Subgroups — C₂×#10598299 #12014187

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 212 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	C₂²×#3951	C₂	1	2	yes	maximal
3	#10598299	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂²×#681	—	2	2	no	—
6	C₂²×C₅×C₁₆	—	2	4	no	—
7	C₂²×C₅×C₁₆	C₄	1	2	yes	—
8	C₂²×C₅×C₁₆	C₄	1	2	yes	—
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×#3951	—	2	2	no	—
11	C₂×#3951	—	2	4	no	—
12	C₂×#3951	C₄	1	8	yes	—
13	C₂×#3951	C₂²	1	4	yes	—
14	C₂×#1381	—	5	1	no	maximal, Sylow
15	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	D₄	1	4	yes	—
21	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₂×#681	—	2	8	no	—
25	C₂×#681	—	2	4	no	—
26	C₂×#681	—	4	2	no	—
27	C₂×#681	—	4	4	no	—
28	C₂²×#168	—	4	2	no	—
29	C₂×C₅×C₁₆	—	2	4	no	—
30	C₂×C₅×C₁₆	—	2	4	no	—
31	C₂×C₅×C₁₆	—	2	4	no	—
32	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
33	#3951	—	2	8	no	—
34	#3951	—	2	4	no	—
35	#3951	C₂×C₄	1	8	yes	—
36	C₂³×C₁₆	D₅	1	1	yes	characteristic
37	C₂²×#108	—	5	2	no	—
38	#1381	—	5	4	no	—
39	C₂⁴×C₅	Q₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₄²	1	2	yes	—
49	C₂×C₅×C₈	C₄²	1	2	yes	—
50	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
51	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
52	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
53	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
54	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
55	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
58	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
61	#681	—	2	8	no	—
62	#681	—	4	8	no	—
63	#681	—	4	4	no	—
64	C₂×#168	—	4	8	no	—
65	C₂×#168	—	4	4	no	—
66	C₅×C₁₆	—	2	4	no	—
67	C₂²×M₁₆	—	10	2	no	—
68	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
69	C₂²×C₁₆	—	2	4	no	—
70	C₂²×C₁₆	Dic₅	1	2	yes	—
71	C₂²×C₁₆	Dic₅	1	2	yes	—
72	C₂²×C₁₆	C₂×D₅	1	2	yes	—
73	C₂×#108	—	5	8	no	—
74	C₂×#108	—	5	4	no	—
75	C₂×#108	—	10	2	no	—
76	C₂×#108	—	10	4	no	—
77	C₂³×C₅	—	2	4	no	—
78	C₂³×C₅	#103	1	4	yes	—
79	C₂³×C₅	#107	1	1	yes	characteristic
80	C₂³×C₅	#107	1	1	yes	characteristic
81	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	#95	1	2	yes	—
83	C₂×C₄×C₅	#95	1	2	yes	—
84	C₂×C₄×C₅	#102	1	4	yes	—
85	C₂×C₄×C₅	#102	1	4	yes	—
86	C₂×C₄×C₅	#103	1	4	yes	—
87	C₂×C₄×C₅	#106	1	2	yes	—
88	C₂×C₄×C₅	#106	1	2	yes	—
89	C₂×C₄×C₅	#107	1	1	yes	characteristic
90	C₂×C₄×C₅	#107	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
95	C₅×C₈	#95	1	2	yes	—
96	C₅×C₈	#95	1	2	yes	—
97	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
98	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
100	#168	—	4	8	no	—
101	C₂³×C₄	D₂₀	1	1	yes	characteristic
102	C₂×M₁₆	—	10	8	no	—
103	C₂×M₁₆	—	10	4	no	—
104	C₂×M₁₆	—	20	4	no	—
105	C₂×M₁₆	—	20	2	no	—
106	C₂²×C₈	—	20	2	no	—
107	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
108	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
109	C₂²×C₈	C₄×D₅	1	4	yes	—
110	C₂²×C₈	D₂₀	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₅	1	4	yes	—
114	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
119	#108	—	5	8	no	—
120	#108	—	10	8	no	—
121	#108	—	10	4	no	—
122	C₂²×C₅	—	2	8	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	#340	1	2	yes	—
126	C₂²×C₅	#340	1	2	yes	—
127	C₂²×C₅	#368	1	4	yes	—
