Subgroups — C₂×#10598320 #12019751

Order 1280.

294 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	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×#64726	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#3926	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598320	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#135	—	2	1	no	—
6	C₅×#508	—	2	1	no	—
7	C₅×#508	—	2	1	no	—
8	C₅×#508	C₄	1	2	yes	—
9	C₅×#508	C₂²	1	2	yes	—
10	C₂×C₅×Q₃₂	C₄	1	1	yes	characteristic
11	C₂×C₅×SD₃₂	—	2	1	no	—
12	C₂×C₅×D₁₆	C₄	1	1	yes	characteristic
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×#3951	—	2	1	no	—
15	#64726	—	2	1	no	—
16	#64726	C₂²	1	2	yes	complemented
17	C₂×#3926	—	2	1	no	—
18	C₂×#3926	—	2	1	no	—
19	C₂×#3926	C₂²	1	2	yes	complemented
20	C₂×#1417	—	5	1	no	maximal, Sylow
21	C₂×C₅×#41	—	4	1	no	—
22	C₅×#135	—	2	2	no	—
23	C₅×#135	—	2	2	no	—
24	C₅×#135	—	4	1	no	—
25	C₅×#135	—	4	1	no	—
26	C₂×C₅×Q₁₆	—	2	1	no	—
27	C₂×C₅×SD₁₆	—	4	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×#681	—	4	1	no	—
31	C₅×Q₃₂	—	2	1	no	—
32	C₅×Q₃₂	C₂×C₄	1	2	yes	—
33	C₅×SD₃₂	—	2	2	no	—
34	C₅×SD₃₂	—	2	2	no	—
35	C₅×D₁₆	—	2	1	no	—
36	C₅×D₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
43	#3951	—	2	2	no	—
44	#3951	—	4	1	no	—
45	#3926	—	2	2	no	—
46	C₂×#508	D₅	1	1	yes	characteristic
47	C₂²×C₃₂	—	5	1	no	—
48	C₂×#368	—	5	1	no	—
49	#1417	—	5	4	no	—
50	C₅×#41	—	4	2	no	—
51	C₅×#41	—	4	2	no	—
52	C₅×#41	—	8	1	no	—
53	C₅×#41	—	8	1	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂×C₅×D₄	—	4	1	no	—
56	C₂×C₅×D₄	—	8	1	no	—
57	C₂²×C₄×C₅	—	8	1	no	—
58	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
59	C₅×Q₁₆	—	2	2	no	—
60	C₅×Q₁₆	—	4	1	no	—
61	C₅×SD₁₆	—	4	2	no	—
62	C₅×SD₁₆	—	4	2	no	—
63	C₅×D₈	—	2	2	no	—
64	C₅×D₈	—	4	1	no	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	D₈	1	1	yes	characteristic
68	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
69	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	#681	—	4	2	no	—
72	#681	—	8	1	no	—
73	C₂×#168	—	8	1	no	—
74	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
75	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
76	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
77	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
78	C₂×#135	—	2	1	no	—
79	#508	—	2	1	no	—
80	#508	—	2	1	no	—
81	#508	Dic₅	1	2	yes	—
82	#508	C₂×D₅	1	2	yes	—
83	C₂×Q₃₂	Dic₅	1	1	yes	characteristic
84	C₂×SD₃₂	—	2	1	no	—
85	C₂×D₁₆	Dic₅	1	1	yes	characteristic
86	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
87	C₂×#108	—	10	1	no	—
88	C₂×C₃₂	—	5	2	no	—
89	C₂×C₃₂	—	10	1	no	—
90	C₂×C₃₂	—	10	1	no	—
91	#368	—	5	2	no	—
92	#368	—	10	1	no	—
93	C₂³×C₅	—	8	1	no	—
94	C₂³×C₅	SD₃₂	1	1	yes	characteristic
95	C₅×Q₈	—	4	2	no	—
96	C₅×Q₈	—	8	1	no	—
97	C₅×D₄	—	4	2	no	—
98	C₅×D₄	—	8	2	no	—
99	C₅×D₄	—	8	2	no	—
100	C₅×D₄	—	8	1	no	—
101	C₂×C₄×C₅	—	8	1	no	—
102	C₂×C₄×C₅	—	8	2	no	—
103	C₂×C₄×C₅	—	8	2	no	—
104	C₂×C₄×C₅	#102	1	1	yes	characteristic
105	C₂×C₄×C₅	#102	1	1	yes	characteristic
106	C₂×C₄×C₅	D₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	D₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅	SD₃₂	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₅×C₈	#102	1	1	yes	characteristic
111	C₅×C₈	#102	1	1	yes	characteristic
112	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
114	#168	—	8	2	no	—
115	C₂×#41	—	4	1	no	—
116	#135	—	2	2	no	—
117	#135	—	2	2	no	—
118	#135	—	4	1	no	—
119	#135	—	4	1	no	—
120	C₂×Q₁₆	—	2	1	no	—
121	C₂×SD₁₆	—	4	1	no	—
122	C₂×D₈	—	2	1	no	—
123	C₂×M₁₆	—	20	1	no	—
124	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
125	Q₃₂	—	2	1	no	—
126	Q₃₂	C₂×Dic₅	1	2	yes	—
127	SD₃₂	—	2	2	no	—
128	SD₃₂	—	2	2	no	—
129	D₁₆	—	2	1	no	—
