Subgroups — #12014190

Order 1280.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	2	yes	maximal
3	#10598299	C₂	1	4	yes	maximal, complemented
4	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂²×#681	—	2	2	no	—
6	C₂×C₅×M₃₂	—	2	4	no	—
7	C₂×C₅×M₃₂	C₄	1	4	yes	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₂×#3951	—	2	2	no	—
10	C₂×#3951	—	2	4	no	—
11	C₂×#3951	C₄	1	8	yes	—
12	C₂×#3951	C₂²	1	4	yes	—
13	#12559	—	5	1	no	maximal, Sylow
14	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	D₄	1	4	yes	—
19	C₂²×C₅×C₈	Q₈	1	2	yes	—
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×#681	—	2	8	no	—
22	C₂×#681	—	2	4	no	—
23	C₂×#681	—	4	2	no	—
24	C₂×#681	—	4	4	no	—
25	C₂²×#168	—	4	2	no	—
26	C₅×M₃₂	—	2	8	no	—
27	C₂×C₅×C₁₆	—	2	4	no	—
28	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
29	#3951	—	2	8	no	—
30	#3951	—	2	4	no	—
31	#3951	C₂×C₄	1	8	yes	—
32	C₂²×M₃₂	D₅	1	1	yes	characteristic
33	C₂²×#108	—	5	2	no	—
34	#1381	—	5	4	no	—
35	C₂⁴×C₅	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅	D₈	1	2	yes	—
39	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₄²	1	4	yes	—
43	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
44	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
45	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
46	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
48	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
50	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
51	#681	—	2	8	no	—
52	#681	—	4	8	no	—
53	#681	—	4	4	no	—
54	C₂×#168	—	4	8	no	—
55	C₂×#168	—	4	4	no	—
56	C₅×C₁₆	—	2	4	no	—
57	C₂²×M₁₆	—	10	2	no	—
58	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
59	C₂×M₃₂	—	2	4	no	—
60	C₂×M₃₂	Dic₅	1	4	yes	—
61	C₂²×C₁₆	C₂×D₅	1	2	yes	—
62	C₂×#108	—	5	8	no	—
63	C₂×#108	—	5	4	no	—
64	C₂×#108	—	10	2	no	—
65	C₂×#108	—	10	4	no	—
66	C₂³×C₅	—	4	2	no	—
67	C₂³×C₅	#103	1	4	yes	—
68	C₂³×C₅	#107	1	2	yes	—
69	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅	#95	1	4	yes	—
71	C₂×C₄×C₅	#102	1	4	yes	—
72	C₂×C₄×C₅	#102	1	4	yes	—
73	C₂×C₄×C₅	#103	1	4	yes	—
74	C₂×C₄×C₅	#106	1	4	yes	—
75	C₂×C₄×C₅	#107	1	2	yes	—
76	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
80	C₅×C₈	#95	1	4	yes	—
81	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
82	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
84	#168	—	4	8	no	—
85	C₂³×C₄	D₂₀	1	1	yes	characteristic
86	C₂×M₁₆	—	10	8	no	—
87	C₂×M₁₆	—	10	4	no	—
88	C₂×M₁₆	—	20	4	no	—
89	C₂×M₁₆	—	20	2	no	—
90	C₂²×C₈	—	20	2	no	—
91	C₂²×C₈	Dic₁₀	1	2	yes	—
92	C₂²×C₈	C₄×D₅	1	4	yes	—
93	C₂²×C₈	D₂₀	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₅	1	2	yes	—
95	C₂²×C₈	C₂²⋊D₅	1	4	yes	—
96	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
97	M₃₂	—	2	8	no	—
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
100	#108	—	5	8	no	—
101	#108	—	10	8	no	—
102	#108	—	10	4	no	—
103	C₂²×C₅	—	2	4	no	—
104	C₂²×C₅	—	2	2	no	—
105	C₂²×C₅	—	4	4	no	—
106	C₂²×C₅	#340	1	4	yes	—
107	C₂²×C₅	C₂×#103	1	2	yes	—
108	C₂²×C₅	C₂×#107	1	1	yes	characteristic
109	C₄×C₅	#340	1	4	yes	—
110	C₄×C₅	C₂×#95	1	1	yes	characteristic
111	C₄×C₅	C₂×#102	1	2	yes	—
112	C₄×C₅	C₂×#106	1	1	yes	characteristic
113	C₂⁴	Dic₂₀	1	1	yes	characteristic
114	C₂²×C₄	#677	1	4	yes	—
115	C₂²×C₄	D₄₀	1	2	yes	—
116	C₂²×C₄	Dic₂₀	1	1	yes	characteristic
117	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
118	C₂²×C₄	#685	1	4	yes	—
119	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
120	M₁₆	—	10	8	no	—
121	M₁₆	—	20	8	no	—
122	M₁₆	—	20	4	no	—
123	C₂×C₈	—	20	8	no	—
124	C₂×C₈	—	20	4	no	—
125	C₂×C₈	C₄×Dic₅	1	4	yes	—
126	C₂×C₈	#683	1	8	yes	—
127	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
128	C₂×C₈	#685	1	4	yes	—
129	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
130	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
132	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
133	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
134	C₁₆	—	2	4	no	—
135	C₂×C₅	—	2	4	no	—
136	C₂×C₅	—	2	2	no	—
137	C₂×C₅	C₂×#340	1	1	yes	characteristic
138	C₂×C₅	#2161	1	2	yes	—
139	C₂³	—	4	2	no	—
140	C₂³	#3948	1	4	yes	—
141	C₂³	#3950	1	2	yes	—
142	C₂³	C₂×Dic₂₀	1	1	yes	characteristic
143	C₂×C₄	#3948	1	4	yes	—
144	C₂×C₄	#3949	1	4	yes	—
145	C₂×C₄	#3950	1	2	yes	—
146	C₂×C₄	#3953	1	4	yes	—
147	C₂×C₄	#3953	1	4	yes	—
148	C₂×C₄	#3963	1	4	yes	—
149	C₂×C₄	C₂×#677	1	2	yes	—
150	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
152	C₂×C₄	C₂×#685	1	2	yes	—
153	C₈	—	20	8	no	—
154	C₈	#3963	1	4	yes	—
155	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
156	C₈	C₂×#683	1	2	yes	—
157	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
158	C₅	#12559	1	1	yes	characteristic, complemented, Sylow
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	#64771	1	4	yes	—
163	C₂²	C₂×#3948	1	2	yes	—
164	C₂²	C₂×#3950	1	1	yes	characteristic
165	C₄	#64771	1	4	yes	—
166	C₄	C₂×#3949	1	1	yes	characteristic
167	C₄	C₂×#3953	1	2	yes	—
168	C₄	C₂×#3963	1	1	yes	characteristic
169	C₂	—	2	4	no	—
170	C₂	—	2	2	no	—
171	C₂	#10602222	1	2	yes	—
172	C₂	C₂×#64771	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂⁴×C₅	Q₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	#12559
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #10598299	C₂	semidirect
158. C₅	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂²×#3951	2
#10598299	2
#12559	5