Subgroups — #11990424

Order 1280.

502 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#3946	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#64674	C₂	1	2	yes	maximal
4	C₅×#1273	C₂	1	1	yes	maximal, characteristic
5	#10598144	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₂×#3946	C₄	1	4	yes	—
9	C₂×C₈×Dic₅	C₄	1	2	yes	—
10	C₂×C₈×Dic₅	C₄	1	2	yes	—
11	C₄²×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×#168	C₂²	1	2	yes	—
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	#64770	C₄	1	4	yes	—
15	#64674	—	2	4	no	—
16	#8260	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
24	C₂²×#168	C₂×C₄	1	4	yes	—
25	C₂²×#168	C₂×C₄	1	2	yes	—
26	C₄×C₅×C₈	—	2	2	no	—
27	C₄×C₅×C₈	—	2	2	no	—
28	C₄×C₅×C₈	—	2	2	no	—
29	#3946	—	2	8	no	—
30	C₈×Dic₅	—	2	4	no	—
31	C₈×Dic₅	C₈	1	8	yes	complemented
32	C₄×#168	—	2	2	no	—
33	C₄×#168	—	2	2	no	—
34	C₄×#168	D₄	1	2	yes	—
35	C₄×#168	Q₈	1	2	yes	—
36	C₄×C₄⋊C₈	—	5	2	no	—
37	C₄²×C₈	—	5	1	no	—
38	C₂×#322	—	5	2	no	—
39	#1273	—	5	1	no	—
40	#1273	D₅	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
43	C₂²×Dic₅	C₄²	1	1	yes	characteristic
44	C₂²×Dic₅	C₄²	1	2	yes	—
45	C₂²×Dic₅	C₄²	1	1	yes	characteristic
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	2	4	no	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
52	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅	#41	1	2	yes	—
55	C₄×Dic₅	—	2	2	no	—
56	C₄×Dic₅	—	2	4	no	—
57	C₄×Dic₅	C₂×C₈	1	4	yes	—
58	C₄×Dic₅	M₁₆	1	8	yes	—
59	C₂×#168	—	2	4	no	—
60	C₂×#168	—	2	4	no	—
61	C₂×#168	—	2	4	no	—
62	C₂×#168	C₄⋊C₄	1	4	yes	—
63	C₂×#168	C₂×C₈	1	4	yes	—
64	C₂×#168	M₁₆	1	4	yes	—
65	C₂×C₄⋊C₈	—	5	4	no	—
66	C₂×C₄×C₈	—	5	2	no	—
67	C₂×C₄×C₈	—	5	1	no	—
68	C₂×C₄×C₈	—	5	2	no	—
69	C₂×C₄×C₈	—	5	2	no	—
70	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	C₄³	—	5	1	no	—
73	#336	—	5	4	no	—
74	#322	—	10	4	no	—
75	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅	#117	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
80	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
83	C₂×C₄×C₅	#131	1	4	yes	—
84	C₂×Dic₅	—	2	2	no	—
85	C₂×Dic₅	—	2	2	no	—
86	C₂×Dic₅	—	2	2	no	—
87	C₂×Dic₅	C₄×C₈	1	2	yes	—
88	C₂×Dic₅	C₄×C₈	1	2	yes	—
89	C₂×Dic₅	#97	1	4	yes	—
90	C₂×Dic₅	#97	1	4	yes	—
91	C₅×C₈	—	2	4	no	—
92	C₅×C₈	—	4	4	no	—
93	#168	—	4	4	no	—
94	#168	C₄⋊C₈	1	8	yes	complemented
95	C₂²×C₈	—	5	2	no	—
96	C₂²×C₈	—	5	4	no	—
97	C₂²×C₈	C₄×D₅	1	4	yes	—
98	C₂²×C₈	C₄×D₅	1	2	yes	—
99	C₂×C₄²	—	5	1	no	—
100	C₂×C₄²	—	5	2	no	—
101	C₂×C₄²	—	5	2	no	—
102	C₂×C₄²	—	5	1	no	—
103	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
104	C₄⋊C₈	—	10	8	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	5	8	no	—
