Subgroups — #11998192

Order 1280.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
3	#10600581	C₂	1	2	yes	maximal
4	C₂×#64676	C₂	1	2	yes	maximal
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3950	C₄	1	1	yes	characteristic
12	C₂×#3948	C₄	1	2	yes	—
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	C₄×Dic₂₀	—	2	2	no	—
15	C₄×Dic₂₀	—	2	2	no	—
16	C₅×#322	—	2	2	no	—
17	#64770	—	2	4	no	—
18	#64678	—	2	2	no	—
19	#64676	—	2	4	no	—
20	#9662	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
28	C₂×#683	—	2	2	no	—
29	C₂×C₄×Dic₅	—	2	2	no	—
30	C₂²×#168	—	2	2	no	—
31	C₂×Dic₂₀	—	2	4	no	—
32	C₂×Dic₂₀	—	2	2	no	—
33	C₄×Dic₁₀	—	2	2	no	—
34	C₄×Dic₁₀	—	2	2	no	—
35	C₄×Dic₁₀	—	2	2	no	—
36	C₄×Dic₁₀	—	2	2	no	—
37	C₄×C₅×C₈	—	2	1	no	—
38	C₄×C₅×C₈	—	2	1	no	—
39	C₄×C₅×C₈	—	2	1	no	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	C₄×C₅×C₈	D₄	1	2	yes	—
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	#3950	—	2	2	no	—
44	#3948	—	2	4	no	—
45	C₄⋊#168	—	2	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₂×C₄×Q₁₆	—	5	1	no	—
48	#1984	—	5	2	no	—
49	C₂×#335	—	5	1	no	—
50	C₂×Q₈⋊C₈	—	5	2	no	—
51	C₂×#322	D₅	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	#41	1	1	yes	characteristic
55	C₂²×Dic₅	—	2	2	no	—
56	C₂²×Dic₅	—	2	2	no	—
57	C₂×Dic₁₀	—	2	2	no	—
58	C₂×Dic₁₀	—	2	2	no	—
59	C₂×Dic₁₀	—	2	2	no	—
60	C₂×Dic₁₀	—	2	2	no	—
61	C₂×Dic₁₀	—	2	2	no	—
62	C₂×Dic₁₀	M₁₆	1	4	yes	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	Q₁₆	1	4	yes	complemented
72	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅	#41	1	1	yes	characteristic
76	C₄⋊Dic₅	—	2	2	no	—
77	C₄⋊Dic₅	—	2	2	no	—
78	#683	—	4	2	no	—
79	#683	—	4	2	no	—
80	C₄×Dic₅	—	4	2	no	—
81	C₄×Dic₅	—	4	2	no	—
82	C₂×#168	—	4	2	no	—
83	C₂×#168	—	4	4	no	—
84	Dic₂₀	—	2	8	no	—
85	C₂²×Q₁₆	—	5	1	no	—
86	C₂×C₄×Q₈	—	5	2	no	—
87	C₄×Q₁₆	—	10	2	no	—
88	C₄×Q₁₆	—	10	2	no	—
89	C₂×#107	—	5	1	no	—
90	C₂×C₄⋊C₈	—	5	2	no	—
91	C₂×#103	—	5	2	no	—
92	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
95	#336	—	10	4	no	—
96	#335	—	10	2	no	—
97	Q₈⋊C₈	—	10	4	no	—
98	#322	—	2	2	no	—
99	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅	#121	1	1	yes	characteristic
104	C₂×C₄×C₅	#121	1	1	yes	characteristic
105	C₂×C₄×C₅	#124	1	1	yes	characteristic
106	C₂×C₄×C₅	#127	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅	#131	1	2	yes	—
109	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₅	#135	1	2	yes	—
111	C₂×Dic₅	—	2	4	no	—
112	C₂×Dic₅	—	4	2	no	—
113	C₂×Dic₅	—	4	2	no	—
114	C₂×Dic₅	—	4	2	no	—
115	C₂×Dic₅	—	4	4	no	—
116	Dic₁₀	—	2	4	no	—
117	Dic₁₀	—	2	4	no	—
118	Dic₁₀	—	4	2	no	—
119	Dic₁₀	—	4	2	no	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	2	4	no	—
123	C₅×C₈	—	4	2	no	—
124	#168	—	8	2	no	—
125	C₂²×Q₈	—	5	2	no	—
126	C₂×Q₁₆	—	10	4	no	—
127	C₂×Q₁₆	—	10	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	10	2	no	—
130	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
132	C₂²×C₈	D₂₀	1	2	yes	—
133	C₄×Q₈	—	10	2	no	—
134	C₄×Q₈	—	10	2	no	—
135	C₄×Q₈	—	10	2	no	—
136	C₄×Q₈	—	10	2	no	—
137	C₂×C₄⋊C₄	—	5	2	no	—
138	C₂×C₄⋊C₄	—	10	2	no	—
139	C₂×C₄²	—	10	2	no	—
140	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
141	#107	—	10	2	no	—
142	C₄⋊C₈	—	10	2	no	—
143	C₄⋊C₈	—	10	2	no	—
