Subgroups — #12008334

Order 1280.

290 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2162	C₂	1	1	yes	maximal, characteristic
2	#10600537	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic
4	#10597990	C₂	1	2	yes	maximal, complemented
5	#10597989	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×#168	C₄	1	1	yes	characteristic
8	C₂×C₄×#168	C₄	1	1	yes	characteristic
9	C₅×C₄⋊C₁₆	C₄	1	2	yes	—
10	C₅×C₂²⋊C₁₆	C₄	1	2	yes	—
11	C₄×C₅×C₁₆	C₂²	1	2	yes	—
12	C₂²⋊#672	C₄	1	2	yes	—
13	C₄⋊#672	C₄	1	2	yes	—
14	C₄×#672	C₂²	1	2	yes	—
15	C₈×#168	C₄	1	1	yes	characteristic
16	C₈×#168	C₄	1	1	yes	characteristic
17	C₈×#168	C₂²	1	1	yes	characteristic
18	C₈×#168	C₂²	1	1	yes	characteristic
19	#11411	—	5	1	no	maximal, Sylow
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×#168	C₈	1	2	yes	—
24	C₂²×#168	C₈	1	2	yes	—
25	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₂×#672	C₂×C₄	1	4	yes	—
31	C₄×#168	C₈	1	2	yes	—
32	C₄×#168	C₈	1	2	yes	—
33	C₄×#168	C₂×C₄	1	1	yes	characteristic
34	C₄×#168	C₂×C₄	1	1	yes	characteristic
35	C₄×#168	C₂×C₄	1	1	yes	characteristic
36	C₄×#168	C₂×C₄	1	1	yes	characteristic
37	#2162	—	5	1	no	—
38	#2162	D₅	1	1	yes	characteristic
39	C₂×C₈²	—	5	1	no	—
40	#1312	—	5	2	no	—
41	#1311	—	5	2	no	—
42	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
48	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅	C₄²	1	1	yes	characteristic
52	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×#168	—	2	4	no	—
56	C₂×#168	—	2	4	no	—
57	C₂×#168	C₂×C₈	1	2	yes	—
58	C₂×#168	C₂×C₈	1	2	yes	—
59	C₂×#168	C₂×C₈	1	2	yes	—
60	C₂×#168	C₂×C₈	1	2	yes	—
61	C₅×C₁₆	—	2	4	no	—
62	#672	—	2	4	no	—
63	C₂×C₄×C₈	—	5	1	no	—
64	C₂×C₄×C₈	—	5	1	no	—
65	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
66	C₄⋊C₁₆	—	5	2	no	—
67	C₄⋊C₁₆	Dic₅	1	2	yes	—
68	C₂²⋊C₁₆	—	5	2	no	—
69	C₂²⋊C₁₆	Dic₅	1	2	yes	—
70	C₄×C₁₆	—	5	2	no	—
71	C₄×C₁₆	C₂×D₅	1	2	yes	—
72	C₈²	—	5	1	no	—
73	C₈²	—	5	1	no	—
74	C₈²	—	5	1	no	—
75	C₈²	—	5	1	no	—
76	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	1	no	—
78	C₂×C₄×C₅	—	2	1	no	—
79	C₂×C₄×C₅	—	2	2	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	M₃₂	1	4	yes	—
84	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
87	C₅×C₈	—	2	1	no	—
88	C₅×C₈	—	2	1	no	—
89	C₅×C₈	#131	1	2	yes	—
90	C₅×C₈	#131	1	2	yes	—
91	#168	—	2	2	no	—
92	#168	—	2	2	no	—
93	#168	—	2	2	no	—
94	#168	M₃₂	1	4	yes	—
95	C₂²×C₈	—	5	2	no	—
96	C₂²×C₈	—	5	2	no	—
97	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
98	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
99	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
100	C₂×C₁₆	—	5	4	no	—
101	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
102	C₄×C₈	—	5	2	no	—
103	C₄×C₈	—	5	2	no	—
104	C₄×C₈	—	5	1	no	—
105	C₄×C₈	—	5	1	no	—
106	C₄×C₈	—	5	1	no	—
107	C₄×C₈	—	5	1	no	—
108	C₄×C₈	C₄×D₅	1	1	yes	characteristic
109	C₄×C₈	C₄×D₅	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
111	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
112	C₂²×C₅	—	2	1	no	—
113	C₂²×C₅	—	2	1	no	—
114	C₂²×C₅	#346	1	2	yes	—
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	#346	1	2	yes	—
119	C₄×C₅	#405	1	1	yes	characteristic
120	C₄×C₅	#405	1	1	yes	characteristic
121	C₄×C₅	C₂×M₃₂	1	2	yes	—
122	C₄×C₅	C₂×M₃₂	1	2	yes	—
123	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
124	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
125	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	5	2	no	—
129	C₂×C₈	—	5	2	no	—
130	C₂×C₈	—	5	2	no	—
131	C₂×C₈	—	5	2	no	—
132	C₂×C₈	—	10	4	no	—
133	C₂×C₈	—	10	4	no	—
134	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
135	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
138	C₄²	C₈×D₅	1	1	yes	characteristic
139	C₄²	C₈×D₅	1	1	yes	characteristic
140	C₄²	C₄×Dic₅	1	1	yes	characteristic
141	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
142	C₁₆	—	2	4	no	—
143	C₁₆	—	10	4	no	—
144	C₂×C₅	—	2	2	no	—
145	C₂×C₅	#1450	1	1	yes	characteristic
146	C₂×C₅	C₂×#346	1	1	yes	characteristic
147	C₂×C₅	C₄×M₃₂	1	1	yes	characteristic
148	C₂³	C₈×Dic₅	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	#3930	1	4	yes	—
153	C₂×C₄	C₈×Dic₅	1	1	yes	characteristic
154	C₂×C₄	C₈×Dic₅	1	1	yes	characteristic
155	C₂×C₄	C₈×Dic₅	1	1	yes	characteristic
156	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
157	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	5	4	no	—
162	C₈	—	10	2	no	—
163	C₈	—	10	2	no	—
164	C₈	—	10	2	no	—
165	C₈	#4053	1	2	yes	—
166	C₈	#4094	1	2	yes	—
167	C₅	#11411	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#64722	1	2	yes	—
171	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	#64722	1	2	yes	—
175	C₄	C₂×#3930	1	2	yes	—
176	C₄	D₅×M₃₂	1	2	yes	—
177	C₄	#65286	1	1	yes	characteristic
178	C₄	#65411	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#10601533	1	1	yes	characteristic
181	C₂	C₂×#64722	1	1	yes	characteristic
182	C₂	Dic₅×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅
commutator, socle	C₂²×C₅	C₂×C₄×C₈
Fitting	C₅×#2162	C₂
prime radical	C₂³×C₅	C₄×C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	#2162	D₅
hypocenter	C₅	#11411
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
4. #10597990	C₂	semidirect
5. #10597989	C₂	semidirect
167. C₅	#11411	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2162	2
#10600537	2
C₂×C₈×#168	2
#10597990	2
#10597989	2
#11411	5