109	C₄×C₈	—	5	2	no	—
110	C₄×C₈	—	5	2	no	—
111	C₄×C₈	—	10	4	no	—
112	C₄×C₈	—	10	2	no	—
113	C₄×C₈	—	10	2	no	—
114	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
116	C₂²×C₅	C₂×#97	1	2	yes	—
117	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
118	C₂²×C₅	#405	1	2	yes	—
119	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
120	C₄×C₅	#431	1	2	yes	—
121	C₄×C₅	#436	1	4	yes	—
122	C₄×C₅	#446	1	4	yes	—
123	Dic₅	—	2	2	no	—
124	Dic₅	—	2	4	no	—
125	Dic₅	#322	1	4	yes	complemented
126	C₂²×C₄	—	5	1	no	—
127	C₂²×C₄	—	5	1	no	—
128	C₂²×C₄	—	5	2	no	—
129	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	5	4	no	—
137	C₂×C₈	—	5	4	no	—
138	C₂×C₈	—	5	4	no	—
139	C₂×C₈	—	10	4	no	—
140	C₂×C₈	—	10	4	no	—
141	C₂×C₈	—	10	4	no	—
142	C₂×C₈	C₈×D₅	1	4	yes	—
143	C₄²	—	5	4	no	—
144	C₄²	—	5	8	no	—
145	C₄²	—	10	4	no	—
146	C₄²	—	10	2	no	—
147	C₄²	D₄×D₅	1	1	yes	characteristic
148	C₄²	#711	1	2	yes	—
149	C₄²	Q₈×D₅	1	1	yes	characteristic
150	C₂×C₅	C₂×#322	1	1	yes	characteristic
151	C₂×C₅	#1450	1	1	yes	characteristic
152	C₂×C₅	#1451	1	2	yes	—
153	C₂×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic
154	C₂×C₅	#1770	1	2	yes	—
155	C₂³	C₄²×D₅	1	1	yes	characteristic
156	C₂×C₄	—	5	2	no	—
157	C₂×C₄	—	5	4	no	—
158	C₂×C₄	—	5	2	no	—
159	C₂×C₄	—	5	4	no	—
160	C₂×C₄	—	10	2	no	—
161	C₂×C₄	—	10	2	no	—
162	C₂×C₄	—	10	2	no	—
163	C₂×C₄	#4023	1	1	yes	characteristic
164	C₂×C₄	#4027	1	2	yes	—
165	C₂×C₄	#4033	1	2	yes	—
166	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
168	C₂×C₄	#4047	1	4	yes	—
169	C₂×C₄	D₅×M₁₆	1	4	yes	—
170	C₂×C₄	D₅×M₁₆	1	2	yes	—
171	C₈	—	2	4	no	—
172	C₈	—	4	4	no	—
173	C₈	—	5	8	no	—
174	C₈	—	20	4	no	—
175	C₅	#8260	1	1	yes	characteristic, complemented, Sylow
176	C₂²	#64938	1	1	yes	characteristic
177	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
178	C₂²	#64978	1	2	yes	—
179	C₂²	D₅×#97	1	2	yes	—
180	C₂²	#64996	1	1	yes	characteristic
181	C₄	—	5	4	no	—
182	C₄	—	10	2	no	—
183	C₄	—	10	4	no	—
184	C₄	#65007	1	2	yes	—
185	C₄	#65018	1	4	yes	—
186	C₄	#65117	1	4	yes	—
187	C₄	D₅×C₄⋊C₈	1	2	yes	—
188	C₂	#10598840	1	1	yes	characteristic
189	C₂	D₅×#322	1	1	yes	characteristic
190	C₂	#10598863	1	2	yes	—
191	C₂	#10599395	1	1	yes	characteristic
192	C₂	#10599445	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₅×#1273	C₂
socle, prime radical	C₂³×C₅	C₂×C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1273	D₅
hypocenter	C₅	#8260
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
31. C₈×Dic₅	C₈	semidirect
94. #168	C₄⋊C₈	semidirect
125. Dic₅	#322	semidirect
175. C₅	#8260	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3946	2
C₄×C₈×Dic₅	2
C₂×#64674	2
C₅×#1273	2
#10598144	2
#8260	5