Subgroups — #11993049

Order 1280.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#64678	C₂	1	2	yes	maximal
3	C₂×#64677	C₂	1	2	yes	maximal
4	#10600187	C₂	1	1	yes	maximal, characteristic
5	#10600185	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#97	C₄	1	2	yes	—
7	C₂×C₅×#97	C₄	1	1	yes	characteristic
8	C₂×C₅×#97	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#168	—	2	2	no	—
12	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
15	#64744	—	2	2	no	—
16	#64678	—	2	4	no	—
17	#64677	—	2	4	no	—
18	#8808	—	5	1	no	maximal, Sylow
19	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	D₄	1	2	yes	—
24	C₂×C₄²×C₅	Q₈	1	2	yes	—
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₂²×#168	—	2	2	no	—
27	C₂²×#168	—	2	1	no	—
28	C₂²×#168	—	2	1	no	—
29	C₅×#97	—	2	4	no	—
30	C₅×#97	—	2	2	no	—
31	C₅×#97	C₈	1	4	yes	complemented
32	C₄×C₅×C₈	D₄	1	4	yes	—
33	C₄×C₅×C₈	Q₈	1	4	yes	—
34	C₄⋊#168	—	2	2	no	—
35	C₄⋊#168	—	2	2	no	—
36	C₄⋊#168	—	2	2	no	—
37	C₄⋊#168	—	2	2	no	—
38	C₄⋊#168	—	4	4	no	—
39	#1842	—	5	1	no	—
40	#1831	—	5	1	no	—
41	C₄×#97	D₅	1	1	yes	characteristic
42	C₂×#335	—	5	2	no	—
43	C₂×#334	—	5	2	no	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
52	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
54	C₂×C₅×C₈	M₁₆	1	4	yes	—
55	C₄²×C₅	—	2	2	no	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	—	2	2	no	—
58	C₄²×C₅	—	4	2	no	—
59	C₄²×C₅	C₂×C₈	1	2	yes	—
60	C₄²×C₅	M₁₆	1	2	yes	—
61	C₄²×C₅	C₂×D₄	1	2	yes	—
62	C₄²×C₅	C₂×Q₈	1	2	yes	—
63	C₂×#168	—	4	2	no	—
64	C₂×#168	—	4	1	no	—
65	C₂×#168	—	4	1	no	—
66	C₂×#168	—	4	4	no	—
67	C₂×#168	—	4	2	no	—
68	C₂×#168	—	4	2	no	—
69	C₂×C₄⋊C₈	—	5	1	no	—
70	C₂×C₄⋊C₈	—	5	2	no	—
71	C₂×C₄⋊C₈	—	5	1	no	—
72	C₂×C₄⋊C₈	—	10	2	no	—
73	C₂×#97	Dic₅	1	2	yes	—
74	C₂×#97	Dic₅	1	1	yes	characteristic
75	C₂×#97	Dic₅	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
77	C₄³	C₂×D₅	1	1	yes	characteristic
78	#336	—	10	2	no	—
79	#335	—	10	4	no	—
80	#334	—	10	4	no	—
81	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅	#127	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₅	#137	1	2	yes	—
98	C₅×C₈	—	2	4	no	—
99	C₅×C₈	C₄⋊C₈	1	8	yes	complemented
100	#168	—	8	2	no	—
101	#168	—	8	1	no	—
102	#168	—	8	1	no	—
103	C₂²×C₈	—	10	1	no	—
104	C₂²×C₈	—	10	1	no	—
105	C₂²×C₈	—	10	2	no	—
106	C₂²×C₈	C₂×Dic₅	1	2	yes	—
107	C₂²×C₈	C₂×Dic₅	1	2	yes	—
108	C₂×C₄²	Dic₁₀	1	2	yes	—
109	C₂×C₄²	D₂₀	1	2	yes	—
110	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
112	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
113	C₄⋊C₈	—	10	2	no	—
114	C₄⋊C₈	—	10	2	no	—
115	C₄⋊C₈	—	10	2	no	—
116	C₄⋊C₈	—	10	2	no	—
117	C₄⋊C₈	—	20	4	no	—
118	#97	—	2	4	no	—
119	#97	—	2	2	no	—
120	#97	#168	1	4	yes	complemented
121	C₄×C₈	Dic₁₀	1	4	yes	—
122	C₄×C₈	D₂₀	1	4	yes	—
