Subgroups — #13043525

Order 1280.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65725	C₂	1	2	yes	maximal, complemented
2	#10610772	C₂	1	1	yes	maximal, characteristic
3	#10604742	C₂	1	2	yes	maximal, complemented
4	#10604150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
6	#10604023	C₂	1	4	yes	maximal, complemented
7	#10604021	C₂	1	4	yes	maximal, complemented
8	C₂×#4053	C₂²	1	2	yes	—
9	#65811	C₄	1	1	yes	characteristic
10	C₂×#3998	C₂²	1	2	yes	—
11	C₂²×#674	C₂²	1	1	yes	characteristic
12	C₂²×#674	C₂²	1	1	yes	characteristic
13	#65725	—	2	8	no	—
14	C₂×#3990	C₂²	1	4	yes	—
15	C₂×D₅⋊C₁₆	C₂²	1	4	yes	—
16	C₅×#423	C₄	1	1	yes	characteristic
17	#65418	C₄	1	2	yes	—
18	C₂×C₄×#168	C₂²	1	1	yes	characteristic
19	#65127	C₄	1	4	yes	—
20	#65123	C₂²	1	4	yes	—
21	#64887	C₂²	1	8	yes	—
22	C₄⋊#674	C₂²	1	4	yes	—
23	C₄×#674	C₂²	1	2	yes	—
24	C₄×#674	C₂²	1	2	yes	—
25	#33434	—	5	1	no	maximal, Sylow
26	C₂×#709	C₂×C₄	1	2	yes	—
27	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	#4076	C₈	1	4	yes	—
30	C₂²×#168	C₂×C₄	1	1	yes	characteristic
31	C₂²×#168	C₂³	1	1	yes	characteristic
32	#4053	—	2	4	no	—
33	#4053	—	2	4	no	—
34	C₂×#676	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₅	C₂³	1	4	yes	—
36	#4021	C₈	1	2	yes	—
37	#4020	C₈	1	1	yes	characteristic
38	C₄×D₂₀	C₂×C₄	1	4	yes	—
39	#4015	C₈	1	1	yes	characteristic
40	#3998	—	2	4	no	—
41	C₂×#674	—	2	2	no	—
42	C₂×#674	D₄	1	4	yes	complemented
43	C₂×#674	C₂³	1	2	yes	—
44	C₂×#674	C₂³	1	4	yes	—
45	C₂×#674	C₂³	1	1	yes	characteristic
46	C₂×#674	C₂³	1	1	yes	characteristic
47	#3990	—	2	8	no	—
48	D₅⋊C₁₆	—	2	8	no	—
49	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	#3952	C₂×C₄	1	8	yes	—
51	C₄×#168	C₂×C₄	1	2	yes	—
52	C₄×#168	C₂³	1	2	yes	—
53	C₂×#504	—	5	2	no	—
54	#2931	—	5	1	no	—
55	#2169	—	5	4	no	—
56	D₄×C₁₆	—	5	4	no	—
57	#2150	—	5	1	no	—
58	#2113	—	5	2	no	—
59	C₂×C₄×C₁₆	—	5	1	no	—
60	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅	C₂×C₈	1	4	yes	—
63	#709	—	2	4	no	—
64	#709	—	2	4	no	—
65	C₂×D₂₀	C₂×C₈	1	2	yes	—
66	C₂×D₂₀	C₂×C₈	1	4	yes	—
67	C₂×C₄×D₅	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
69	C₅×M₁₆	—	2	4	no	—
70	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
71	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₅	C₂²×C₄	1	2	yes	—
74	#685	C₂×C₈	1	8	yes	—
75	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
76	C₂×#168	—	2	2	no	—
77	C₂×#168	C₂²×C₄	1	2	yes	—
78	C₂×#168	C₂²×C₄	1	1	yes	characteristic
79	C₂×#168	C₂×D₄	1	2	yes	—
80	C₂×#168	#41	1	2	yes	—
81	C₂×#168	C₂⁴	1	1	yes	characteristic
82	#676	—	2	8	no	—
83	C₈×D₅	—	2	4	no	—
84	C₈×D₅	—	2	4	no	—
85	#674	—	2	4	no	—
86	#674	—	2	2	no	—
87	#674	C₂×D₄	1	2	yes	—
88	#674	C₂×D₄	1	2	yes	—
89	C₂×#131	—	5	2	no	—
90	#532	—	5	1	no	—
91	#504	—	10	8	no	—
92	C₂×M₃₂	—	5	2	no	—
93	C₂×M₃₂	—	5	4	no	—
94	C₂²×C₁₆	—	5	1	no	—
95	C₂²×C₁₆	—	5	1	no	—
96	C₂²×C₁₆	—	5	4	no	—
97	#436	—	5	4	no	—
98	C₈×D₄	—	5	4	no	—
99	#423	AGL(1,𝔽₅)	1	1	yes	characteristic
100	#408	—	5	2	no	—
101	C₂×C₄×C₈	—	5	1	no	—
102	C₄⋊C₁₆	—	5	4	no	—
103	C₂²⋊C₁₆	—	5	8	no	—
104	C₄×C₁₆	—	5	2	no	—
105	C₄×C₁₆	—	5	2	no	—
106	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
107	C₂²×D₅	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅	#131	1	2	yes	—
114	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
115	C₂²⋊D₅	—	2	8	no	—
116	C₂×Dic₅	C₂²×C₈	1	4	yes	—
117	D₂₀	—	2	4	no	—
118	D₂₀	—	2	8	no	—
119	C₄×D₅	—	2	4	no	—
120	C₄×D₅	—	2	4	no	—
121	Dic₁₀	—	2	4	no	—
122	C₅×C₈	—	2	4	no	—
123	#168	—	2	2	no	—
124	#168	C₄×D₄	1	2	yes	—
125	#168	C₂²×D₄	1	1	yes	characteristic
126	#168	C₂×#41	1	1	yes	characteristic
127	C₂×#41	—	5	2	no	—
128	#131	—	10	4	no	—
129	#131	—	10	4	no	—
