Subgroups — #13043442

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65286	C₂	1	1	yes	maximal, characteristic
2	#10604872	C₂	1	2	yes	maximal
3	#10604368	C₂	1	8	yes	maximal, complemented
4	#10604150	C₂	1	2	yes	maximal, complemented
5	C₄×#3998	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#117	C₄	1	1	yes	characteristic
7	C₂²×#681	C₄	1	1	yes	characteristic
8	C₂³×#168	C₂²	1	1	yes	characteristic
9	C₂×#3998	C₂²	1	4	yes	—
10	#65286	C₄	1	4	yes	—
11	#65286	C₂²	1	4	yes	—
12	C₂×#3935	C₄	1	1	yes	characteristic
13	C₂×#3935	C₄	1	1	yes	characteristic
14	C₂×C₄×#168	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	C₂²⋊#674	C₂²	1	8	yes	—
17	#64859	C₂²	1	4	yes	complemented
18	C₄⋊#674	C₂²	1	8	yes	complemented
19	C₄×#674	C₂²	1	4	yes	complemented
20	#33419	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₅×#117	C₈	1	4	yes	—
23	C₅×#117	C₂×C₄	1	4	yes	—
24	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
25	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×#681	—	2	4	no	—
29	C₂×#681	C₂×C₄	1	4	yes	—
30	C₂²×#168	—	2	4	no	—
31	C₂²×#168	C₂×C₄	1	2	yes	—
32	C₂²×#168	C₂×C₄	1	1	yes	characteristic
33	C₂²×#168	C₂×C₄	1	1	yes	characteristic
34	C₂²×#168	C₂×C₄	1	1	yes	characteristic
35	C₂²×#168	C₂³	1	2	yes	—
36	C₂²×#168	C₂³	1	1	yes	characteristic
37	#3998	—	2	8	no	—
38	C₂×#674	C₂³	1	8	yes	—
39	#3935	C₂×C₄	1	4	yes	—
40	#3935	C₂×C₄	1	4	yes	—
41	C₄×#168	C₂×C₄	1	2	yes	—
42	C₄×#168	C₂×C₄	1	2	yes	—
43	C₄×#168	C₂³	1	2	yes	—
44	C₄×#168	C₂³	1	2	yes	—
45	C₂×#405	—	5	1	no	—
46	#2164	—	5	8	no	—
47	#2150	—	5	2	no	—
48	#2112	—	5	2	no	—
49	C₄×M₃₂	—	5	2	no	—
50	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
60	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
61	C₄²×C₅	C₂×C₈	1	2	yes	—
62	C₄²×C₅	C₂×C₈	1	2	yes	—
63	C₄²×C₅	C₂²×C₄	1	2	yes	—
64	C₄²×C₅	C₂²×C₄	1	2	yes	—
65	#681	—	2	8	no	—
66	C₂×#168	—	2	2	no	—
67	C₂×#168	—	2	2	no	—
68	C₂×#168	—	2	4	no	—
69	C₂×#168	—	2	4	no	—
70	C₂×#168	M₁₆	1	4	yes	—
71	C₂×#168	C₂²×C₄	1	2	yes	—
72	C₂×#168	C₂²×C₄	1	2	yes	—
73	C₂×#168	C₂²×C₄	1	1	yes	characteristic
74	C₂×#168	C₂²×C₄	1	2	yes	—
75	C₂×#168	#41	1	4	yes	—
76	C₂×#168	C₂⁴	1	1	yes	characteristic
77	#674	—	2	8	no	—
78	C₂²×M₁₆	—	5	1	no	—
79	C₂³×C₈	—	5	1	no	—
80	C₂×#117	AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₂×M₃₂	—	5	4	no	—
82	#405	—	5	4	no	—
83	#405	—	5	4	no	—
84	C₂×#97	—	5	1	no	—
85	C₂×#97	—	5	1	no	—
86	C₂×C₄×C₈	—	5	1	no	—
87	C₂×C₄×C₈	—	5	1	no	—
88	C₄⋊C₁₆	—	5	8	no	—
89	C₂²⋊C₁₆	—	5	8	no	—
90	#346	—	5	4	no	—
91	C₄×C₁₆	—	5	4	no	—
92	C₂³×C₅	—	2	2	no	—
93	C₂³×C₅	—	4	2	no	—
94	C₂³×C₅	C₂²×C₈	1	2	yes	—
95	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	#117	1	2	yes	—
103	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
104	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
109	#168	—	2	2	no	—
110	#168	—	2	2	no	—
111	#168	#117	1	2	yes	—
112	#168	C₂×M₁₆	1	4	yes	—
113	#168	C₂×#41	1	2	yes	—
114	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×M₁₆	—	5	4	no	—
116	C₂×M₁₆	—	10	4	no	—
117	C₂²×C₈	—	5	1	no	—
118	C₂²×C₈	—	5	1	no	—
119	C₂²×C₈	—	5	2	no	—
120	C₂²×C₈	—	5	1	no	—