123	C₂²×C₅	#384	1	1	yes	characteristic
124	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₅	#423	1	1	yes	characteristic
126	C₂²×C₅	#428	1	2	yes	—
127	C₂²×C₅	#430	1	2	yes	—
128	C₄×C₅	—	2	4	no	—
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	—	4	2	no	—
131	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
132	C₄×C₅	#423	1	2	yes	—
133	C₄×C₅	#496	1	1	yes	characteristic
134	C₄×C₅	#497	1	1	yes	characteristic
135	C₄×C₅	#501	1	2	yes	—
136	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
137	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
138	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	20	2	no	—
144	C₂×C₈	—	20	1	no	—
145	C₂×C₈	—	20	1	no	—
146	C₂×C₈	—	20	4	no	—
147	C₂×C₈	—	20	2	no	—
148	C₂×C₈	—	20	2	no	—
149	C₂×C₈	C₂×#168	1	4	yes	—
150	C₂×C₈	#681	1	4	yes	—
151	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
152	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	4	2	no	—
157	C₄²	C₂×#168	1	2	yes	—
158	C₄²	#681	1	2	yes	—
159	C₄²	C₂×Dic₁₀	1	2	yes	—
160	C₄²	C₂×D₂₀	1	2	yes	—
161	C₂×C₅	#1565	1	2	yes	—
162	C₂×C₅	#1569	1	2	yes	—
163	C₂×C₅	#1842	1	1	yes	characteristic
164	C₂×C₅	#1847	1	1	yes	characteristic
165	C₂×C₅	#1849	1	1	yes	characteristic
166	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	C₄⋊#168	1	4	yes	—
174	C₂×C₄	#4015	1	2	yes	—
175	C₂×C₄	#4015	1	1	yes	characteristic
176	C₂×C₄	#4020	1	1	yes	characteristic
177	C₂×C₄	#4054	1	2	yes	—
178	C₂×C₄	#4055	1	2	yes	—
179	C₂×C₄	C₂²×#168	1	1	yes	characteristic
180	C₂×C₄	C₂×#681	1	2	yes	—
181	C₂×C₄	C₂×#681	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	40	2	no	—
185	C₈	—	40	1	no	—
186	C₈	—	40	1	no	—
187	C₈	C₄⋊#168	1	8	yes	complemented
188	C₅	#8808	1	1	yes	characteristic, complemented, Sylow
189	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
190	C₂²	#65213	1	1	yes	characteristic
191	C₂²	#65224	1	1	yes	characteristic
192	C₂²	#65410	1	2	yes	—
193	C₂²	#65414	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	#64991	1	2	yes	—
198	C₄	#65001	1	1	yes	characteristic
199	C₄	#65004	1	1	yes	characteristic
200	C₄	C₂×C₄⋊#168	1	2	yes	—
201	C₄	#65213	1	2	yes	—
202	C₂	#10600187	1	1	yes	characteristic
203	C₂	#10600609	1	1	yes	characteristic
204	C₂	#10600635	1	1	yes	characteristic
205	C₂	#10600643	1	2	yes	—
206	C₂	#10600644	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#97	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#97	D₅
hypocenter	C₅	#8808
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
31. C₅×#97	C₈	semidirect
99. C₅×C₈	C₄⋊C₈	semidirect
120. #97	#168	semidirect
187. C₈	C₄⋊#168	semidirect
188. C₅	#8808	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#97	2
C₂×#64678	2
C₂×#64677	2
#10600187	2
#10600185	2
#8808	5