144	#103	—	10	4	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	D₂₀	1	2	yes	—
149	C₄×C₈	D₂₀	1	2	yes	—
150	C₄×C₈	C₂²⋊D₅	1	2	yes	—
151	C₂²×C₅	#390	1	1	yes	characteristic
152	C₂²×C₅	#435	1	2	yes	—
153	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
154	C₂²×C₅	#444	1	2	yes	—
155	C₄×C₅	—	2	1	no	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	#407	1	1	yes	characteristic
158	C₄×C₅	#408	1	1	yes	characteristic
159	C₄×C₅	#435	1	2	yes	—
160	C₄×C₅	#467	1	2	yes	—
161	C₄×C₅	#494	1	1	yes	characteristic
162	C₄×C₅	#495	1	1	yes	characteristic
163	Dic₅	—	4	4	no	—
164	Dic₅	—	8	2	no	—
165	C₂×Q₈	—	5	4	no	—
166	C₂×Q₈	—	10	2	no	—
167	C₂×Q₈	—	10	2	no	—
168	C₂×Q₈	—	10	2	no	—
169	C₂×Q₈	—	10	2	no	—
170	C₂×Q₈	—	10	2	no	—
171	C₂²×C₄	—	10	2	no	—
172	C₂²×C₄	—	10	2	no	—
173	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
175	C₂²×C₄	#709	1	1	yes	characteristic
176	Q₁₆	—	10	8	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	20	2	no	—
184	C₂×C₈	—	20	4	no	—
185	C₂×C₈	Dic₂₀	1	4	yes	complemented
186	C₂×C₈	#685	1	2	yes	—
187	C₂×C₈	#685	1	2	yes	—
188	C₄⋊C₄	—	10	2	no	—
189	C₄⋊C₄	—	10	2	no	—
190	C₄⋊C₄	—	20	2	no	—
191	C₄⋊C₄	—	20	2	no	—
192	C₄²	—	20	2	no	—
193	C₄²	—	20	2	no	—
194	C₄²	C₂×D₂₀	1	1	yes	characteristic
195	C₄²	C₂×D₂₀	1	1	yes	characteristic
196	C₄²	#709	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
198	C₂×C₅	#1584	1	4	yes	—
199	C₂×C₅	#1955	1	1	yes	characteristic
200	C₂×C₅	#1969	1	1	yes	characteristic
201	C₂×C₅	#1975	1	1	yes	characteristic
202	C₂³	C₄×D₂₀	1	1	yes	characteristic
203	Q₈	—	10	4	no	—
204	Q₈	—	10	4	no	—
205	Q₈	—	20	2	no	—
206	Q₈	—	20	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	10	4	no	—
209	C₂×C₄	—	20	2	no	—
210	C₂×C₄	—	20	2	no	—
211	C₂×C₄	—	20	2	no	—
212	C₂×C₄	—	20	4	no	—
213	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
214	C₂×C₄	#4020	1	1	yes	characteristic
215	C₂×C₄	#4021	1	1	yes	characteristic
216	C₂×C₄	#4050	1	2	yes	—
217	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
218	C₂×C₄	D₅×M₁₆	1	2	yes	—
219	C₂×C₄	#4053	1	2	yes	—
220	C₂×C₄	C₂×#685	1	1	yes	characteristic
221	C₂×C₄	#4076	1	1	yes	characteristic
222	C₂×C₄	#4076	1	1	yes	characteristic
223	C₈	—	2	4	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	40	2	no	—
228	C₅	#9662	1	1	yes	characteristic, complemented, Sylow
229	C₂²	C₄×Dic₂₀	1	2	yes	—
230	C₂²	#64995	1	2	yes	—
231	C₂²	#65006	1	2	yes	—
232	C₂²	#65228	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	20	4	no	—
236	C₄	—	40	2	no	—
237	C₄	#64985	1	1	yes	characteristic
238	C₄	#64988	1	1	yes	characteristic
239	C₄	#64995	1	2	yes	—
240	C₄	#65405	1	2	yes	—
241	C₄	#65415	1	1	yes	characteristic
242	C₄	#65418	1	1	yes	characteristic
243	C₂	#10598879	1	4	yes	—
244	C₂	#10600568	1	1	yes	characteristic
245	C₂	#10600630	1	1	yes	characteristic
246	C₂	#10600638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#322	C₂
socle, prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#322	D₅
hypocenter	C₅	#9662
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
71. C₂×C₅×C₈	Q₁₆	semidirect
185. C₂×C₈	Dic₂₀	semidirect
228. C₅	#9662	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₀	2
C₂×C₅×#322	2
#10600581	2
C₂×#64676	2
C₂×#64678	2
#9662	5