Subgroups — C₂×#10602919 #12098404

Order 1280.

6110 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊#168	C₂	1	6	yes	maximal, complemented
3	#10602919	C₂	1	8	yes	maximal, direct product factor
4	C₂⁶×C₅	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₅	—	2	24	no	—
6	C₂⁴×C₄×C₅	C₄	1	3	yes	—
7	C₂⁴×C₄×C₅	C₄	1	4	yes	—
8	C₂⁴×C₄×C₅	C₂²	1	4	yes	—
9	C₂⁴×C₄×C₅	C₂²	1	3	yes	—
10	C₂³×#168	C₂²	1	4	yes	complemented
11	C₂×C₂²⋊#168	—	2	24	no	—
12	C₂×C₂²⋊#168	C₂²	1	24	yes	complemented
13	C₂×#1779	—	5	1	no	maximal, Sylow
14	C₂⁵×C₅	—	2	12	no	—
15	C₂⁵×C₅	—	2	12	no	—
16	C₂⁵×C₅	C₈	1	8	yes	complemented
17	C₂⁵×C₅	C₂×C₄	1	4	yes	—
18	C₂⁵×C₅	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₅	—	2	128	no	—
20	C₂³×C₄×C₅	—	2	12	no	—
21	C₂³×C₄×C₅	—	2	24	no	—
22	C₂³×C₄×C₅	—	2	48	no	—
23	C₂³×C₄×C₅	—	2	48	no	—
24	C₂³×C₄×C₅	—	2	24	no	—
25	C₂³×C₄×C₅	C₂×C₄	1	12	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₅	C₂×C₄	1	6	yes	—
28	C₂³×C₄×C₅	D₄	1	24	yes	—
29	C₂³×C₄×C₅	C₂³	1	6	yes	—
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×#168	—	2	24	no	—
32	C₂²×#168	C₂³	1	8	yes	complemented
33	C₂²⋊#168	—	2	48	no	—
34	C₂⁵×C₄	D₅	1	1	yes	characteristic
35	C₂²×C₂²⋊C₈	—	5	6	no	—
36	#1779	—	5	8	no	—
37	C₂⁴×C₅	—	2	32	no	—
38	C₂⁴×C₅	—	2	96	no	—
39	C₂⁴×C₅	—	2	96	no	—
40	C₂⁴×C₅	—	2	24	no	—
41	C₂⁴×C₅	—	2	12	no	—
42	C₂⁴×C₅	—	2	24	no	—
43	C₂⁴×C₅	—	2	12	no	—
44	C₂⁴×C₅	C₂²⋊C₄	1	12	yes	—
45	C₂⁴×C₅	C₂×C₈	1	12	yes	—
46	C₂⁴×C₅	C₂×C₈	1	16	yes	complemented
47	C₂⁴×C₅	M₁₆	1	12	yes	—
48	C₂⁴×C₅	C₂²×C₄	1	6	yes	—
49	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	—	2	96	no	—
51	C₂²×C₄×C₅	—	2	128	no	—
52	C₂²×C₄×C₅	—	2	128	no	—
53	C₂²×C₄×C₅	—	2	24	no	—
54	C₂²×C₄×C₅	—	2	12	no	—
55	C₂²×C₄×C₅	—	2	48	no	—
56	C₂²×C₄×C₅	—	2	24	no	—
57	C₂²×C₄×C₅	—	2	96	no	—
58	C₂²×C₄×C₅	—	2	24	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	12	yes	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	24	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
64	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
66	C₂×#168	—	2	24	no	—
67	C₂×#168	—	4	16	no	—
68	C₂⁶	Dic₅	1	1	yes	characteristic
69	C₂⁴×C₄	—	2	24	no	—
70	C₂⁴×C₄	Dic₅	1	3	yes	—
71	C₂⁴×C₄	Dic₅	1	4	yes	—
72	C₂⁴×C₄	C₂×D₅	1	4	yes	—
73	C₂⁴×C₄	C₂×D₅	1	3	yes	—
74	C₂³×C₈	—	5	4	no	—
75	C₂×C₂²⋊C₈	—	5	24	no	—
76	C₂×C₂²⋊C₈	—	10	24	no	—
77	C₂³×C₅	—	2	32	no	—
78	C₂³×C₅	—	2	96	no	—
79	C₂³×C₅	—	2	256	no	—
80	C₂³×C₅	—	2	12	no	—
81	C₂³×C₅	—	2	12	no	—
82	C₂³×C₅	—	2	24	no	—
83	C₂³×C₅	—	2	48	no	—
84	C₂³×C₅	—	2	96	no	—
