Subgroups — C₂×#10598260 #12009115

Order 1280.

346 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×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
3	C₂×#64681	C₂	1	1	yes	maximal, characteristic
4	#10598260	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₅×#446	—	2	1	no	—
7	C₅×#446	—	2	1	no	—
8	C₅×#446	C₄	1	2	yes	—
9	C₅×#446	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×#97	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂²×#672	—	2	1	no	—
15	C₂²×#672	—	2	1	no	—
16	C₄⋊#672	—	2	1	no	—
17	C₄⋊#672	C₂²	1	2	yes	—
18	#64681	—	2	1	no	—
19	#64681	C₂²	1	2	yes	—
20	C₂×#1338	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₄×C₅×Q₈	—	2	1	no	—
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	C₈	1	2	yes	—
28	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₅	—	2	1	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₅×#97	—	2	2	no	—
38	C₅×#97	—	2	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂×#672	—	2	4	no	—
44	C₂×#672	—	2	2	no	—
45	C₂×#672	—	2	2	no	—
46	C₂×#672	—	4	2	no	—
47	C₂×#446	D₅	1	1	yes	characteristic
48	C₂×C₄⋊C₁₆	—	5	1	no	—
49	C₂×#346	—	5	1	no	—
50	#1338	—	5	4	no	—
51	C₂×C₅×Q₈	—	2	1	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	SD₁₆	1	2	yes	—
65	C₂×C₅×C₈	Q₁₆	1	2	yes	—
66	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₄⋊C₄	—	2	1	no	—
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₅	M₁₆	1	1	yes	characteristic
77	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
78	#672	—	2	4	no	—
79	#672	—	4	2	no	—
80	C₂×C₄×Q₈	Dic₅	1	1	yes	characteristic
81	C₂²×C₁₆	—	10	1	no	—
82	C₂²×C₁₆	—	10	1	no	—
83	#446	—	2	1	no	—
84	#446	—	2	1	no	—
85	#446	Dic₅	1	2	yes	—
86	#446	C₂×D₅	1	2	yes	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
89	C₂×#97	—	2	1	no	—
90	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
91	C₄⋊C₁₆	—	5	2	no	—
92	C₄⋊C₁₆	—	10	1	no	—
93	#346	—	5	2	no	—
94	#346	—	10	1	no	—
95	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
96	C₅×Q₈	—	4	2	no	—
97	C₅×Q₈	—	4	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	#103	1	2	yes	—
108	C₂×C₄×C₅	#104	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	4	2	no	—
115	C₅×C₈	#103	1	2	yes	—
116	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
117	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
118	C₂²×Q₈	#168	1	1	yes	characteristic
119	C₂²×C₈	—	2	1	no	—
120	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
121	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	#168	1	2	yes	—
125	C₄×Q₈	C₂×Dic₅	1	2	yes	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
130	C₂×C₁₆	—	10	4	no	—
131	C₂×C₁₆	—	10	2	no	—
132	C₂×C₁₆	—	10	2	no	—
133	C₂×C₁₆	—	20	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
138	#97	—	2	2	no	—
139	#97	—	2	2	no	—
140	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
144	C₂²×C₅	Q₈⋊C₈	1	2	yes	—
145	C₂²×C₅	#349	1	2	yes	—
146	C₂²×C₅	#350	1	2	yes	—
147	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	4	2	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	Q₈⋊C₈	1	2	yes	—
155	C₄×C₅	C₂×#103	1	1	yes	characteristic
156	C₄×C₅	C₂×#104	1	1	yes	characteristic
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂×#168	1	2	yes	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×#168	1	1	yes	characteristic
163	C₂²×C₄	#681	1	1	yes	characteristic
164	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	#688	1	2	yes	—
169	C₂×C₈	#689	1	2	yes	—
170	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	C₂×#168	1	2	yes	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	C₂×#168	1	1	yes	characteristic
181	C₄²	#681	1	1	yes	characteristic
182	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
183	C₄²	C₂²×Dic₅	1	1	yes	characteristic
184	C₁₆	—	10	4	no	—
185	C₁₆	—	20	2	no	—
186	C₂×C₅	#1338	1	4	yes	complemented
187	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
188	C₂×C₅	C₂×#349	1	1	yes	characteristic
189	C₂×C₅	C₂×#350	1	1	yes	characteristic
190	C₂³	C₂²⋊#168	1	1	yes	characteristic
191	Q₈	—	4	2	no	—
192	Q₈	—	4	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
200	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
201	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
202	C₂×C₄	#3967	1	2	yes	—
203	C₂×C₄	#3969	1	2	yes	—
204	C₂×C₄	C₂²×#168	1	1	yes	characteristic
205	C₂×C₄	C₂×#681	1	1	yes	characteristic
206	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	4	2	no	—
210	C₈	#3967	1	2	yes	—
211	C₈	C₂×#688	1	1	yes	characteristic
212	C₈	C₂×#689	1	1	yes	characteristic
213	C₅	C₂×#1338	1	1	yes	characteristic, complemented, Sylow
214	C₂²	Q₈⋊#168	1	2	yes	—
215	C₂²	#64773	1	2	yes	—
216	C₂²	#64781	1	2	yes	—
217	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	Q₈⋊#168	1	2	yes	—
225	C₄	C₂×#3967	1	1	yes	characteristic
226	C₄	C₂×#3969	1	1	yes	characteristic
227	C₂	#10598260	1	4	yes	direct product factor
228	C₂	C₂×Q₈⋊#168	1	1	yes	characteristic
229	C₂	C₂×#64773	1	1	yes	characteristic
230	C₂	C₂×#64781	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×#446	C₂
socle, prime radical	C₂³×C₅	C₂²⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×#446	D₅
hypocenter	C₅	C₂×#1338
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
4. #10598260	C₂	direct (N × H)
186. C₂×C₅	#1338	semidirect
213. C₅	C₂×#1338	semidirect
227. C₂	#10598260	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×C₅×#446	2
C₂×C₄⋊#672	2
C₂×#64681	2
#10598260	2
C₂×#1338	5

Subgroups — C2×#10598260 #12009115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598260 #12009115