Subgroups — #12009112

Order 1280.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#10600536	C₂	1	1	yes	maximal, characteristic
4	#10597997	C₂	1	4	yes	maximal, complemented
5	#65804	C₄	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	—	2	1	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	#65391	—	2	1	no	—
9	#65390	—	2	1	no	—
10	C₂×#3944	—	2	1	no	—
11	#65213	C₄	1	1	yes	characteristic
12	#64968	—	2	1	no	—
13	#64968	C₄	1	2	yes	—
14	C₈×Dic₁₀	—	2	1	no	—
15	C₈×Dic₁₀	C₂²	1	2	yes	—
16	C₄×C₅×C₁₆	—	2	1	no	—
17	C₄×C₅×C₁₆	C₂²	1	2	yes	—
18	C₄⋊#672	—	2	1	no	—
19	C₄⋊#672	C₂²	1	2	yes	—
20	#11608	—	5	1	no	maximal, Sylow
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	#4072	—	2	1	no	—
25	#4070	—	2	1	no	—
26	#4070	C₈	1	2	yes	—
27	C₂×#681	—	2	1	no	—
28	#4016	—	2	1	no	—
29	#4015	C₈	1	1	yes	characteristic
30	#4015	C₈	1	1	yes	characteristic
31	C₄×Dic₁₀	—	2	1	no	—
32	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₁₆	—	2	4	no	—
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×C₈	D₄	1	1	yes	characteristic
37	C₄×C₅×C₈	D₄	1	1	yes	characteristic
38	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
39	#3947	—	2	2	no	—
40	#3946	—	2	2	no	—
41	#3946	—	2	2	no	—
42	C₈×Dic₅	—	2	2	no	—
43	#3944	—	4	2	no	—
44	C₂×#672	—	2	2	no	—
45	C₄⋊#168	—	2	1	no	—
46	C₄⋊#168	C₂×C₄	1	2	yes	—
47	#2960	—	5	1	no	—
48	#2150	—	5	1	no	—
49	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
50	Q₈⋊C₁₆	—	5	4	no	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
54	C₂×Dic₁₀	—	2	1	no	—
55	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	SD₁₆	1	2	yes	—
60	C₂×C₅×C₈	Q₁₆	1	2	yes	—
61	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₅	M₁₆	1	1	yes	characteristic
66	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₅	—	2	2	no	—
68	C₄⋊Dic₅	—	2	1	no	—
69	C₄⋊Dic₅	—	2	1	no	—
70	C₄⋊Dic₅	—	2	1	no	—
71	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
72	#683	—	2	2	no	—
73	#683	—	2	2	no	—
74	C₄×Dic₅	—	2	2	no	—
75	#681	—	4	2	no	—
76	C₂×#168	—	2	2	no	—
77	C₅×C₁₆	—	2	4	no	—
78	#672	—	4	2	no	—
79	#533	—	5	1	no	—
80	C₂×M₃₂	—	10	1	no	—
81	C₂²×C₁₆	—	2	1	no	—
82	#446	—	5	2	no	—
83	#446	—	10	1	no	—
84	C₈×Q₈	—	5	2	no	—
85	C₈×Q₈	—	10	1	no	—
86	#424	—	10	1	no	—
87	#423	—	5	1	no	—
88	#405	—	10	1	no	—
89	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	C₄⋊C₁₆	—	5	2	no	—
91	C₄⋊C₁₆	—	10	1	no	—
92	C₄×C₁₆	—	2	1	no	—
93	C₄×C₁₆	C₂×D₅	1	2	yes	—
94	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₅	#103	1	2	yes	—
101	C₂×C₄×C₅	#104	1	2	yes	—
102	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
105	C₂×Dic₅	—	2	2	no	—
106	C₂×Dic₅	—	2	2	no	—
107	Dic₁₀	—	2	4	no	—
108	Dic₁₀	—	4	2	no	—
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	—	2	1	no	—
111	C₅×C₈	#103	1	2	yes	—
112	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
113	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
114	#168	—	4	2	no	—
115	C₂×M₁₆	—	10	1	no	—
116	C₂²×C₈	D₂₀	1	1	yes	characteristic
117	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
118	C₄⋊Q₈	—	5	1	no	—
119	C₄⋊Q₈	—	5	1	no	—
120	#125	—	10	1	no	—
121	C₂²⋊Q₈	—	5	2	no	—
122	C₂²⋊Q₈	—	10	1	no	—
123	C₄×Q₈	—	5	2	no	—
124	C₄×Q₈	—	10	1	no	—
125	#117	—	10	1	no	—