130	C₂×M₁₆	—	5	4	no	—
131	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
132	C₂²×C₈	—	5	1	no	—
133	C₂²×C₈	—	5	1	no	—
134	C₂²×C₈	—	5	4	no	—
135	C₄⋊Q₈	—	5	1	no	—
136	#127	—	5	1	no	—
137	#124	—	5	2	no	—
138	#121	—	5	4	no	—
139	C₄×D₄	—	5	4	no	—
140	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
141	M₃₂	—	10	8	no	—
142	M₃₂	—	10	4	no	—
143	C₂×C₁₆	—	5	1	no	—
144	C₂×C₁₆	—	5	1	no	—
145	C₂×C₁₆	—	5	4	no	—
146	C₂×C₁₆	—	5	2	no	—
147	C₂×C₁₆	—	5	4	no	—
148	C₂×C₁₆	—	10	8	no	—
149	C₂×C₁₆	—	10	2	no	—
150	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
151	C₂²⋊C₈	—	5	8	no	—
152	C₄×C₈	—	5	2	no	—
153	C₄×C₈	—	5	2	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	C₈×D₄	1	2	yes	—
156	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
157	C₂×D₅	—	2	8	no	—
158	C₂×D₅	—	2	4	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	C₈×D₄	1	2	yes	—
161	C₄×C₅	#504	1	4	yes	complemented
162	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₅	C₂×#131	1	1	yes	characteristic
164	Dic₅	—	2	4	no	—
165	#41	—	10	4	no	—
166	#41	—	10	4	no	—
167	C₂×Q₈	—	5	2	no	—
168	C₂×D₄	—	5	4	no	—
169	C₂×D₄	—	5	4	no	—
170	C₂×D₄	—	5	2	no	—
171	C₂²×C₄	—	5	4	no	—
172	C₂²×C₄	D₅⋊C₈	1	2	yes	—
173	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
174	M₁₆	—	2	4	no	—
175	M₁₆	—	10	8	no	—
176	C₂×C₈	—	5	2	no	—
177	C₂×C₈	—	5	2	no	—
178	C₂×C₈	—	5	1	no	—
179	C₂×C₈	—	5	1	no	—
180	C₂×C₈	—	5	2	no	—
181	C₂×C₈	—	10	4	no	—
182	C₂×C₈	—	10	4	no	—
183	C₂×C₈	—	10	2	no	—
184	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	4	yes	—
185	C₄⋊C₄	—	5	4	no	—
186	C₂²⋊C₄	—	5	8	no	—
187	C₄²	D₅⋊C₈	1	1	yes	characteristic
188	C₄²	D₅⋊C₈	1	1	yes	characteristic
189	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
190	C₁₆	—	5	2	no	—
191	C₁₆	—	5	2	no	—
192	C₁₆	—	10	4	no	—
193	C₁₆	—	10	2	no	—
194	C₂×C₅	—	2	2	no	—
195	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
196	C₂×C₅	C₂×#504	1	2	yes	—
197	D₅	—	4	4	no	—
198	C₂³	—	5	4	no	—
199	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
200	Q₈	—	10	4	no	—
201	D₄	—	10	8	no	—
202	D₄	—	10	4	no	—
203	D₄	—	10	8	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	5	4	no	—
206	C₂×C₄	—	10	4	no	—
207	C₂×C₄	—	10	4	no	—
208	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
209	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
210	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
211	C₂×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
212	C₂×C₄	#4133	1	2	yes	—
213	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₈	—	5	1	no	—
216	C₈	—	5	1	no	—
217	C₈	—	5	2	no	—
218	C₈	—	10	2	no	—
219	C₅	#33434	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	—	10	8	no	—
222	C₂²	—	10	4	no	—
223	C₂²	#65702	1	2	yes	—
224	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	10	4	no	—
227	C₄	#65702	1	2	yes	—
228	C₄	#65725	1	4	yes	complemented
229	C₄	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₄	C₂×#4133	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	20	4	no	—
233	C₂	C₂×#65702	1	1	yes	characteristic
234	C₂	C₂×#65725	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#423	C₄
prime radical	#65811	C₄
cosocle	C₂×#168	C₂⁴
hypercenter	#423	AGL(1,𝔽₅)
hypocenter	C₅	#33434
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×#65725	C₂	semidirect
3. #10604742	C₂	semidirect
4. #10604150	C₂	semidirect
5. C₂×C₄×#674	C₂	semidirect
6. #10604023	C₂	semidirect
7. #10604021	C₂	semidirect
42. C₂×#674	D₄	semidirect
161. C₄×C₅	#504	semidirect
219. C₅	#33434	semidirect
228. C₄	#65725	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65725	2
#10610772	2
#10604742	2
#10604150	2
C₂×C₄×#674	2
#10604023	2
#10604021	2
#33434	5