Finite Group Databasebeta

Subgroups — #13025100

Order 1280.

232 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 110 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	#10604152	C₂	1	1	yes	maximal, characteristic
2	#10604150	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#168	C₂	1	1	yes	maximal, characteristic
4	#10598524	C₂	1	2	yes	maximal, complemented
5	#10598522	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	#64859	—	2	1	no	—
12	C₄⋊#674	—	2	1	no	—
13	C₄⋊#674	C₂²	1	2	yes	—
14	C₄⋊#674	C₂²	1	1	yes	characteristic
15	C₄⋊#674	C₂²	1	1	yes	characteristic
16	#64675	C₄	1	2	yes	—
17	C₈×#168	C₂²	1	2	yes	—
18	#12017	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	C₈	1	2	yes	—
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	2	no	—
25	C₂×#674	—	2	1	no	—
26	C₂×#674	—	2	1	no	—
27	C₅×#97	C₈	1	1	yes	characteristic
28	C₅×#97	C₈	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²⋊#168	—	2	1	no	—
31	C₄⋊#168	—	2	1	no	—
32	C₄×#168	C₂×C₄	1	2	yes	—
33	C₄×#168	C₂×C₄	1	1	yes	characteristic
34	C₄×#168	D₄	1	1	yes	characteristic
35	C₄×#168	Q₈	1	1	yes	characteristic
36	C₄×#168	C₂³	1	1	yes	characteristic
37	#2163	—	5	1	no	—
38	#2150	—	5	1	no	—
39	C₈×M₁₆	—	5	1	no	—
40	#1370	—	5	2	no	—
41	#1367	—	5	2	no	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
45	C₅×M₁₆	—	2	4	no	—
46	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
48	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₅	M₁₆	1	1	yes	characteristic
51	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×#168	—	2	1	no	—
53	C₂×#168	—	2	1	no	—
54	C₂×#168	—	2	2	no	—
55	C₂×#168	—	4	1	no	—
56	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#168	C₂×D₄	1	1	yes	characteristic
59	C₂×#168	C₂×Q₈	1	1	yes	characteristic
60	#674	—	4	1	no	—
61	#674	—	4	2	no	—
62	#674	—	4	1	no	—
63	C₂×M₃₂	—	10	1	no	—
64	#431	—	5	1	no	—
65	C₄×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₂×C₄×C₈	—	5	1	no	—
67	C₄⋊C₁₆	—	5	1	no	—
68	C₄⋊C₁₆	—	5	2	no	—
69	C₄⋊C₁₆	—	5	1	no	—
70	C₄⋊C₁₆	—	10	1	no	—
71	C₂²⋊C₁₆	—	10	1	no	—
72	#346	—	10	1	no	—
73	#322	—	5	2	no	—
74	C₈²	—	5	2	no	—
75	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	2	1	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	4	1	no	—
80	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
86	C₅×C₈	—	2	2	no	—
87	C₅×C₈	M₃₂	1	4	yes	complemented
88	#168	—	2	1	no	—
89	#168	—	2	1	no	—
90	#168	—	2	1	no	—
91	#168	—	4	2	no	—
92	#168	AGL(1,ℤ/8)	1	1	yes	characteristic
93	#168	#137	1	1	yes	characteristic
94	C₂×M₁₆	#170	1	2	yes	—
95	C₂²×C₈	—	5	1	no	—
96	C₂²×C₈	—	5	1	no	—
97	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	M₃₂	—	20	2	no	—
99	C₂×C₁₆	—	10	1	no	—
100	C₂×C₁₆	—	10	1	no	—
101	C₂×C₁₆	—	10	2	no	—
102	C₄⋊C₈	—	10	1	no	—
103	C₂²⋊C₈	—	10	1	no	—
104	#97	#170	1	1	yes	characteristic
105	#97	#170	1	1	yes	characteristic
106	C₄×C₈	—	5	1	no	—
107	C₄×C₈	—	5	1	no	—
108	C₄×C₈	—	5	1	no	—
109	C₄×C₈	—	5	1	no	—
110	C₄×C₈	—	5	2	no	—
111	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
112	C₂²×C₅	—	2	1	no	—
113	C₂²×C₅	—	2	1	no	—
114	C₂²×C₅	—	2	1	no	—
115	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₄×C₅	—	2	1	no	—
117	C₄×C₅	—	2	1	no	—
118	C₄×C₅	—	2	1	no	—
119	C₄×C₅	—	2	1	no	—
120	C₄×C₅	#428	1	1	yes	characteristic
121	C₄×C₅	#430	1	1	yes	characteristic
122	C₄×C₅	C₂×M₃₂	1	2	yes	—
123	C₂²×C₄	#700	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₂²×C₄	C₂×#170	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	5	1	no	—
128	C₂×C₈	—	5	1	no	—
129	C₂×C₈	—	5	1	no	—
130	C₂×C₈	—	5	1	no	—
131	C₂×C₈	—	10	1	no	—
132	C₂×C₈	—	10	1	no	—
133	C₂×C₈	—	10	2	no	—
134	C₂×C₈	—	20	1	no	—
135	C₂×C₈	C₂×#170	1	2	yes	—
136	C₂×C₈	C₂×#170	1	2	yes	—
137	C₄²	#700	1	1	yes	characteristic
138	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₄²	C₂×#170	1	1	yes	characteristic
140	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₁₆	—	20	2	no	—
142	C₁₆	—	20	1	no	—
143	C₁₆	—	20	1	no	—
144	C₂×C₅	—	4	1	no	—
145	C₂×C₅	#1565	1	1	yes	characteristic
146	C₂×C₅	#2150	1	1	yes	characteristic
147	C₂×C₅	#2153	1	1	yes	characteristic
148	C₂³	C₄⋊#170	1	1	yes	characteristic
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊#170	1	1	yes	characteristic
154	C₂×C₄	C₄⋊#170	1	1	yes	characteristic
155	C₂×C₄	C₄⋊#170	1	1	yes	characteristic
156	C₂×C₄	C₂×#700	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×C₄	C₂²×#170	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	5	1	no	—
161	C₈	—	5	1	no	—
162	C₈	—	10	1	no	—
163	C₈	—	10	1	no	—
164	C₈	—	10	1	no	—
165	C₈	—	20	2	no	—
166	C₈	#3998	1	4	yes	complemented
167	C₅	#12017	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂×C₄⋊#170	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#65727	1	1	yes	characteristic
177	C₄	#65729	1	1	yes	characteristic
178	C₄	C₂×#3998	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#10604150	1	1	yes	characteristic
181	C₂	#10604738	1	1	yes	characteristic
182	C₂	#10604751	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	#12017
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. #10598524	C₂	semidirect
5. #10598522	C₂	semidirect
87. C₅×C₈	M₃₂	semidirect
166. C₈	#3998	semidirect
167. C₅	#12017	semidirect

Maximal subgroups

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

subgroup	index
#10604152	2
#10604150	2
M₁₆×#168	2
#10598524	2
#10598522	2
#12017	5