Finite Group Databasebeta

Subgroups — #13025105

Order 1280.

236 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 98 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	#10604151	C₂	1	1	yes	maximal, characteristic
2	#10604150	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#168	C₂	1	1	yes	maximal, characteristic
4	#10598525	C₂	1	2	yes	maximal, complemented
5	#10598523	C₂	1	2	yes	maximal, complemented
6	C₂×#3998	—	2	1	no	—
7	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
8	#65214	C₄	1	1	yes	characteristic
9	C₂×C₄×#168	C₂²	1	1	yes	characteristic
10	C₂²⋊#674	—	2	1	no	—
11	C₄⋊#674	—	2	1	no	—
12	C₄⋊#674	C₂²	1	2	yes	—
13	C₄⋊#674	C₂²	1	1	yes	characteristic
14	C₄⋊#674	C₂²	1	1	yes	characteristic
15	C₄×#674	—	2	1	no	—
16	#64675	C₄	1	2	yes	—
17	C₈×#168	C₂²	1	2	yes	—
18	#12018	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	—	2	1	no	—
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×#168	D₄	1	1	yes	characteristic
22	C₂²×#168	Q₈	1	1	yes	characteristic
23	#3998	—	4	2	no	—
24	C₂×#674	—	2	1	no	—
25	C₂×#674	—	2	1	no	—
26	C₂×#674	—	2	2	no	—
27	C₅×#97	—	2	1	no	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²⋊#168	C₈	1	2	yes	—
30	C₄⋊#168	C₈	1	2	yes	—
31	C₄×#168	C₂×C₄	1	1	yes	characteristic
32	C₄×#168	C₂×C₄	1	2	yes	—
33	C₄×#168	D₄	1	1	yes	characteristic
34	C₄×#168	Q₈	1	1	yes	characteristic
35	C₄×#168	C₂³	1	1	yes	characteristic
36	#2162	—	5	1	no	—
37	#2150	—	5	1	no	—
38	C₈×M₁₆	—	5	1	no	—
39	#1370	—	5	2	no	—
40	#1367	—	5	2	no	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
44	C₅×M₁₆	—	2	4	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₅	M₁₆	1	1	yes	characteristic
49	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×#168	—	2	1	no	—
51	C₂×#168	—	2	1	no	—
52	C₂×#168	—	4	1	no	—
53	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
54	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
55	C₂×#168	C₂×C₈	1	4	yes	—
56	C₂×#168	C₂×D₄	1	1	yes	characteristic
57	C₂×#168	C₂×Q₈	1	1	yes	characteristic
58	#674	—	4	1	no	—
59	#674	—	4	2	no	—
60	#674	—	4	1	no	—
61	C₂×M₃₂	—	10	1	no	—
62	#431	—	5	1	no	—
63	C₄×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₂×C₄×C₈	—	5	1	no	—
65	C₄⋊C₁₆	—	5	1	no	—
66	C₄⋊C₁₆	—	5	2	no	—
67	C₄⋊C₁₆	—	5	1	no	—
68	C₄⋊C₁₆	—	10	1	no	—
69	C₂²⋊C₁₆	—	10	1	no	—
70	C₄×C₁₆	—	10	1	no	—
71	#322	—	5	2	no	—
72	C₈²	—	5	2	no	—
73	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₅×C₈	—	2	4	no	—
85	#168	—	2	1	no	—
86	#168	—	2	1	no	—
87	#168	—	2	1	no	—
88	#168	—	2	4	no	—
89	#168	AGL(1,ℤ/8)	1	1	yes	characteristic
90	#168	#137	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂²×C₈	—	5	1	no	—
93	C₂²×C₈	—	5	1	no	—
94	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
95	M₃₂	—	20	2	no	—
96	C₂×C₁₆	—	10	1	no	—
97	C₂×C₁₆	—	10	1	no	—
98	C₂×C₁₆	—	10	2	no	—
99	C₄⋊C₈	—	5	2	no	—
100	C₂²⋊C₈	—	5	2	no	—
101	#97	—	2	1	no	—
102	C₄×C₈	—	5	1	no	—
103	C₄×C₈	—	5	1	no	—
104	C₄×C₈	—	5	1	no	—
105	C₄×C₈	—	5	2	no	—
106	C₄×C₈	—	5	1	no	—
107	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	—	2	1	no	—
110	C₂²×C₅	—	2	1	no	—
111	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₅	—	2	1	no	—
113	C₄×C₅	—	2	1	no	—
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	#364	1	2	yes	—
116	C₄×C₅	#428	1	1	yes	characteristic
117	C₄×C₅	#430	1	1	yes	characteristic
118	C₄×C₅	#504	1	2	yes	—
119	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₂²×C₄	#704	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	5	1	no	—
125	C₂×C₈	—	5	4	no	—
126	C₂×C₈	—	5	1	no	—
127	C₂×C₈	—	5	1	no	—
128	C₂×C₈	—	5	1	no	—
129	C₂×C₈	—	10	1	no	—
130	C₂×C₈	—	10	1	no	—
131	C₂×C₈	—	20	1	no	—
132	C₄²	D₅⋊C₈	1	1	yes	characteristic
133	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₄²	#704	1	1	yes	characteristic
135	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₁₆	—	20	2	no	—
137	C₁₆	—	20	1	no	—
138	C₁₆	—	20	1	no	—
139	C₂×C₅	—	4	1	no	—
140	C₂×C₅	#1565	1	1	yes	characteristic
141	C₂×C₅	#2151	1	1	yes	characteristic
142	C₂×C₅	C₂×#364	1	1	yes	characteristic
143	C₂³	#4004	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	#4004	1	1	yes	characteristic
149	C₂×C₄	#4004	1	1	yes	characteristic
150	C₂×C₄	#4004	1	1	yes	characteristic
151	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂×C₄	C₂×#704	1	1	yes	characteristic
154	C₈	—	2	4	no	—
155	C₈	—	5	1	no	—
156	C₈	—	5	1	no	—
157	C₈	—	10	4	no	—
158	C₈	—	10	1	no	—
159	C₈	—	10	1	no	—
160	C₈	—	10	1	no	—
161	C₅	#12018	1	1	yes	characteristic, complemented, Sylow
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	C₂×#4004	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	#64861	1	2	yes	—
170	C₄	#65725	1	2	yes	—
171	C₄	#65727	1	1	yes	characteristic
172	C₄	#65729	1	1	yes	characteristic
173	C₂	—	4	1	no	—
174	C₂	C₂×#64861	1	1	yes	characteristic
175	C₂	#10604739	1	1	yes	characteristic
176	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	C₄×C₅×M₁₆	C₄
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₄⋊C₈
cosocle	C₄×#168	C₂³
hypercenter	C₄×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#12018
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10598525	C₂	semidirect
5. #10598523	C₂	semidirect
161. C₅	#12018	semidirect

Maximal subgroups

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

subgroup	index
#10604151	2
#10604150	2
M₁₆×#168	2
#10598525	2
#10598523	2
#12018	5