85	C₂³×C₅	—	2	24	no	—
86	C₂³×C₅	—	2	48	no	—
87	C₂³×C₅	C₂²⋊C₈	1	48	yes	complemented
88	C₂³×C₅	C₂×C₂²⋊C₄	1	12	yes	—
89	C₂³×C₅	C₂×C₂²⋊C₄	1	6	yes	—
90	C₂³×C₅	C₂²×C₈	1	12	yes	—
91	C₂³×C₅	C₂²×C₈	1	2	yes	—
92	C₂³×C₅	C₂×M₁₆	1	12	yes	—
93	C₂³×C₅	C₂×M₁₆	1	6	yes	—
94	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	96	no	—
96	C₂×C₄×C₅	—	2	12	no	—
97	C₂×C₄×C₅	—	2	24	no	—
98	C₂×C₄×C₅	—	2	48	no	—
99	C₂×C₄×C₅	—	2	48	no	—
100	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
101	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
102	C₂×C₄×C₅	#120	1	16	yes	—
103	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
104	#168	—	4	8	no	—
105	C₂⁵	—	2	12	no	—
106	C₂⁵	—	2	12	no	—
107	C₂⁵	#168	1	8	yes	complemented
108	C₂⁵	C₂×Dic₅	1	4	yes	—
109	C₂⁵	C₂×Dic₅	1	3	yes	—
110	C₂³×C₄	—	2	128	no	—
111	C₂³×C₄	—	2	12	no	—
112	C₂³×C₄	—	2	24	no	—
113	C₂³×C₄	—	2	48	no	—
114	C₂³×C₄	—	2	48	no	—
115	C₂³×C₄	—	2	24	no	—
116	C₂³×C₄	C₂×Dic₅	1	12	yes	—
117	C₂³×C₄	C₂×Dic₅	1	3	yes	—
118	C₂³×C₄	C₂×Dic₅	1	6	yes	—
119	C₂³×C₄	C₂²⋊D₅	1	24	yes	—
120	C₂³×C₄	C₂²×D₅	1	6	yes	—
121	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
122	C₂²×C₈	—	5	8	no	—
123	C₂²×C₈	—	10	24	no	—
124	C₂²⋊C₈	—	10	48	no	—
125	C₂²×C₅	—	2	128	no	—
126	C₂²×C₅	—	2	24	no	—
127	C₂²×C₅	—	2	12	no	—
128	C₂²×C₅	—	2	24	no	—
129	C₂²×C₅	—	2	12	no	—
130	C₂²×C₅	—	2	96	no	—
131	C₂²×C₅	#379	1	4	yes	—
132	C₂²×C₅	C₂×C₂²⋊C₈	1	12	yes	—
133	C₂²×C₅	C₂×C₂²⋊C₈	1	24	yes	complemented
134	C₂²×C₅	#407	1	12	yes	—
135	C₂²×C₅	C₂²×C₂²⋊C₄	1	3	yes	—
136	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
137	C₂²×C₅	C₂²×M₁₆	1	3	yes	—
138	C₄×C₅	—	2	12	no	—
139	C₄×C₅	#379	1	4	yes	—
140	C₄×C₅	C₂×#120	1	4	yes	—
141	C₂⁴	—	2	32	no	—
142	C₂⁴	—	2	96	no	—
143	C₂⁴	—	2	24	no	—
144	C₂⁴	—	2	12	no	—
145	C₂⁴	—	2	96	no	—
146	C₂⁴	—	2	24	no	—
147	C₂⁴	—	2	12	no	—
148	C₂⁴	C₂×#168	1	12	yes	—
149	C₂⁴	C₂×#168	1	16	yes	complemented
150	C₂⁴	#681	1	12	yes	—
151	C₂⁴	C₂²⋊Dic₅	1	12	yes	—
152	C₂⁴	C₂²×Dic₅	1	6	yes	—
153	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
154	C₂²×C₄	—	2	96	no	—
155	C₂²×C₄	—	2	128	no	—
156	C₂²×C₄	—	2	128	no	—
157	C₂²×C₄	—	2	24	no	—
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	2	48	no	—
160	C₂²×C₄	—	2	24	no	—
161	C₂²×C₄	—	2	96	no	—
162	C₂²×C₄	—	2	24	no	—
163	C₂²×C₄	C₂²⋊Dic₅	1	12	yes	—
164	C₂²×C₄	C₂²⋊Dic₅	1	24	yes	—
165	C₂²×C₄	C₂²×Dic₅	1	6	yes	—
166	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₅	1	24	yes	—
168	C₂²×C₄	C₂×C₂²⋊D₅	1	12	yes	—
169	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
