	M₆₄	—	5	4	no	—
63	M₆₄	—	10	2	no	—
64	C₂×C₃₂	—	10	2	no	—
65	C₂³×C₅	C₂×C₁₆	1	2	yes	—
66	C₂³×C₅	C₂×C₁₆	1	1	yes	characteristic
67	C₅×Q₈	—	2	2	no	—
68	C₅×D₄	—	2	4	no	—
69	C₅×D₄	—	2	2	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
73	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
74	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅	M₃₂	1	2	yes	—
76	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	C₂²⋊C₈	1	2	yes	—
79	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
81	C₂×#41	#170	1	1	yes	characteristic
82	#131	—	2	2	no	—
83	#131	—	2	2	no	—
84	C₂×M₁₆	#170	1	1	yes	characteristic
85	C₂×M₁₆	#170	1	2	yes	—
86	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
87	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
88	M₃₂	—	10	4	no	—
89	M₃₂	—	10	2	no	—
90	C₂×C₁₆	—	5	2	no	—
91	C₂×C₁₆	—	5	2	no	—
92	C₂×C₁₆	—	5	1	no	—
93	C₂×C₁₆	—	5	1	no	—
94	C₂×C₁₆	—	5	2	no	—
95	C₂×C₁₆	—	10	2	no	—
96	C₂×C₁₆	—	10	2	no	—
97	C₃₂	—	10	4	no	—
98	C₂²×C₅	—	2	2	no	—
99	C₂²×C₅	—	4	1	no	—
100	C₂²×C₅	—	4	2	no	—
101	C₂²×C₅	C₂²⋊C₁₆	1	2	yes	—
102	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
105	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	#674	1	1	yes	characteristic
110	C₂×D₄	#674	1	2	yes	—
111	C₂×D₄	#674	1	1	yes	characteristic
112	C₂²×C₄	C₂×#170	1	2	yes	—
113	C₂²×C₄	C₂×#170	1	1	yes	characteristic
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	C₂×#170	1	2	yes	—
119	C₂×C₈	C₂×#170	1	1	yes	characteristic
120	C₂×C₈	#704	1	2	yes	—
121	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
122	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
123	C₁₆	—	5	2	no	—
124	C₁₆	—	5	1	no	—
125	C₁₆	—	5	1	no	—
126	C₁₆	—	10	2	no	—
127	C₂×C₅	—	2	1	no	—
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	—	4	2	no	—
130	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
131	C₂³	C₂×#674	1	2	yes	—
132	C₂³	C₂×#674	1	1	yes	characteristic
133	Q₈	—	2	2	no	—
134	D₄	—	2	4	no	—
135	D₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂×#674	1	2	yes	—
139	C₂×C₄	C₂×#674	1	1	yes	characteristic
140	C₂×C₄	#3998	1	2	yes	—
141	C₂×C₄	C₂²⋊#170	1	2	yes	—
142	C₂×C₄	C₂²×#170	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	C₂²⋊#170	1	2	yes	—
145	C₈	C₂×#704	1	1	yes	characteristic
146	C₈	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₅	#13556	1	1	yes	characteristic, complemented, Sylow
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	2	no	—
151	C₂²	C₂²⋊#674	1	2	yes	—
152	C₂²	C₂²×#674	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	C₂²⋊#674	1	2	yes	—
155	C₄	C₂×#3998	1	1	yes	characteristic
156	C₄	C₂×C₂²⋊#170	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₂×C₂²⋊#674	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×#131	C₄
socle	C₂×C₅	C₂×C₂²⋊C₁₆
prime radical	C₂×C₅×#41	C₈
cosocle	C₂×#672	C₂³
hypercenter	C₂×#131	AGL(1,𝔽₅)
hypocenter	C₅	#13556
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#64877	C₂	semidirect
3. #10598759	C₂	semidirect
147. C₅	#13556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65429	2
C₂×#64877	2
#10598759	2
#13556	5