Subgroups — C₂×#10600230 #12099447

Order 1280.

1006 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#168	C₂	1	2	yes	maximal
3	C₂×#64744	C₂	1	4	yes	maximal
4	#10600230	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄²×C₅	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂³×#168	C₂²	1	4	yes	—
8	C₅×#376	C₄	1	4	yes	—
9	C₅×#376	C₂²	1	4	yes	—
10	C₂×C₅×#95	C₄	1	4	yes	—
11	C₂×C₄⋊#168	—	2	8	no	—
12	C₂×C₄⋊#168	C₂²	1	8	yes	—
13	#64744	C₂²	1	16	yes	—
14	C₂×#1832	—	5	1	no	maximal, Sylow
15	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	—	2	8	no	—
19	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅	D₄	1	4	yes	—
22	C₂×C₄²×C₅	Q₈	1	4	yes	—
23	C₂×C₄²×C₅	C₂³	1	4	yes	—
24	C₂×C₄²×C₅	C₂³	1	2	yes	—
25	C₂²×#168	—	2	8	no	—
26	C₂²×#168	—	2	16	no	—
27	C₂²×#168	C₂³	1	8	yes	—
28	C₅×#95	C₂×C₄	1	16	yes	—
29	C₄⋊#168	—	2	16	no	—
30	C₂²×C₄⋊C₈	—	5	2	no	—
31	C₂×#376	D₅	1	1	yes	characteristic
32	#1832	—	5	8	no	—
33	C₂×#336	—	5	4	no	—
34	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	—	2	8	no	—
36	C₂²×C₄×C₅	—	2	8	no	—
37	C₂²×C₄×C₅	—	2	16	no	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
40	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₅	#41	1	8	yes	—
45	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
46	C₄²×C₅	—	2	8	no	—
47	C₄²×C₅	—	2	8	no	—
48	C₄²×C₅	C₄⋊C₄	1	8	yes	—
49	C₄²×C₅	C₂×D₄	1	4	yes	—
50	C₄²×C₅	C₂×Q₈	1	4	yes	—
51	C₂×#168	—	2	8	no	—
52	C₂×#168	—	2	16	no	—
53	C₂×#168	—	4	16	no	—
54	C₂³×C₈	—	5	4	no	—
55	C₂²×C₄²	C₂×D₅	1	2	yes	—
56	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	5	8	no	—
58	C₂×C₄⋊C₈	—	10	8	no	—
59	#376	Dic₅	1	4	yes	—
60	#376	C₂×D₅	1	4	yes	—
61	C₂×#95	Dic₅	1	4	yes	—
62	#336	—	5	16	no	—
63	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
64	C₂³×C₅	#117	1	4	yes	—
65	C₂³×C₅	#131	1	8	yes	—
66	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	8	no	—
68	C₂×C₄×C₅	—	2	16	no	—
69	C₂×C₄×C₅	—	2	8	no	—
70	C₂×C₄×C₅	—	2	16	no	—
71	C₂×C₄×C₅	—	4	16	no	—
72	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
73	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₅	#117	1	4	yes	—
75	C₂×C₄×C₅	#124	1	8	yes	—
76	C₂×C₄×C₅	#125	1	8	yes	—
77	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×#41	1	4	yes	—
80	#168	—	4	8	no	—
81	C₂³×C₄	C₂×Dic₅	1	4	yes	—
82	C₂³×C₄	C₂×Dic₅	1	2	yes	—
83	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
84	C₂²×C₈	—	5	8	no	—
85	C₂²×C₈	—	10	8	no	—
86	C₂²×C₈	—	10	16	no	—
87	C₂×C₄²	—	2	8	no	—
88	C₂×C₄²	Dic₁₀	1	4	yes	—
89	C₂×C₄²	D₂₀	1	4	yes	—
90	C₂×C₄²	C₂×Dic₅	1	4	yes	—
91	C₂×C₄²	C₂×Dic₅	1	2	yes	—
92	C₂×C₄²	C₂²×D₅	1	4	yes	—
93	C₂×C₄²	C₂²×D₅	1	2	yes	—
94	C₄⋊C₈	—	10	16	no	—
95	#95	C₂×Dic₅	1	16	yes	—
96	C₂²×C₅	#382	1	4	yes	—
97	C₂²×C₅	#424	1	8	yes	—
98	C₂²×C₅	#433	1	16	yes	—
99	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₅	C₂×#117	1	2	yes	—
101	C₂²×C₅	C₂×#131	1	4	yes	—
102	C₄×C₅	—	2	8	no	—
103	C₄×C₅	—	4	8	no	—
104	C₄×C₅	#382	1	4	yes	—
105	C₄×C₅	C₂×#124	1	2	yes	—
106	C₄×C₅	C₂×#125	1	2	yes	—
107	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
108	C₂²×C₄	—	2	8	no	—
109	C₂²×C₄	—	2	8	no	—
110	C₂²×C₄	—	2	16	no	—
111	C₂²×C₄	C₄⋊Dic₅	1	8	yes	—
112	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
113	C₂²×C₄	C₂×D₂₀	1	2	yes	—
114	C₂²×C₄	#709	1	8	yes	—
115	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
116	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
117	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
118	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
119	C₂×C₈	—	10	8	no	—
120	C₂×C₈	—	10	16	no	—
121	C₂×C₈	—	20	16	no	—
122	C₄²	—	2	8	no	—
123	C₄²	—	2	8	no	—
124	C₄²	C₄⋊Dic₅	1	8	yes	—
125	C₄²	C₂×Dic₁₀	1	4	yes	—
126	C₄²	C₂×D₂₀	1	4	yes	—
127	C₂×C₅	#1832	1	8	yes	complemented
128	C₂×C₅	C₂×#382	1	1	yes	characteristic
129	C₂×C₅	C₂×#424	1	2	yes	—
130	C₂×C₅	C₂×#433	1	4	yes	—
131	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
132	C₂³	#4072	1	4	yes	—
133	C₂³	#4094	1	8	yes	—
134	C₂³	C₂³×Dic₅	1	1	yes	characteristic
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	—	4	16	no	—
140	C₂×C₄	#4016	1	8	yes	—
141	C₂×C₄	#4021	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₅	1	8	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
144	C₂×C₄	#4072	1	4	yes	—
145	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
146	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
147	C₂×C₄	C₂×#709	1	4	yes	—
148	C₈	—	20	8	no	—
149	C₅	C₂×#1832	1	1	yes	characteristic, complemented, Sylow
150	C₂²	#65225	1	4	yes	—
151	C₂²	#65288	1	8	yes	—
152	C₂²	#65289	1	16	yes	—
153	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
154	C₂²	C₂×#4072	1	2	yes	—
155	C₂²	C₂×#4094	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	4	8	no	—
158	C₄	#65225	1	4	yes	—
159	C₄	C₂×#4016	1	2	yes	—
160	C₄	C₂×#4021	1	2	yes	—
161	C₂	#10600230	1	8	yes	direct product factor
162	C₂	C₂×#65225	1	1	yes	characteristic
163	C₂	C₂×#65288	1	2	yes	—
164	C₂	C₂×#65289	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂³×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#376	C₂
socle, prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#376	D₅
hypocenter	C₅	C₂×#1832
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
4. #10600230	C₂	direct (N × H)
127. C₂×C₅	#1832	semidirect
149. C₅	C₂×#1832	semidirect
161. C₂	#10600230	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×#376	2
C₂²×C₄⋊#168	2
C₂×#64744	2
#10600230	2
C₂×#1832	5

Subgroups — C2×#10600230 #12099447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600230 #12099447