128	C₂²×C₅	C₂×#103	1	2	yes	—
129	C₂²×C₅	C₂×#107	1	1	yes	characteristic
130	C₄×C₅	#340	1	2	yes	—
131	C₄×C₅	#340	1	2	yes	—
132	C₄×C₅	C₂×#95	1	1	yes	characteristic
133	C₄×C₅	C₂×#102	1	2	yes	—
134	C₄×C₅	C₂×#106	1	1	yes	characteristic
135	C₂⁴	Dic₂₀	1	1	yes	characteristic
136	C₂²×C₄	#677	1	4	yes	—
137	C₂²×C₄	D₄₀	1	1	yes	characteristic
138	C₂²×C₄	D₄₀	1	1	yes	characteristic
139	C₂²×C₄	Dic₂₀	1	1	yes	characteristic
140	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
141	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
142	C₂²×C₄	#685	1	4	yes	—
143	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
144	M₁₆	—	10	8	no	—
145	M₁₆	—	20	8	no	—
146	M₁₆	—	20	4	no	—
147	C₂×C₈	—	20	8	no	—
148	C₂×C₈	—	20	4	no	—
149	C₂×C₈	C₄×Dic₅	1	2	yes	—
150	C₂×C₈	C₄×Dic₅	1	2	yes	—
151	C₂×C₈	#683	1	4	yes	—
152	C₂×C₈	#683	1	4	yes	—
153	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
154	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
155	C₂×C₈	#685	1	4	yes	—
156	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
157	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
158	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
160	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
161	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂×C₅	—	2	2	no	—
164	C₂×C₅	—	2	2	no	—
165	C₂×C₅	#1381	1	4	yes	complemented
166	C₂×C₅	C₂×#340	1	1	yes	characteristic
167	C₂×C₅	C₂×#368	1	2	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	#3948	1	4	yes	—
170	C₂³	#3950	1	1	yes	characteristic
171	C₂³	#3950	1	1	yes	characteristic
172	C₂³	C₂×Dic₂₀	1	1	yes	characteristic
173	C₂×C₄	#3948	1	4	yes	—
174	C₂×C₄	#3949	1	2	yes	—
175	C₂×C₄	#3949	1	2	yes	—
176	C₂×C₄	#3950	1	1	yes	characteristic
177	C₂×C₄	#3950	1	1	yes	characteristic
178	C₂×C₄	#3953	1	4	yes	—
179	C₂×C₄	#3953	1	4	yes	—
180	C₂×C₄	#3963	1	2	yes	—
181	C₂×C₄	#3963	1	2	yes	—
182	C₂×C₄	C₂×#677	1	2	yes	—
183	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
185	C₂×C₄	C₂×#685	1	2	yes	—
186	C₈	—	20	8	no	—
187	C₈	#3963	1	2	yes	—
188	C₈	#3963	1	2	yes	—
189	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
190	C₈	C₂×#683	1	2	yes	—
191	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
192	C₅	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#64726	1	4	yes	—
197	C₂²	#64771	1	2	yes	—
198	C₂²	#64771	1	2	yes	—
199	C₂²	C₂×#3948	1	2	yes	—
200	C₂²	C₂×#3950	1	1	yes	characteristic
201	C₄	#64771	1	2	yes	—
202	C₄	#64771	1	2	yes	—
203	C₄	C₂×#3949	1	1	yes	characteristic
204	C₄	C₂×#3953	1	2	yes	—
205	C₄	C₂×#3963	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#10598299	1	4	yes	direct product factor
209	C₂	C₂×#64726	1	2	yes	—
210	C₂	C₂×#64771	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄₀
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂³×C₅×C₁₆	C₂
socle	C₂³×C₅	#107
prime radical	C₂⁴×C₅	Q₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	C₂×#1381
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #10598299	C₂	direct (N × H)
165. C₂×C₅	#1381	semidirect
192. C₅	C₂×#1381	semidirect
208. C₂	#10598299	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
C₂²×#3951	2
#10598299	2
C₂×#1381	5

Subgroups — C2×#10598299 #12014187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598299 #12014187