Subgroups — #12005105

Order 1280.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10613793	C₂	1	2	yes	maximal
3	C₂×#64777	C₂	1	4	yes	maximal
4	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂×#4072	C₂²	1	4	yes	—
7	C₄×C₅×M₁₆	—	2	2	no	—
8	C₄×C₅×M₁₆	—	2	2	no	—
9	C₂×C₅×#97	C₄	1	2	yes	complemented
10	C₂×C₄×C₅×C₈	C₄	1	2	yes	complemented
11	#65501	—	2	4	no	—
12	#65225	—	2	2	no	—
13	#65224	—	2	2	no	—
14	C₄×C₄⋊Dic₅	C₄	1	8	yes	complemented
15	#64777	—	2	8	no	—
16	#10837	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	D₄	1	2	yes	—
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	4	no	—
20	C₂×C₅×M₁₆	—	2	4	no	—
21	C₂×C₅×M₁₆	D₄	1	4	yes	—
22	C₂×C₅×M₁₆	Q₈	1	4	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	Q₈	1	2	yes	—
30	C₂×C₂²⋊Dic₅	—	2	4	no	—
31	#4072	—	2	8	no	—
32	#4072	—	2	8	no	—
33	C₂×C₄⋊Dic₅	—	2	8	no	—
34	C₂×C₄⋊Dic₅	—	2	4	no	—
35	C₂×C₄×Dic₅	C₂×C₄	1	8	yes	—
36	C₅×#97	—	2	4	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	#3963	—	2	8	no	—
40	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
41	#2285	—	5	2	no	—
42	C₂×H(3,ℤ/4)	—	5	4	no	—
43	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	—	4	4	no	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
47	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₅	#41	1	4	yes	—
56	C₂²×Dic₅	—	2	8	no	—
57	C₅×M₁₆	—	2	8	no	—
58	C₅×M₁₆	—	2	8	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	complemented
62	C₄²×C₅	C₄²	1	2	yes	—
63	C₄²×C₅	C₄²	1	2	yes	—
64	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₅	C₄⋊C₄	1	4	yes	—
68	C₄²×C₅	C₄⋊C₄	1	4	yes	—
69	C₂²⋊Dic₅	—	4	4	no	—
70	C₂²⋊Dic₅	—	4	4	no	—
71	C₄⋊Dic₅	—	4	16	no	—
72	C₄⋊Dic₅	—	4	8	no	—
73	C₄×Dic₅	—	2	16	no	—
74	C₂²×M₁₆	C₂×D₅	1	2	yes	—
75	C₂×#117	—	5	4	no	—
76	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
77	C₄×M₁₆	—	2	2	no	—
78	C₄×M₁₆	—	2	2	no	—
79	C₂×#97	Dic₅	1	2	yes	complemented
80	C₂×C₄×C₈	Dic₅	1	2	yes	complemented
81	#384	—	10	2	no	—
82	#382	—	10	2	no	—
83	#380	—	10	4	no	—
84	C₄×C₄⋊C₄	—	5	8	no	—
85	H(3,ℤ/4)	—	10	8	no	—
86	C₂³×C₅	—	4	2	no	—
87	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂³×C₅	#124	1	2	yes	—
90	C₂³×C₅	#127	1	1	yes	characteristic
91	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅	—	2	8	no	—
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	4	4	no	—
95	C₂×C₄×C₅	—	4	2	no	—
96	C₂×C₄×C₅	#95	1	2	yes	—
97	C₂×C₄×C₅	#95	1	4	yes	—
98	C₂×C₄×C₅	#95	1	2	yes	—
99	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅	#117	1	2	yes	—
104	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
105	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
106	C₂×C₄×C₅	#121	1	4	yes	—
107	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₅	#125	1	2	yes	—
109	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
110	C₂×Dic₅	—	4	16	no	—
111	C₂×Dic₅	—	4	8	no	—
112	C₅×C₈	—	2	8	no	—
113	C₂³×C₄	D₂₀	1	2	yes	—
114	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	—	2	4	no	—
117	C₂×M₁₆	Dic₁₀	1	4	yes	—
118	C₂×M₁₆	D₂₀	1	4	yes	—
119	C₂²×C₈	C₂×Dic₅	1	4	yes	—
120	#117	—	10	8	no	—
121	#117	—	10	8	no	—
122	C₂×C₄⋊C₄	—	10	8	no	—
123	C₂×C₄⋊C₄	—	10	4	no	—
124	C₂×C₂²⋊C₄	—	10	4	no	—
125	C₂×C₄²	—	5	8	no	—
126	C₂×C₄²	Dic₁₀	1	2	yes	—
127	C₂×C₄²	C₄×D₅	1	4	yes	—
128	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
129	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
132	#97	—	2	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	#95	—	10	8	no	—
136	C₂²×C₅	—	2	4	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	—	4	2	no	—
