Subgroups — C₂×#10602192 #12251934

Order 1280.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#65390	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	#10602192	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64722	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₆×Dic₅	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#4072	C₄	1	1	yes	characteristic
9	C₂²×#681	C₄	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₄	1	8	yes	complemented
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
14	#65390	C₄	1	4	yes	—
15	#65390	C₂²	1	4	yes	—
16	C₂×#3947	C₄	1	2	yes	—
17	C₂×C₈×Dic₅	C₂²	1	2	yes	—
18	C₂×#3944	C₄	1	8	yes	complemented
19	C₂×#3944	C₂²	1	4	yes	—
20	C₂²×#672	C₂²	1	2	yes	—
21	#64722	C₂²	1	8	yes	complemented
22	C₁₆×Dic₅	C₂²	1	8	yes	complemented
23	C₂×#2108	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₂×C₂²⋊Dic₅	C₈	1	2	yes	—
32	#4072	C₈	1	4	yes	—
33	#4072	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊Dic₅	C₈	1	2	yes	—
35	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
36	C₂×#681	C₈	1	8	yes	—
37	C₂×#681	C₂×C₄	1	4	yes	—
38	C₂²×#168	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
40	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₅×C₁₆	C₂×C₄	1	16	yes	complemented
42	C₂×C₅×C₁₆	C₂³	1	2	yes	—
43	C₂×C₅×C₁₆	C₂³	1	2	yes	—
44	#3947	C₂×C₄	1	8	yes	—
45	C₈×Dic₅	C₂×C₄	1	4	yes	—
46	C₈×Dic₅	C₂³	1	4	yes	—
47	#3944	C₂×C₄	1	16	yes	complemented
48	C₂×#672	C₂×C₄	1	8	yes	—
49	C₂×#672	C₂³	1	4	yes	—
50	C₂²×M₃₂	—	5	1	no	—
51	C₂³×C₁₆	D₅	1	1	yes	characteristic
52	C₂×#405	—	5	1	no	—
53	#2108	—	5	8	no	—
54	C₂×#346	—	5	2	no	—
55	C₂×C₄×C₁₆	—	5	2	no	—
56	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄²	1	4	yes	—
58	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
63	C₂²×Dic₅	C₂×C₈	1	4	yes	—
64	C₂×C₅×C₈	C₄²	1	4	yes	—
65	C₂×C₅×C₈	C₄²	1	8	yes	—
66	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
73	C₂²⋊Dic₅	C₂×C₈	1	8	yes	—
74	C₄⋊Dic₅	C₂×C₈	1	8	yes	—
75	C₄×Dic₅	C₂×C₈	1	4	yes	—
76	C₄×Dic₅	C₂²×C₄	1	4	yes	—
77	#681	C₂×C₈	1	16	yes	—
78	C₂×#168	C₂×C₈	1	8	yes	—
79	C₂×#168	C₂²×C₄	1	4	yes	—
80	C₅×C₁₆	C₂²×C₄	1	4	yes	—
81	C₅×C₁₆	C₂²×C₄	1	4	yes	—
82	#672	C₂²×C₄	1	8	yes	—
83	C₂²×M₁₆	—	5	1	no	—
84	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
85	C₂×#117	—	5	1	no	—
86	C₂×M₃₂	—	5	8	no	—
87	C₂×M₃₂	—	5	4	no	—
88	C₂²×C₁₆	—	5	2	no	—
89	C₂²×C₁₆	Dic₅	1	8	yes	complemented
90	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₁₆	C₂×D₅	1	4	yes	—
92	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
93	#405	—	5	4	no	—
94	#405	—	5	4	no	—
95	C₂×#97	—	5	2	no	—
96	C₂×C₄×C₈	—	5	2	no	—
97	#346	—	5	8	no	—
98	C₄×C₁₆	—	5	8	no	—
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	C₄×C₈	1	4	yes	—
101	C₂³×C₅	C₂²×C₈	1	2	yes	—
102	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	C₄×C₈	1	8	yes	—
104	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
105	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₅	—	2	8	no	—
113	C₂×Dic₅	C₂²×C₈	1	8	yes	—
114	C₅×C₈	C₂×C₄²	1	2	yes	—
115	C₅×C₈	C₂×C₄²	1	4	yes	—
116	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
117	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
118	#168	C₂²×C₈	1	8	yes	—
119	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
120	C₂×M₁₆	—	5	8	no	—
121	C₂×M₁₆	—	5	4	no	—
122	C₂²×C₈	—	5	2	no	—
123	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
124	C₂²×C₈	C₄×D₅	1	2	yes	—
125	C₂²×C₈	C₄×D₅	1	4	yes	—
126	C₂²×C₈	C₂×Dic₅	1	4	yes	—
127	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₅	1	2	yes	—
129	#117	—	5	4	no	—
130	#117	—	5	4	no	—
131	C₂×C₄⋊C₄	—	5	2	no	—
132	C₂×C₂²⋊C₄	—	5	2	no	—
133	C₂×C₄²	—	5	2	no	—
134	M₃₂	—	5	16	no	—
135	C₂×C₁₆	—	5	8	no	—
136	C₂×C₁₆	—	5	4	no	—
137	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
138	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