126	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
127	M₃₂	—	20	2	no	—
128	C₂×C₁₆	—	2	4	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	10	2	no	—
131	C₄⋊C₈	—	5	2	no	—
132	C₄⋊C₈	—	10	2	no	—
133	C₄⋊C₈	—	10	2	no	—
134	C₄⋊C₈	—	10	1	no	—
135	#97	—	10	2	no	—
136	C₄×C₈	—	10	2	no	—
137	C₄×C₈	C₄×D₅	1	1	yes	characteristic
138	C₄×C₈	D₂₀	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
140	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	Q₈⋊C₈	1	2	yes	—
144	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	Q₈⋊C₈	1	2	yes	—
148	C₄×C₅	#350	1	2	yes	—
149	C₄×C₅	C₂×#103	1	1	yes	characteristic
150	C₄×C₅	C₂×#104	1	1	yes	characteristic
151	C₄×C₅	#504	1	2	yes	—
152	Dic₅	—	4	2	no	—
153	Dic₅	—	4	2	no	—
154	C₂×Q₈	—	5	2	no	—
155	C₂×Q₈	—	10	1	no	—
156	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
157	C₂²×C₄	#676	1	1	yes	characteristic
158	C₂²×C₄	#685	1	1	yes	characteristic
159	M₁₆	—	20	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	10	2	no	—
162	C₂×C₈	#677	1	2	yes	—
163	C₂×C₈	Dic₂₀	1	2	yes	—
164	C₂×C₈	#685	1	1	yes	characteristic
165	C₂×C₈	#685	1	1	yes	characteristic
166	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	5	2	no	—
169	C₄⋊C₄	—	10	1	no	—
170	C₄⋊C₄	—	10	1	no	—
171	C₄⋊C₄	—	10	2	no	—
172	C₄⋊C₄	—	10	1	no	—
173	C₄⋊C₄	—	10	2	no	—
174	C₂²⋊C₄	—	10	2	no	—
175	C₄²	—	10	2	no	—
176	C₄²	C₈×D₅	1	1	yes	characteristic
177	C₄²	#676	1	1	yes	characteristic
178	C₄²	#685	1	1	yes	characteristic
179	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
180	C₁₆	—	2	4	no	—
181	C₁₆	—	20	2	no	—
182	C₂×C₅	—	2	2	no	—
183	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
184	C₂×C₅	#2113	1	1	yes	characteristic
185	C₂×C₅	C₂×#350	1	1	yes	characteristic
186	C₂³	#3952	1	1	yes	characteristic
187	Q₈	—	10	4	no	—
188	Q₈	—	20	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	10	2	no	—
192	C₂×C₄	—	10	2	no	—
193	C₂×C₄	#3937	1	2	yes	—
194	C₂×C₄	#3948	1	2	yes	—
195	C₂×C₄	#3952	1	1	yes	characteristic
196	C₂×C₄	#3952	1	1	yes	characteristic
197	C₂×C₄	#3952	1	1	yes	characteristic
198	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
199	C₂×C₄	C₂×#676	1	1	yes	characteristic
200	C₂×C₄	C₂×#685	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	20	2	no	—
204	C₈	#3948	1	2	yes	—
205	C₈	C₂×#677	1	1	yes	characteristic
206	C₈	C₂×Dic₂₀	1	1	yes	characteristic
207	C₅	#11608	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#64676	1	2	yes	—
211	C₂²	C₂×#3952	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	20	2	no	—
215	C₄	—	20	2	no	—
216	C₄	#64676	1	2	yes	—
217	C₄	#64728	1	2	yes	—
218	C₄	#65190	1	2	yes	—
219	C₄	C₂×#3937	1	1	yes	characteristic
220	C₄	C₂×#3948	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#64676	1	1	yes	characteristic
223	C₂	#10602202	1	1	yes	characteristic
224	C₂	C₂×#64728	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂²⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	#11608
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10597997	C₂	semidirect
207. C₅	#11608	semidirect

Maximal subgroups

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

subgroup	index
#10610176	2
C₂×C₄×C₅×C₁₆	2
#10600536	2
#10597997	2
#11608	5