Subgroups — C₂×#10604368 #13043431

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₂×C₂²⋊#674	C₂	1	2	yes	maximal
3	#10604368	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#674	C₂	1	2	yes	maximal, complemented
5	C₂×#64859	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#117	C₄	1	1	yes	characteristic
8	C₂²×#681	C₄	1	1	yes	characteristic
9	C₂³×#168	C₂²	1	1	yes	characteristic
10	C₂²×#674	C₂²	1	2	yes	—
11	C₂²×#674	C₂²	1	2	yes	—
12	#65286	C₄	1	4	yes	—
13	#65286	C₂²	1	4	yes	—
14	C₂×#3935	C₄	1	1	yes	characteristic
15	C₂×#3935	C₄	1	1	yes	characteristic
16	C₂×C₄×#168	C₂²	1	1	yes	characteristic
17	C₂×C₄×#168	C₂²	1	1	yes	characteristic
18	C₂²⋊#674	C₂²	1	8	yes	—
19	#64859	C₂²	1	4	yes	complemented
20	C₄⋊#674	C₂²	1	8	yes	complemented
21	C₄×#674	C₂²	1	4	yes	complemented
22	C₂×#2164	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×#117	C₈	1	4	yes	—
25	C₅×#117	C₂×C₄	1	4	yes	—
26	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
27	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×#681	—	2	4	no	—
31	C₂×#681	C₂×C₄	1	4	yes	—
32	C₂²×#168	—	2	4	no	—
33	C₂²×#168	C₂×C₄	1	2	yes	—
34	C₂²×#168	C₂×C₄	1	1	yes	characteristic
35	C₂²×#168	C₂×C₄	1	1	yes	characteristic
36	C₂²×#168	C₂×C₄	1	1	yes	characteristic
37	C₂²×#168	C₂³	1	2	yes	—
38	C₂²×#168	C₂³	1	1	yes	characteristic
39	C₂×#674	—	2	4	no	—
40	C₂×#674	—	2	4	no	—
41	C₂×#674	C₂³	1	4	yes	—
42	C₂×#674	C₂³	1	4	yes	—
43	#3935	C₂×C₄	1	4	yes	—
44	#3935	C₂×C₄	1	4	yes	—
45	C₄×#168	C₂×C₄	1	2	yes	—
46	C₄×#168	C₂×C₄	1	2	yes	—
47	C₄×#168	C₂³	1	2	yes	—
48	C₄×#168	C₂³	1	2	yes	—
49	C₂×#405	—	5	1	no	—
50	#2164	—	5	8	no	—
51	C₂×C₄⋊C₁₆	—	5	2	no	—
52	C₂×C₂²⋊C₁₆	—	5	2	no	—
53	C₂×#346	—	5	1	no	—
54	C₂×C₄×C₁₆	—	5	1	no	—
55	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
65	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
66	C₄²×C₅	C₂×C₈	1	2	yes	—
67	C₄²×C₅	C₂×C₈	1	2	yes	—
68	C₄²×C₅	C₂²×C₄	1	2	yes	—
69	C₄²×C₅	C₂²×C₄	1	2	yes	—
70	#681	—	2	8	no	—
71	C₂×#168	—	2	2	no	—
72	C₂×#168	—	2	2	no	—
73	C₂×#168	—	2	4	no	—
74	C₂×#168	—	2	4	no	—
75	C₂×#168	M₁₆	1	4	yes	—
76	C₂×#168	C₂²×C₄	1	2	yes	—
77	C₂×#168	C₂²×C₄	1	2	yes	—
78	C₂×#168	C₂²×C₄	1	1	yes	characteristic
79	C₂×#168	C₂²×C₄	1	2	yes	—
80	C₂×#168	#41	1	4	yes	—
81	C₂×#168	C₂⁴	1	1	yes	characteristic
82	#674	—	2	4	no	—
83	#674	—	2	4	no	—
84	C₂²×M₁₆	—	5	1	no	—
85	C₂³×C₈	—	5	1	no	—
86	C₂×#117	AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂²×C₁₆	—	5	2	no	—
88	C₂²×C₁₆	—	5	2	no	—
89	#405	—	5	4	no	—
90	#405	—	5	4	no	—
91	C₂×#97	—	5	1	no	—
92	C₂×#97	—	5	1	no	—
93	C₂×C₄×C₈	—	5	1	no	—
94	C₂×C₄×C₈	—	5	1	no	—
95	C₄⋊C₁₆	—	5	8	no	—
96	C₂²⋊C₁₆	—	5	8	no	—
97	#346	—	5	4	no	—
98	C₄×C₁₆	—	5	4	no	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	—	2	4	no	—
101	C₂³×C₅	C₂²×C₈	1	2	yes	—
102	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	#117	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
116	#168	—	2	2	no	—
117	#168	—	2	2	no	—
118	#168	#117	1	2	yes	—
119	#168	C₂×M₁₆	1	4	yes	—
120	#168	C₂×#41	1	2	yes	—
121	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₂×M₁₆	—	5	4	no	—
123	C₂×M₁₆	—	10	4	no	—
124	C₂²×C₈	—	5	1	no	—
125	C₂²×C₈	—	5	1	no	—
126	C₂²×C₈	—	5	2	no	—
127	C₂²×C₈	—	5	2	no	—
128	C₂²×C₈	—	5	1	no	—
129	C₂²×C₈	—	5	1	no	—
130	C₂²×C₈	—	10	4	no	—
131	#117	#170	1	4	yes	—
132	#117	C₂×AGL(1,𝔽₅)	1	4	yes	—
