Finite Group Databasebeta

Subgroups — #13027434

Order 1280.

308 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 96 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	#10615060	C₂	1	1	yes	maximal, characteristic
2	#10604370	C₂	1	1	yes	maximal, characteristic
3	#10604369	C₂	1	1	yes	maximal, characteristic
4	#10598729	C₂	1	2	yes	maximal
5	#10598699	C₂	1	2	yes	maximal
6	C₅×#533	C₄	1	1	yes	characteristic
7	C₂×#3998	C₂²	1	2	yes	—
8	C₂×#3998	C₂²	1	2	yes	—
9	#65313	—	2	2	no	—
10	#65313	—	2	2	no	—
11	#65288	C₄	1	1	yes	characteristic
12	#65288	C₄	1	1	yes	characteristic
13	#65286	C₂²	1	1	yes	characteristic
14	#65286	C₂²	1	1	yes	characteristic
15	#65213	C₄	1	1	yes	characteristic
16	C₂×#3935	C₂²	1	1	yes	characteristic
17	#64861	—	2	2	no	—
18	#64860	—	2	2	no	—
19	#13269	—	5	1	no	maximal, Sylow
20	C₅×C₄⋊Q₈	—	2	1	no	—
21	C₅×#125	—	2	1	no	—
22	C₅×C₂²⋊Q₈	—	2	2	no	—
23	C₄×C₅×Q₈	—	2	2	no	—
24	C₅×#117	C₂×C₄	1	1	yes	characteristic
25	C₅×#117	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×#681	C₂×C₄	1	1	yes	characteristic
28	C₂×#681	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₂²×#168	C₂³	1	1	yes	characteristic
31	#3998	—	2	2	no	—
32	#3998	—	2	2	no	—
33	#3998	—	2	2	no	—
34	#3998	—	2	2	no	—
35	C₂×#674	—	2	2	no	—
36	C₂×#674	Q₈	1	4	yes	complemented
37	C₄⋊#168	—	2	2	no	—
38	C₄⋊#168	—	2	2	no	—
39	C₄⋊#168	—	2	2	no	—
40	#3935	—	2	1	no	—
41	#3935	—	2	1	no	—
42	#3935	—	2	1	no	—
43	#3935	D₄	1	2	yes	—
44	C₄×#168	—	2	1	no	—
45	C₄×#168	D₄	1	2	yes	—
46	#2961	—	5	1	no	—
47	#2153	—	5	1	no	—
48	#2110	—	5	1	no	—
49	#1378	—	5	2	no	—
50	#1375	—	5	2	no	—
51	C₂×C₅×Q₈	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	1	no	—
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₅×C₄⋊C₄	—	2	4	no	—
55	C₅×C₄⋊C₄	—	2	2	no	—
56	C₅×C₄⋊C₄	—	2	1	no	—
57	C₅×C₄⋊C₄	—	2	1	no	—
58	C₅×C₂²⋊C₄	—	2	1	no	—
59	C₅×C₂²⋊C₄	M₁₆	1	2	yes	—
60	C₄²×C₅	—	2	1	no	—
61	C₄²×C₅	—	2	1	no	—
62	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅	M₁₆	1	1	yes	characteristic
64	C₄²×C₅	M₁₆	1	1	yes	characteristic
65	#681	—	2	2	no	—
66	#681	—	2	2	no	—
67	C₂×#168	—	2	1	no	—
68	C₂×#168	—	2	1	no	—
69	C₂×#168	—	2	2	no	—
70	C₂×#168	C₂²⋊C₄	1	2	yes	—
71	C₂×#168	C₂×D₄	1	1	yes	characteristic
72	C₂×#168	C₂×D₄	1	1	yes	characteristic
73	C₂×#168	C₂×Q₈	1	2	yes	—
74	C₂×#168	#41	1	2	yes	—
75	#674	—	2	4	no	—
76	#674	—	4	2	no	—
77	#533	AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂×M₃₂	—	5	2	no	—
79	C₂×M₃₂	—	5	2	no	—
80	#446	—	10	2	no	—
81	#446	—	10	2	no	—
82	#424	—	5	1	no	—
83	#424	—	5	1	no	—
84	#423	—	5	1	no	—
85	#405	—	5	1	no	—
86	#405	—	5	1	no	—
87	C₂×#97	—	5	1	no	—
88	#364	—	10	2	no	—
89	#347	—	10	2	no	—
90	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₅×Q₈	—	8	2	no	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	2	1	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	8	1	no	—
97	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅	#131	1	2	yes	—
101	#168	—	2	2	no	—
102	#168	—	4	1	no	—