121	C₂²×C₈	—	5	2	no	—
122	C₂²×C₈	—	5	1	no	—
123	C₂²×C₈	—	10	4	no	—
124	#117	#170	1	4	yes	—
125	#117	C₂×AGL(1,𝔽₅)	1	4	yes	—
126	C₂×C₄⋊C₄	#170	1	2	yes	—
127	C₂×C₂²⋊C₄	#170	1	2	yes	—
128	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
130	M₃₂	—	10	8	no	—
131	C₂×C₁₆	—	5	8	no	—
132	#97	—	5	4	no	—
133	#97	—	5	4	no	—
134	C₄×C₈	—	5	2	no	—
135	C₄×C₈	—	5	2	no	—
136	C₄×C₈	—	5	2	no	—
137	C₄×C₈	—	5	2	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	4	4	no	—
142	C₂²×C₅	#431	1	2	yes	—
143	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₅	—	2	4	no	—
145	C₄×C₅	—	2	4	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	#431	1	2	yes	—
148	C₄×C₅	C₂×#117	1	1	yes	characteristic
149	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
150	C₂⁴	C₂×#170	1	1	yes	characteristic
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂×#170	1	1	yes	characteristic
154	C₂²×C₄	C₂×#170	1	2	yes	—
155	C₂²×C₄	C₂×#170	1	2	yes	—
156	C₂²×C₄	C₂×#170	1	2	yes	—
157	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
158	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
159	M₁₆	—	10	8	no	—
160	C₂×C₈	—	5	4	no	—
161	C₂×C₈	—	5	4	no	—
162	C₂×C₈	—	5	1	no	—
163	C₂×C₈	—	5	1	no	—
164	C₂×C₈	—	5	2	no	—
165	C₂×C₈	—	5	2	no	—
166	C₂×C₈	—	5	2	no	—
167	C₂×C₈	—	10	4	no	—
168	C₂×C₈	—	10	4	no	—
169	C₂×C₈	—	10	2	no	—
170	C₂×C₈	—	10	2	no	—
171	C₄⋊C₄	C₂×#170	1	8	yes	—
172	C₂²⋊C₄	C₂×#170	1	8	yes	—
173	C₄²	C₂×#170	1	2	yes	—
174	C₄²	C₂×#170	1	2	yes	—
175	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
176	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
177	C₁₆	—	10	8	no	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	—	4	2	no	—
180	C₂×C₅	C₂×#431	1	1	yes	characteristic
181	C₂×C₅	#3407	1	2	yes	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	C₂²×#170	1	2	yes	—
185	C₂³	C₂²×#170	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
193	C₂×C₄	#4130	1	2	yes	—
194	C₂×C₄	C₂²×#170	1	4	yes	—
195	C₂×C₄	C₂²×#170	1	4	yes	—
196	C₂×C₄	C₂²×#170	1	2	yes	—
197	C₂×C₄	C₂²×#170	1	1	yes	characteristic
198	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₈	—	5	4	no	—
200	C₈	—	5	2	no	—
201	C₈	—	5	2	no	—
202	C₈	—	10	2	no	—
203	C₈	—	10	2	no	—
204	C₅	#33419	1	1	yes	characteristic, complemented, Sylow
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#65748	1	2	yes	—
210	C₂²	C₂³×#170	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	4	no	—
214	C₄	#65748	1	2	yes	—
215	C₄	C₂×D₅⋊M₁₆	1	1	yes	characteristic
216	C₄	C₂×#4130	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#65748	1	1	yes	characteristic
220	C₂	#10617530	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₂×C₅×#117	C₄
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×#117	AGL(1,𝔽₅)
hypocenter	C₅	#33419
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #10604368	C₂	semidirect
4. #10604150	C₂	semidirect
5. C₄×#3998	C₂	semidirect
17. #64859	C₂²	semidirect
18. C₄⋊#674	C₂²	semidirect
19. C₄×#674	C₂²	semidirect
204. C₅	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65286	2
#10604872	2
#10604368	2
#10604150	2
C₄×#3998	2
#33419	5