139	C₂²×C₅	—	4	1	no	—
140	C₂²×C₅	—	4	1	no	—
141	C₂²×C₅	C₂×#95	1	1	yes	characteristic
142	C₂²×C₅	#382	1	1	yes	characteristic
143	C₂²×C₅	#384	1	1	yes	characteristic
144	C₂²×C₅	#390	1	2	yes	—
145	C₂²×C₅	#391	1	2	yes	—
146	C₄×C₅	—	2	4	no	—
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	H(3,ℤ/4)	1	4	yes	complemented
149	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
150	C₄×C₅	#380	1	2	yes	—
151	C₄×C₅	#389	1	4	yes	—
152	Dic₅	—	8	8	no	—
153	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	—	10	8	no	—
156	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
157	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
158	C₂²×C₄	#685	1	4	yes	—
159	C₂²×C₄	#685	1	4	yes	—
160	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
161	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
163	C₂²×C₄	C₂×D₂₀	1	2	yes	—
164	C₂²×C₄	#709	1	4	yes	—
165	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
167	M₁₆	—	2	8	no	—
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	C₄⋊Dic₅	1	8	yes	complemented
172	C₄⋊C₄	—	20	16	no	—
173	C₄⋊C₄	—	20	8	no	—
174	C₂²⋊C₄	—	20	4	no	—
175	C₂²⋊C₄	—	20	4	no	—
176	C₄²	—	10	16	no	—
177	C₄²	C₄×Dic₅	1	2	yes	—
178	C₄²	C₄×Dic₅	1	2	yes	—
179	C₄²	#683	1	4	yes	—
180	C₄²	#683	1	4	yes	—
181	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
182	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₅	1	2	yes	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	—	2	1	no	—
186	C₂×C₅	—	2	1	no	—
187	C₂×C₅	—	2	2	no	—
188	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
189	C₂×C₅	C₂×H(3,ℤ/4)	1	2	yes	—
190	C₂³	—	4	2	no	—
191	C₂³	#4015	1	1	yes	characteristic
192	C₂³	#4020	1	1	yes	characteristic
193	C₂³	#4021	1	2	yes	—
194	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
195	C₂³	C₂×#685	1	2	yes	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	20	16	no	—
201	C₂×C₄	—	20	8	no	—
202	C₂×C₄	#3963	1	2	yes	—
203	C₂×C₄	#3963	1	4	yes	—
204	C₂×C₄	#3963	1	2	yes	—
205	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
206	C₂×C₄	#4015	1	2	yes	—
207	C₂×C₄	#4016	1	2	yes	—
208	C₂×C₄	C₄×D₂₀	1	4	yes	—
209	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
210	C₂×C₄	C₂×#683	1	2	yes	—
211	C₂×C₄	#4070	1	4	yes	—
212	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
213	C₂×C₄	#4072	1	2	yes	—
214	C₂×C₄	#4076	1	4	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
216	C₈	—	2	8	no	—
217	C₅	#10837	1	1	yes	characteristic, complemented, Sylow
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	2	no	—
223	C₂²	#65218	1	2	yes	—
224	C₂²	#65224	1	1	yes	characteristic
225	C₂²	#65225	1	1	yes	characteristic
226	C₂²	#65228	1	2	yes	—
227	C₂²	C₂×#3963	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	40	8	no	—
231	C₄	#64777	1	4	yes	complemented
232	C₄	#65217	1	4	yes	—
233	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
234	C₄	#65501	1	2	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	2	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	#10602328	1	1	yes	characteristic
240	C₂	C₂×#64777	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#10837
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
9. C₂×C₅×#97	C₄	semidirect
10. C₂×C₄×C₅×C₈	C₄	semidirect
14. C₄×C₄⋊Dic₅	C₄	semidirect
61. C₂×C₅×C₈	C₄⋊C₄	semidirect
79. C₂×#97	Dic₅	semidirect
80. C₂×C₄×C₈	Dic₅	semidirect
148. C₄×C₅	H(3,ℤ/4)	semidirect
171. C₂×C₈	C₄⋊Dic₅	semidirect
217. C₅	#10837	semidirect
231. C₄	#64777	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×M₁₆	2
#10613793	2
C₂×#64777	2
#10837	5