103	#168	—	4	1	no	—
104	#168	C₂²⋊Q₈	1	2	yes	—
105	#168	#124	1	1	yes	characteristic
106	#168	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×M₁₆	—	5	1	no	—
108	C₂×M₁₆	—	5	1	no	—
109	C₂²×C₈	—	5	1	no	—
110	C₂²×C₈	—	5	1	no	—
111	C₄⋊Q₈	—	2	1	no	—
112	#125	—	2	1	no	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	#117	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
116	#117	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
118	M₃₂	—	10	2	no	—
119	M₃₂	—	10	2	no	—
120	M₃₂	—	10	2	no	—
121	M₃₂	—	10	2	no	—
122	C₂×C₁₆	—	5	4	no	—
123	C₂×C₁₆	—	10	2	no	—
124	C₄⋊C₈	—	10	2	no	—
125	C₄⋊C₈	—	10	2	no	—
126	C₄⋊C₈	—	10	2	no	—
127	#97	—	5	2	no	—
128	#97	—	10	1	no	—
129	#97	—	10	1	no	—
130	#97	—	10	1	no	—
131	C₄×C₈	—	5	2	no	—
132	C₄×C₈	—	10	1	no	—
133	C₂²×C₅	—	4	1	no	—
134	C₂²×C₅	#407	1	1	yes	characteristic
135	C₂²×C₅	#446	1	2	yes	—
136	C₄×C₅	—	4	1	no	—
137	C₄×C₅	—	4	1	no	—
138	C₄×C₅	—	4	2	no	—
139	C₄×C₅	—	4	2	no	—
140	C₄×C₅	#391	1	1	yes	characteristic
141	C₄×C₅	#408	1	1	yes	characteristic
142	C₄×C₅	#446	1	2	yes	—
143	C₂×Q₈	—	2	2	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
146	M₁₆	—	10	2	no	—
147	M₁₆	—	10	2	no	—
148	C₂×C₈	—	5	2	no	—
149	C₂×C₈	—	5	2	no	—
150	C₂×C₈	—	5	1	no	—
151	C₂×C₈	—	5	1	no	—
152	C₂×C₈	—	5	2	no	—
153	C₂×C₈	—	10	2	no	—
154	C₂×C₈	—	10	1	no	—
155	C₂×C₈	—	10	1	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₂²⋊C₄	#704	1	2	yes	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	#704	1	1	yes	characteristic
165	C₄²	#704	1	1	yes	characteristic
166	C₄²	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
167	C₁₆	—	10	4	no	—
168	C₁₆	—	20	2	no	—
169	C₂×C₅	—	2	1	no	—
170	C₂×C₅	—	2	2	no	—
171	C₂×C₅	#1985	1	1	yes	characteristic
172	C₂³	C₂×#704	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	#4133	1	2	yes	—
180	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
181	C₂×C₄	C₂×#704	1	1	yes	characteristic
182	C₂×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₈	—	5	2	no	—
184	C₈	—	5	1	no	—
185	C₈	—	5	1	no	—
186	C₈	—	10	2	no	—
187	C₈	—	20	1	no	—
188	C₈	—	20	1	no	—
189	C₅	#13269	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	4	1	no	—
191	C₂²	#65788	1	2	yes	—
192	C₂²	#65798	1	1	yes	characteristic
193	C₄	—	4	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	#65787	1	1	yes	characteristic
198	C₄	#65788	1	2	yes	—
199	C₄	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	#10604985	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×AGL(1,𝔽₅))⋊Q₈
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#533	C₄
socle	C₂×C₅	#1985
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×#168	C₂³
hypercenter	#533	AGL(1,𝔽₅)
hypocenter	C₅	#13269
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
36. C₂×#674	Q₈	semidirect
189. C₅	#13269	semidirect

Maximal subgroups

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

subgroup	index
#10615060	2
#10604370	2
#10604369	2
#10598729	2
#10598699	2
#13269	5