Finite Group Databasebeta

Subgroups — #13025036

Order 1280.

242 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 116 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	#10604150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#674	C₂	1	1	yes	maximal, characteristic
3	#10601533	C₂	1	1	yes	maximal, characteristic
4	#10598523	C₂	1	4	yes	maximal, complemented
5	C₂×#3998	—	2	1	no	—
6	C₂²×#674	—	2	1	no	—
7	C₅×#431	C₄	1	1	yes	characteristic
8	#65214	C₄	1	1	yes	characteristic
9	C₂×C₄×#168	C₂²	1	1	yes	characteristic
10	C₄⋊#674	—	2	1	no	—
11	C₄⋊#674	—	2	1	no	—
12	C₄⋊#674	C₂²	1	2	yes	—
13	C₄⋊#674	C₂²	1	2	yes	—
14	#64674	C₄	1	2	yes	—
15	C₈×#168	C₂²	1	2	yes	—
16	#11997	—	5	1	no	maximal, Sylow
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×#168	D₄	1	1	yes	characteristic
19	C₂²×#168	Q₈	1	1	yes	characteristic
20	#3998	—	4	2	no	—
21	C₂×#674	—	2	2	no	—
22	C₂×#674	—	2	2	no	—
23	C₂×#674	—	4	2	no	—
24	C₅×C₄⋊C₈	—	2	1	no	—
25	C₅×C₂²⋊C₈	—	2	1	no	—
26	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²⋊#168	C₈	1	2	yes	—
28	C₄⋊#168	C₈	1	2	yes	—
29	C₄×#168	C₂×C₄	1	1	yes	characteristic
30	C₄×#168	C₂×C₄	1	2	yes	—
31	C₄×#168	D₄	1	1	yes	characteristic
32	C₄×#168	Q₈	1	1	yes	characteristic
33	C₄×#168	C₂³	1	1	yes	characteristic
34	#2150	—	5	1	no	—
35	C₂×C₄⋊C₁₆	—	5	1	no	—
36	#1450	—	5	1	no	—
37	#1367	—	5	4	no	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
41	C₂×C₅×C₈	—	2	2	no	—
42	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₅	M₁₆	1	1	yes	characteristic
45	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×#168	—	2	2	no	—
47	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#168	C₂×C₈	1	4	yes	—
50	C₂×#168	SD₁₆	1	2	yes	—
51	C₂×#168	SD₁₆	1	2	yes	—
52	C₂×#168	C₂×D₄	1	1	yes	characteristic
53	C₂×#168	C₂×Q₈	1	1	yes	characteristic
54	#674	—	4	2	no	—
55	#674	—	4	2	no	—
56	C₂×M₃₂	—	10	1	no	—
57	C₂²×C₁₆	—	10	1	no	—
58	#431	—	5	1	no	—
59	#431	AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₂×C₄×C₈	—	5	1	no	—
61	C₄⋊C₁₆	—	5	2	no	—
62	C₄⋊C₁₆	—	5	2	no	—
63	C₄⋊C₁₆	—	10	1	no	—
64	C₄⋊C₁₆	—	10	1	no	—
65	#322	—	5	2	no	—
66	C₈²	—	5	2	no	—
67	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	4	1	no	—
70	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₅	#106	1	2	yes	—
74	C₂×C₄×C₅	#108	1	2	yes	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
78	C₅×C₈	—	4	2	no	—
79	#168	—	2	1	no	—
80	#168	—	2	1	no	—
81	#168	—	2	4	no	—
82	#168	#106	1	2	yes	—
83	#168	C₂×SD₁₆	1	1	yes	characteristic
84	#168	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²×C₈	—	5	1	no	—
86	C₂²×C₈	—	5	1	no	—
87	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
88	M₃₂	—	20	2	no	—
89	C₂×C₁₆	—	10	2	no	—
90	C₂×C₁₆	—	10	2	no	—
91	C₂×C₁₆	—	20	2	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	5	2	no	—
94	C₂²⋊C₈	—	2	1	no	—
95	C₂²⋊C₈	—	5	2	no	—
96	C₄×C₈	—	5	1	no	—
97	C₄×C₈	—	5	1	no	—
98	C₄×C₈	—	5	1	no	—
99	C₄×C₈	—	5	2	no	—
100	C₄×C₈	—	5	1	no	—
101	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
102	C₂²×C₅	—	2	1	no	—
103	C₂²×C₅	—	2	1	no	—
104	C₂²×C₅	#334	1	2	yes	—
105	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	—	2	1	no	—
109	C₄×C₅	#334	1	2	yes	—
110	C₄×C₅	C₂×#106	1	1	yes	characteristic
111	C₄×C₅	C₂×#108	1	1	yes	characteristic
112	C₄×C₅	#504	1	2	yes	—
113	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂²×C₄	#704	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	5	1	no	—
118	C₂×C₈	—	5	1	no	—
119	C₂×C₈	—	5	2	no	—
120	C₂×C₈	—	5	2	no	—
121	C₂×C₈	—	5	4	no	—
122	C₂×C₈	—	5	1	no	—
123	C₂×C₈	—	5	1	no	—
124	C₂×C₈	—	10	2	no	—
125	C₄²	D₅⋊C₈	1	1	yes	characteristic
126	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₄²	#704	1	1	yes	characteristic
128	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₁₆	—	20	2	no	—
130	C₁₆	—	20	2	no	—
131	C₂×C₅	—	2	2	no	—
132	C₂×C₅	C₂×#334	1	1	yes	characteristic
133	C₂×C₅	#2151	1	1	yes	characteristic
134	C₂×C₅	#2153	1	1	yes	characteristic
135	C₂³	#4004	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	#3993	1	2	yes	—
139	C₂×C₄	#3996	1	2	yes	—
140	C₂×C₄	#4004	1	1	yes	characteristic
141	C₂×C₄	#4004	1	1	yes	characteristic
142	C₂×C₄	#4004	1	1	yes	characteristic
143	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₂×C₄	C₂×#704	1	1	yes	characteristic
146	C₈	—	4	2	no	—
147	C₈	—	5	1	no	—
148	C₈	—	5	1	no	—
149	C₈	—	5	2	no	—
150	C₈	—	10	1	no	—
151	C₈	—	10	1	no	—
152	C₈	—	10	4	no	—
153	C₅	#11997	1	1	yes	characteristic, complemented, Sylow
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#64883	1	2	yes	—
157	C₂²	C₂×#4004	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#64883	1	2	yes	—
162	C₄	C₂×#3993	1	1	yes	characteristic
163	C₄	C₂×#3996	1	1	yes	characteristic
164	C₄	#65725	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#10604370	1	1	yes	characteristic
167	C₂	C₂×#64883	1	1	yes	characteristic
168	C₂	#10604746	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#431	C₄
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₄⋊C₈
cosocle	C₄×#168	C₂³
hypercenter	#431	AGL(1,𝔽₅)
hypocenter	C₅	#11997
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
4. #10598523	C₂	semidirect
153. C₅	#11997	semidirect

Maximal subgroups

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

subgroup	index
#10604150	2
C₂×C₄⋊#674	2
#10601533	2
#10598523	2
#11997	5