139	C₂×C₁₆	C₂×Dic₅	1	16	yes	complemented
140	C₂×C₁₆	C₂²×D₅	1	2	yes	—
141	C₂×C₁₆	C₂²×D₅	1	2	yes	—
142	#97	—	5	8	no	—
143	C₄×C₈	—	5	4	no	—
144	C₄×C₈	—	5	4	no	—
145	C₂²×C₅	—	2	8	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
148	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
149	C₂²×C₅	#504	1	4	yes	—
150	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
151	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
152	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
153	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
154	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
155	Dic₅	—	2	8	no	—
156	C₂⁴	C₈×D₅	1	1	yes	characteristic
157	C₂²×C₄	—	5	4	no	—
158	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
159	C₂²×C₄	C₈×D₅	1	2	yes	—
160	C₂²×C₄	C₈×D₅	1	4	yes	—
161	C₂²×C₄	C₄×Dic₅	1	4	yes	—
162	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
164	M₁₆	—	5	16	no	—
165	C₂×C₈	—	5	4	no	—
166	C₂×C₈	—	5	8	no	—
167	C₂×C₈	C₄×Dic₅	1	4	yes	—
168	C₂×C₈	C₄×Dic₅	1	8	yes	—
169	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
170	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
171	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
172	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
173	C₂×C₈	C₂²×Dic₅	1	2	yes	—
174	C₂×C₈	C₂²×Dic₅	1	4	yes	—
175	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
176	C₄⋊C₄	—	5	8	no	—
177	C₂²⋊C₄	—	5	8	no	—
178	C₄²	—	5	4	no	—
179	C₄²	—	5	4	no	—
180	C₁₆	—	5	8	no	—
181	C₁₆	C₂²×Dic₅	1	4	yes	—
182	C₁₆	C₂²×Dic₅	1	4	yes	—
183	C₂×C₅	—	2	4	no	—
184	C₂×C₅	#2108	1	4	yes	complemented
185	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
186	C₂×C₅	C₂×#504	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₈×Dic₅	1	4	yes	—
189	C₂³	C₂×C₈×D₅	1	2	yes	—
190	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
191	C₂×C₄	—	5	8	no	—
192	C₂×C₄	—	10	8	no	—
193	C₂×C₄	C₈×Dic₅	1	8	yes	—
194	C₂×C₄	C₈×Dic₅	1	4	yes	—
195	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
196	C₂×C₄	C₂×C₈×D₅	1	4	yes	—
197	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
198	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
199	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
200	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
201	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
202	C₈	—	5	8	no	—
203	C₈	C₂×C₄×Dic₅	1	2	yes	—
204	C₈	C₂×C₄×Dic₅	1	4	yes	—
205	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
206	C₈	C₂³×Dic₅	1	1	yes	characteristic
207	C₅	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	#65190	1	4	yes	—
211	C₂²	C₂×C₈×Dic₅	1	4	yes	—
212	C₂²	C₂×C₈×Dic₅	1	2	yes	—
213	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
214	C₄	—	10	8	no	—
215	C₄	C₂×C₈×Dic₅	1	2	yes	—
216	C₄	C₂×C₈×Dic₅	1	4	yes	—
217	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
218	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#10602192	1	4	yes	direct product factor
221	C₂	C₂×#65190	1	2	yes	—
222	C₂	C₂²×C₈×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂³×C₅×C₁₆	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	C₂×#2108
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10602192	C₂	direct (N × H)
5. C₂×#64722	C₂	semidirect
6. C₂×C₁₆×Dic₅	C₂	semidirect
10. C₂²×C₅×C₁₆	C₄	semidirect
18. C₂×#3944	C₄	semidirect
21. #64722	C₂²	semidirect
22. C₁₆×Dic₅	C₂²	semidirect
41. C₂×C₅×C₁₆	C₂×C₄	semidirect
47. #3944	C₂×C₄	semidirect
89. C₂²×C₁₆	Dic₅	semidirect
139. C₂×C₁₆	C₂×Dic₅	semidirect
184. C₂×C₅	#2108	semidirect
207. C₅	C₂×#2108	semidirect
220. C₂	#10602192	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂×#65390	2
C₂²×#3944	2
#10602192	2
C₂×#64722	2
C₂×C₁₆×Dic₅	2
C₂×#2108	5

Subgroups — C2×#10602192 #12251934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10602192 #12251934