170	C₂×C₈	—	10	24	no	—
171	C₂×C₈	—	20	16	no	—
172	C₂×C₅	—	2	24	no	—
173	C₂×C₅	#1779	1	8	yes	complemented
174	C₂×C₅	C₂×#379	1	1	yes	characteristic
175	C₂×C₅	C₂²×C₂²⋊C₈	1	3	yes	—
176	C₂×C₅	C₂×#407	1	3	yes	—
177	C₂³	—	2	32	no	—
178	C₂³	—	2	256	no	—
179	C₂³	—	2	96	no	—
180	C₂³	—	2	24	no	—
181	C₂³	—	2	12	no	—
182	C₂³	—	2	12	no	—
183	C₂³	—	2	48	no	—
184	C₂³	—	2	96	no	—
185	C₂³	—	2	48	no	—
186	C₂³	—	2	24	no	—
187	C₂³	C₂²⋊#168	1	48	yes	complemented
188	C₂³	C₂²×#168	1	12	yes	—
189	C₂³	C₂²×#168	1	2	yes	—
190	C₂³	C₂×#681	1	12	yes	—
191	C₂³	C₂×#681	1	6	yes	—
192	C₂³	C₂×C₂²⋊Dic₅	1	12	yes	—
193	C₂³	C₂×C₂²⋊Dic₅	1	6	yes	—
194	C₂³	C₂³×Dic₅	1	1	yes	characteristic
195	C₂×C₄	—	2	96	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	2	48	no	—
199	C₂×C₄	—	2	48	no	—
200	C₂×C₄	C₂×C₂²⋊Dic₅	1	12	yes	—
201	C₂×C₄	C₂×C₂²⋊Dic₅	1	6	yes	—
202	C₂×C₄	#4099	1	16	yes	—
203	C₂×C₄	C₂²×C₂²⋊D₅	1	6	yes	—
204	C₈	—	20	8	no	—
205	C₅	C₂×#1779	1	1	yes	characteristic, complemented, Sylow
206	C₂²	—	2	128	no	—
207	C₂²	—	2	12	no	—
208	C₂²	—	2	96	no	—
209	C₂²	—	2	24	no	—
210	C₂²	—	2	12	no	—
211	C₂²	—	2	24	no	—
212	C₂²	C₂×C₂²⋊#168	1	12	yes	—
213	C₂²	C₂×C₂²⋊#168	1	24	yes	complemented
214	C₂²	#65496	1	12	yes	—
215	C₂²	#65536	1	4	yes	—
216	C₂²	C₂³×#168	1	1	yes	characteristic
217	C₂²	C₂²×#681	1	3	yes	—
218	C₂²	C₂²×C₂²⋊Dic₅	1	3	yes	—
219	C₄	—	2	12	no	—
220	C₄	#65536	1	4	yes	—
221	C₄	C₂×#4099	1	4	yes	—
222	C₂	—	2	24	no	—
223	C₂	#10602919	1	8	yes	direct product factor
224	C₂	C₂²×C₂²⋊#168	1	3	yes	—
225	C₂	C₂×#65496	1	3	yes	—
226	C₂	C₂×#65536	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂⁵×C₄×C₅	C₂
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂⁶×C₅	C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂⁵×C₄	D₅
hypocenter	C₅	C₂×#1779
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂²⋊#168	C₂	semidirect
3. #10602919	C₂	direct (N × H)
10. C₂³×#168	C₂²	semidirect
12. C₂×C₂²⋊#168	C₂²	semidirect
16. C₂⁵×C₅	C₈	semidirect
32. C₂²×#168	C₂³	semidirect
46. C₂⁴×C₅	C₂×C₈	semidirect
87. C₂³×C₅	C₂²⋊C₈	semidirect
107. C₂⁵	#168	semidirect
133. C₂²×C₅	C₂×C₂²⋊C₈	semidirect
149. C₂⁴	C₂×#168	semidirect
173. C₂×C₅	#1779	semidirect
187. C₂³	C₂²⋊#168	semidirect
205. C₅	C₂×#1779	semidirect
213. C₂²	C₂×C₂²⋊#168	semidirect
223. C₂	#10602919	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₅	2
C₂²×C₂²⋊#168	2
#10602919	2
C₂×#1779	5

Subgroups — C2×#10602919 #12098404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10602919 #12098404