Subgroups — #12010889

Order 1280.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#10600522	C₂	1	1	yes	maximal, characteristic
3	C₂×#64678	C₂	1	1	yes	maximal, characteristic
4	#10597992	C₂	1	2	yes	maximal, complemented
5	#10597991	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	#65214	—	2	1	no	—
10	C₂×C₄⋊#168	—	2	1	no	—
11	C₅×#346	C₄	1	2	yes	—
12	C₄×C₅×C₁₆	C₂²	1	2	yes	—
13	#64678	—	2	1	no	—
14	#64678	C₂²	1	2	yes	—
15	#64678	C₂²	1	1	yes	characteristic
16	#64678	C₂²	1	1	yes	characteristic
17	#64677	—	2	1	no	—
18	#12009	—	5	1	no	maximal, Sylow
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₂²×#168	—	4	1	no	—
23	C₅×M₃₂	—	2	2	no	—
24	C₅×M₃₂	C₈	1	4	yes	complemented
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
30	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂²⋊#168	—	4	1	no	—
32	C₄⋊#168	—	2	1	no	—
33	C₄⋊#168	—	2	1	no	—
34	C₄⋊#168	—	2	2	no	—
35	C₄⋊#168	—	4	1	no	—
36	C₄×#168	—	4	1	no	—
37	C₄×M₃₂	D₅	1	1	yes	characteristic
38	#1564	—	5	1	no	—
39	C₂×#335	—	5	1	no	—
40	#1372	—	5	2	no	—
41	#1371	—	5	2	no	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
49	C₂×C₅×C₈	M₁₆	1	2	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₅	D₈	1	1	yes	characteristic
54	C₄²×C₅	Q₁₆	1	1	yes	characteristic
55	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×#168	—	4	1	no	—
57	C₂×#168	—	4	1	no	—
58	C₂×#168	—	4	2	no	—
59	C₂×#168	—	8	2	no	—
60	C₅×C₁₆	—	2	2	no	—
61	C₅×C₁₆	C₂×C₈	1	4	yes	complemented
62	C₂×M₃₂	Dic₅	1	1	yes	characteristic
63	C₂×M₃₂	Dic₅	1	1	yes	characteristic
64	#431	—	10	1	no	—
65	C₂×C₄⋊C₈	—	10	1	no	—
66	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	#346	Dic₅	1	2	yes	—
68	C₄×C₁₆	C₂×D₅	1	2	yes	—
69	#335	—	5	2	no	—
70	#335	—	5	1	no	—
71	#335	—	5	1	no	—
72	#335	—	10	1	no	—
73	#334	—	10	1	no	—
74	C₂³×C₅	#107	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	4	1	no	—
78	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₅	#107	1	1	yes	characteristic
80	C₂×C₄×C₅	#107	1	1	yes	characteristic
81	C₂×C₄×C₅	#107	1	1	yes	characteristic
82	C₂×C₄×C₅	#108	1	2	yes	—
83	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
86	C₅×C₈	—	2	1	no	—
87	C₅×C₈	—	2	1	no	—
88	C₅×C₈	C₄⋊C₈	1	2	yes	—
89	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
90	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
91	#168	—	8	1	no	—
92	#168	—	8	2	no	—
93	#168	—	8	1	no	—
94	C₂²×C₈	—	20	1	no	—
95	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
96	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
97	C₂×C₄²	D₂₀	1	1	yes	characteristic
98	M₃₂	—	2	2	no	—
99	M₃₂	#168	1	4	yes	complemented
100	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
101	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
102	C₄⋊C₈	—	10	1	no	—
103	C₄⋊C₈	—	10	1	no	—
104	C₄⋊C₈	—	10	2	no	—
105	C₄⋊C₈	—	20	1	no	—
106	C₂²⋊C₈	—	20	1	no	—
107	C₄×C₈	—	20	1	no	—
108	C₄×C₈	Dic₁₀	1	1	yes	characteristic
109	C₄×C₈	D₂₀	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	#335	1	2	yes	—
115	C₂²×C₅	C₂×#107	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	#335	1	2	yes	—
120	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₄×C₅	C₂×#108	1	1	yes	characteristic
122	C₄×C₅	#509	1	1	yes	characteristic
123	C₄×C₅	#510	1	1	yes	characteristic
124	C₂²×C₄	D₄₀	1	1	yes	characteristic
125	C₂²×C₄	Dic₂₀	1	1	yes	characteristic
126	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	20	1	no	—
129	C₂×C₈	—	20	1	no	—
130	C₂×C₈	—	20	2	no	—
131	C₂×C₈	—	40	2	no	—
132	C₂×C₈	C₂×#168	1	2	yes	—
133	C₂×C₈	#681	1	2	yes	—
134	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
135	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
138	C₄²	D₄₀	1	1	yes	characteristic
139	C₄²	Dic₂₀	1	1	yes	characteristic
140	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
141	C₄²	C₂×D₂₀	1	1	yes	characteristic
142	C₁₆	—	2	2	no	—
143	C₁₆	C₂×#168	1	4	yes	complemented
144	C₂×C₅	—	2	2	no	—
145	C₂×C₅	C₂×#335	1	1	yes	characteristic
146	C₂×C₅	#2159	1	1	yes	characteristic
147	C₂×C₅	#2161	1	1	yes	characteristic
148	C₂³	#3950	1	1	yes	characteristic
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	C₄⋊#168	1	2	yes	—
153	C₂×C₄	#3950	1	1	yes	characteristic
154	C₂×C₄	#3950	1	1	yes	characteristic
155	C₂×C₄	#3950	1	1	yes	characteristic
156	C₂×C₄	#3951	1	2	yes	—
157	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	40	2	no	—
163	C₈	—	40	1	no	—
164	C₈	—	40	1	no	—
165	C₈	C₄⋊#168	1	2	yes	—
166	C₈	C₂²×#168	1	1	yes	characteristic
167	C₈	C₂×#681	1	1	yes	characteristic
168	C₅	#12009	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#64678	1	2	yes	—
172	C₂²	C₂×#3950	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#64678	1	2	yes	—
177	C₄	#65197	1	1	yes	characteristic
178	C₄	#65198	1	1	yes	characteristic
179	C₄	C₂×C₄⋊#168	1	1	yes	characteristic
180	C₄	C₂×#3951	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#64678	1	1	yes	characteristic
183	C₂	#10602221	1	1	yes	characteristic
184	C₂	#10602222	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀
commutator	C₅×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	#107
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#12009
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10597992	C₂	semidirect
5. #10597991	C₂	semidirect
24. C₅×M₃₂	C₈	semidirect
61. C₅×C₁₆	C₂×C₈	semidirect
99. M₃₂	#168	semidirect
143. C₁₆	C₂×#168	semidirect
168. C₅	#12009	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×M₃₂	2
#10600522	2
C₂×#64678	2
#10597992	2
#10597991	2
#12009	5