130	D₁₆	C₂×Dic₅	1	2	yes	—
131	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
132	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
133	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
134	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
135	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
136	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
137	#108	—	10	2	no	—
138	#108	—	20	1	no	—
139	C₃₂	—	10	2	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	8	2	no	—
142	C₂²×C₅	—	16	2	no	—
143	C₂²×C₅	#357	1	1	yes	characteristic
144	C₂²×C₅	#357	1	1	yes	characteristic
145	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
146	C₄×C₅	—	8	2	no	—
147	C₄×C₅	#357	1	1	yes	characteristic
148	C₄×C₅	#357	1	1	yes	characteristic
149	C₄×C₅	C₂×#102	1	1	yes	characteristic
150	C₄×C₅	C₂×D₁₆	1	1	yes	characteristic
151	#41	—	4	2	no	—
152	#41	—	4	2	no	—
153	#41	—	8	1	no	—
154	#41	—	8	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂×D₄	—	4	1	no	—
157	C₂×D₄	—	8	1	no	—
158	C₂²×C₄	—	8	1	no	—
159	C₂²×C₄	#686	1	1	yes	characteristic
160	Q₁₆	—	2	2	no	—
161	Q₁₆	—	4	1	no	—
162	SD₁₆	—	4	2	no	—
163	SD₁₆	—	4	2	no	—
164	D₈	—	2	2	no	—
165	D₈	—	4	1	no	—
166	M₁₆	—	20	2	no	—
167	M₁₆	—	40	1	no	—
168	C₂×C₈	—	40	1	no	—
169	C₂×C₈	#686	1	1	yes	characteristic
170	C₂×C₈	#687	1	1	yes	characteristic
171	C₂×C₈	#687	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	C₁₆	C₂²⋊Dic₅	1	1	yes	characteristic
176	C₁₆	C₂²⋊Dic₅	1	1	yes	characteristic
177	C₁₆	C₂²×Dic₅	1	1	yes	characteristic
178	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
179	C₂×C₅	—	2	1	no	—
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	—	16	2	no	—
182	C₂×C₅	#1417	1	2	yes	complemented
183	C₂×C₅	C₂×#357	1	1	yes	characteristic
184	C₂³	—	8	1	no	—
185	C₂³	#3959	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	8	1	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	D₄	—	8	2	no	—
191	D₄	—	8	1	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	#3958	1	1	yes	characteristic
196	C₂×C₄	#3958	1	1	yes	characteristic
197	C₂×C₄	#3959	1	1	yes	characteristic
198	C₂×C₄	#3964	1	1	yes	characteristic
199	C₂×C₄	#3964	1	1	yes	characteristic
200	C₂×C₄	C₂×#686	1	1	yes	characteristic
201	C₈	—	40	2	no	—
202	C₈	#3964	1	1	yes	characteristic
203	C₈	#3964	1	1	yes	characteristic
204	C₈	C₂×#687	1	1	yes	characteristic
205	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
206	C₅	C₂×#1417	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	2	2	no	—
208	C₂²	—	8	2	no	—
209	C₂²	—	16	2	no	—
210	C₂²	#64782	1	1	yes	characteristic
211	C₂²	#64782	1	1	yes	characteristic
212	C₂²	C₂×#3959	1	1	yes	characteristic
213	C₄	—	8	2	no	—
214	C₄	#64782	1	1	yes	characteristic
215	C₄	#64782	1	1	yes	characteristic
216	C₄	C₂×#3958	1	1	yes	characteristic
217	C₄	C₂×#3964	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	—	16	2	no	—
221	C₂	#10598320	1	2	yes	direct product factor
222	C₂	C₂×#64782	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3958
commutator	C₅×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅
Fitting, prime radical	C₂×C₅×#508	C₂
socle	C₂²×C₅	#357
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	C₂×#508	D₅
hypocenter	C₅	C₂×#1417
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
2. C₂×#64726	C₂	semidirect
3. C₂²×#3926	C₂	semidirect
4. #10598320	C₂	direct (N × H)
16. #64726	C₂²	semidirect
19. C₂×#3926	C₂²	semidirect
182. C₂×C₅	#1417	semidirect
206. C₅	C₂×#1417	semidirect
221. C₂	#10598320	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×#508	2
C₂×#64726	2
C₂²×#3926	2
#10598320	2
C₂×#1417	5

Subgroups — C2×#10598320 #12019751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598320 #12019751