Subgroups — #12015725

Order 1280.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1844	C₂	1	1	yes	maximal, characteristic
2	C₂×#64707	C₂	1	4	yes	maximal
3	#10600585	C₂	1	1	yes	maximal, characteristic
4	C₄×#3944	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#106	—	2	1	no	—
6	C₂×C₅×#106	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#384	—	2	1	no	—
9	C₂×#3951	—	2	1	no	—
10	C₂×#3951	C₂²	1	2	yes	—
11	C₂×#3944	C₂²	1	2	yes	—
12	#65213	—	2	1	no	—
13	#64707	—	2	8	no	—
14	#64681	C₄	1	2	yes	complemented
15	C₄×#672	C₄	1	2	yes	complemented
16	#12903	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	4	2	no	—
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₂×#681	—	2	2	no	—
23	C₅×#106	—	2	4	no	—
24	C₅×#106	—	4	2	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
28	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
29	#3951	—	2	4	no	—
30	#3951	—	4	2	no	—
31	#3944	D₄	1	4	yes	—
32	#3944	Q₈	1	4	yes	—
33	C₂×#672	C₂×C₄	1	4	yes	—
34	C₄⋊#168	—	4	2	no	—
35	C₂×#365	—	5	4	no	—
36	C₄×M₃₂	—	5	1	no	—
37	#1848	—	5	1	no	—
38	#1844	D₅	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	4	2	no	—
40	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
41	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	SD₁₆	1	2	yes	—
45	C₂×C₅×C₈	SD₁₆	1	2	yes	—
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
48	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
50	C₅×C₄⋊C₄	—	4	2	no	—
51	C₅×C₄⋊C₄	—	4	2	no	—
52	C₅×C₄⋊C₄	—	8	4	no	—
53	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₅	SD₁₆	1	2	yes	—
56	#681	—	4	2	no	—
57	#681	—	4	2	no	—
58	C₂×#168	—	4	2	no	—
59	#672	C₄⋊C₄	1	8	yes	complemented
60	C₂×M₃₂	—	5	2	no	—
61	C₂×#108	—	5	2	no	—
62	C₂×#108	—	10	1	no	—
63	C₂×#106	—	2	1	no	—
64	C₂×#106	C₂×D₅	1	2	yes	—
65	#423	—	10	1	no	—
66	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	#384	—	2	1	no	—
68	#365	—	10	8	no	—
69	#346	—	5	2	no	—
70	C₄×C₁₆	—	5	2	no	—
71	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	4	no	—
73	C₂×C₄×C₅	—	8	4	no	—
74	C₂×C₄×C₅	—	8	2	no	—
75	C₂×C₄×C₅	#106	1	2	yes	—
76	C₂×C₄×C₅	#106	1	2	yes	—
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	#127	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
83	C₅×C₈	#106	1	2	yes	—
84	C₅×C₈	#106	1	2	yes	—
85	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
86	C₅×C₈	C₄⋊Q₈	1	2	yes	—
87	#168	—	8	2	no	—
88	C₂×M₁₆	—	10	2	no	—
89	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
94	M₃₂	—	5	4	no	—
95	M₃₂	—	5	4	no	—
96	C₂×C₁₆	—	5	4	no	—
97	#108	—	10	4	no	—
98	#108	—	20	2	no	—
99	#106	—	2	4	no	—
100	#106	—	4	2	no	—
101	C₄⋊C₈	—	20	2	no	—
102	C₄×C₈	Dic₁₀	1	1	yes	characteristic
103	C₄×C₈	Dic₁₀	1	1	yes	characteristic
104	C₄×C₈	C₄×D₅	1	1	yes	characteristic
105	C₄×C₈	C₄×D₅	1	1	yes	characteristic
106	C₂²×C₅	—	2	2	no	—
107	C₂²×C₅	C₂×#106	1	1	yes	characteristic
108	C₂²×C₅	#492	1	1	yes	characteristic
109	C₂²×C₅	#498	1	1	yes	characteristic
110	C₄×C₅	—	2	2	no	—
111	C₄×C₅	—	16	2	no	—
112	C₄×C₅	#365	1	4	yes	complemented
113	C₄×C₅	#384	1	1	yes	characteristic
114	C₄×C₅	C₂×#106	1	1	yes	characteristic
115	C₄×C₅	#492	1	1	yes	characteristic
116	C₄×C₅	#498	1	1	yes	characteristic
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	#687	1	2	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
120	M₁₆	—	20	2	no	—
121	M₁₆	—	20	2	no	—
122	C₂×C₈	—	20	2	no	—
123	C₂×C₈	#683	1	1	yes	characteristic
124	C₂×C₈	#683	1	1	yes	characteristic
125	C₂×C₈	#687	1	2	yes	—
126	C₂×C₈	#688	1	2	yes	—
127	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
129	C₂×C₈	D₄×D₅	1	2	yes	—
130	C₂×C₈	Q₈×D₅	1	2	yes	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	8	4	no	—
134	C₄²	#683	1	1	yes	characteristic
135	C₄²	#683	1	1	yes	characteristic
136	C₄²	#688	1	2	yes	—
137	C₁₆	—	5	8	no	—
138	C₂×C₅	—	2	1	no	—
139	C₂×C₅	—	2	1	no	—
140	C₂×C₅	#1844	1	1	yes	characteristic
141	C₂×C₅	C₂×#365	1	2	yes	—
142	C₂³	C₂×#687	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	#3940	1	2	yes	—
147	C₂×C₄	#3940	1	2	yes	—
148	C₂×C₄	C₂×#683	1	1	yes	characteristic
149	C₂×C₄	C₂×#687	1	1	yes	characteristic
150	C₂×C₄	#4086	1	1	yes	characteristic
151	C₂×C₄	C₂×#688	1	1	yes	characteristic
152	C₂×C₄	C₂×#688	1	1	yes	characteristic
153	C₂×C₄	#4090	1	1	yes	characteristic
154	C₈	—	40	2	no	—
155	C₈	#3940	1	2	yes	—
156	C₈	#3940	1	2	yes	—
157	C₈	#4034	1	2	yes	—
158	C₈	D₅×C₄⋊C₄	1	2	yes	—
159	C₅	#12903	1	1	yes	characteristic, complemented, Sylow
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#3940	1	1	yes	characteristic
162	C₂²	#65365	1	1	yes	characteristic
163	C₂²	#65368	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	16	2	no	—
166	C₄	#64707	1	4	yes	complemented
167	C₄	C₂×#3940	1	1	yes	characteristic
168	C₄	#65265	1	1	yes	characteristic
169	C₄	#65368	1	1	yes	characteristic
170	C₄	#65374	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#10600369	1	1	yes	characteristic
174	C₂	C₂×#64707	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#3940
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1844	C₂
socle	C₂²×C₅	C₂×#106
prime radical	C₂³×C₅	C₂×SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1844	D₅
hypocenter	C₅	#12903
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
14. #64681	C₄	semidirect
15. C₄×#672	C₄	semidirect
59. #672	C₄⋊C₄	semidirect
112. C₄×C₅	#365	semidirect
159. C₅	#12903	semidirect
166. C₄	#64707	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1844	2
C₂×#64707	2
#10600585	2
C₄×#3944	2
#12903	5