133	C₂×C₄⋊C₄	#170	1	2	yes	—
134	C₂×C₂²⋊C₄	#170	1	2	yes	—
135	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₂×C₁₆	—	5	4	no	—
138	C₂×C₁₆	—	5	4	no	—
139	C₂×C₁₆	—	10	4	no	—
140	C₂×C₁₆	—	10	4	no	—
141	#97	—	5	4	no	—
142	#97	—	5	4	no	—
143	C₄×C₈	—	5	2	no	—
144	C₄×C₈	—	5	2	no	—
145	C₄×C₈	—	5	2	no	—
146	C₄×C₈	—	5	2	no	—
147	C₂²×C₅	—	2	4	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	4	4	no	—
150	C₂²×C₅	#431	1	2	yes	—
151	C₂²×C₅	#504	1	4	yes	—
152	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
153	C₄×C₅	—	2	4	no	—
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	#431	1	2	yes	—
157	C₄×C₅	C₂×#117	1	1	yes	characteristic
158	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
159	C₂⁴	C₂×#170	1	1	yes	characteristic
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂×#170	1	2	yes	—
163	C₂²×C₄	C₂×#170	1	2	yes	—
164	C₂²×C₄	C₂×#170	1	1	yes	characteristic
165	C₂²×C₄	C₂×#170	1	2	yes	—
166	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
167	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
168	M₁₆	—	10	8	no	—
169	C₂×C₈	—	5	4	no	—
170	C₂×C₈	—	5	4	no	—
171	C₂×C₈	—	5	1	no	—
172	C₂×C₈	—	5	1	no	—
173	C₂×C₈	—	5	2	no	—
174	C₂×C₈	—	5	2	no	—
175	C₂×C₈	—	5	2	no	—
176	C₂×C₈	—	10	4	no	—
177	C₂×C₈	—	10	4	no	—
178	C₂×C₈	—	10	2	no	—
179	C₂×C₈	—	10	2	no	—
180	C₄⋊C₄	C₂×#170	1	8	yes	—
181	C₂²⋊C₄	C₂×#170	1	8	yes	—
182	C₄²	C₂×#170	1	2	yes	—
183	C₄²	C₂×#170	1	2	yes	—
184	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
185	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
186	C₁₆	—	10	4	no	—
187	C₁₆	—	10	4	no	—
188	C₂×C₅	—	4	2	no	—
189	C₂×C₅	#2164	1	4	yes	complemented
190	C₂×C₅	C₂×#431	1	1	yes	characteristic
191	C₂×C₅	C₂×#504	1	2	yes	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂²×#170	1	2	yes	—
195	C₂³	C₂²×#170	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
203	C₂×C₄	#4130	1	2	yes	—
204	C₂×C₄	C₂²×#170	1	4	yes	—
205	C₂×C₄	C₂²×#170	1	4	yes	—
206	C₂×C₄	C₂²×#170	1	2	yes	—
207	C₂×C₄	C₂²×#170	1	1	yes	characteristic
208	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₈	—	5	4	no	—
210	C₈	—	5	2	no	—
211	C₈	—	5	2	no	—
212	C₈	—	10	2	no	—
213	C₈	—	10	2	no	—
214	C₅	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	#65748	1	2	yes	—
219	C₂²	#65791	1	4	yes	—
220	C₂²	C₂³×#170	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	4	no	—
224	C₄	#65748	1	2	yes	—
225	C₄	C₂×D₅⋊M₁₆	1	1	yes	characteristic
226	C₄	C₂×#4130	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	#10604368	1	4	yes	direct product factor
229	C₂	C₂×#65748	1	1	yes	characteristic
230	C₂	C₂×#65791	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₅×#117	C₄
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×#117	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#2164
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #10604368	C₂	direct (N × H)
4. C₂×C₄⋊#674	C₂	semidirect
5. C₂×#64859	C₂	semidirect
6. C₂×C₄×#674	C₂	semidirect
19. #64859	C₂²	semidirect
20. C₄⋊#674	C₂²	semidirect
21. C₄×#674	C₂²	semidirect
189. C₂×C₅	#2164	semidirect
214. C₅	C₂×#2164	semidirect
228. C₂	#10604368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#65286	2
C₂×C₂²⋊#674	2
#10604368	2
C₂×C₄⋊#674	2
C₂×#64859	2
C₂×C₄×#674	2
C₂×#2164	5

Subgroups — C2×#10604368 #13043431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10604368 #13043431