Subgroups — #12014697

Order 1280.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic
2	#10602197	C₂	1	1	yes	maximal, characteristic
3	C₂×#64722	C₂	1	1	yes	maximal, characteristic
4	#10601495	C₂	1	2	yes	maximal
5	#10598220	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#108	C₂²	1	2	yes	—
8	#65411	—	2	2	no	—
9	#65395	C₂²	1	2	yes	—
10	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂²×#672	C₂²	1	1	yes	characteristic
12	C₅×#368	C₄	1	4	yes	complemented
13	#64775	—	2	2	no	—
14	#64725	C₄	1	2	yes	complemented
15	#64724	C₄	1	2	yes	complemented
16	#64722	C₄	1	4	yes	complemented
17	#12671	—	5	1	no	maximal, Sylow
18	C₂×C₅×M₁₆	—	2	2	no	—
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	#4072	—	2	2	no	—
21	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
22	C₂²×#168	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
27	C₅×#108	—	2	2	no	—
28	C₅×#108	C₂×C₄	1	4	yes	—
29	#3950	C₂×C₄	1	4	yes	—
30	#3949	—	2	2	no	—
31	#3947	—	4	2	no	—
32	C₈×Dic₅	—	4	2	no	—
33	C₈×Dic₅	C₂×C₄	1	2	yes	—
34	C₈×Dic₅	Q₈	1	2	yes	—
35	C₂×#672	—	2	1	no	—
36	C₂×#672	—	2	1	no	—
37	C₂×#672	C₂×C₄	1	2	yes	—
38	C₂×#368	D₅	1	1	yes	characteristic
39	#2158	—	5	1	no	—
40	C₂×#346	—	5	1	no	—
41	#1772	—	5	2	no	—
42	#1382	—	5	2	no	—
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅	—	2	1	no	—
45	C₅×M₁₆	—	2	4	no	—
46	C₅×M₁₆	—	4	2	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₅×C₈	#41	1	1	yes	characteristic
53	C₂×C₅×C₈	#41	1	1	yes	characteristic
54	C₂²⋊Dic₅	—	4	2	no	—
55	C₄⋊Dic₅	—	2	4	no	—
56	C₄×Dic₅	—	2	1	no	—
57	C₄×Dic₅	—	4	2	no	—
58	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
59	C₂×#168	C₄⋊C₄	1	2	yes	—
60	C₂×#168	D₈	1	2	yes	—
61	C₂×#168	Q₁₆	1	2	yes	—
62	C₅×C₁₆	C₄²	1	2	yes	—
63	C₅×C₁₆	C₄²	1	2	yes	—
64	#672	—	2	2	no	—
65	C₂²×C₁₆	—	5	1	no	—
66	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
67	C₂×#108	C₂×D₅	1	2	yes	—
68	#427	—	5	2	no	—
69	#405	—	10	2	no	—
70	C₂×C₄×C₈	—	5	1	no	—
71	#368	Dic₅	1	4	yes	complemented
72	#367	—	5	2	no	—
73	#366	—	5	2	no	—
74	#346	—	5	4	no	—
75	#339	—	10	2	no	—
76	C₂³×C₅	#135	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	#135	1	1	yes	characteristic
83	C₂×Dic₅	—	2	2	no	—
84	C₂×Dic₅	—	4	2	no	—
85	C₂×Dic₅	—	4	4	no	—
86	C₅×C₈	—	4	4	no	—
87	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
88	C₅×C₈	#117	1	1	yes	characteristic
89	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
90	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
91	#168	#107	1	4	yes	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	—	5	1	no	—
94	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
95	#117	—	10	2	no	—
96	C₂×C₄²	—	5	1	no	—
97	C₂×C₁₆	—	5	2	no	—
98	C₂×C₁₆	—	10	1	no	—
99	C₂×C₁₆	—	10	1	no	—
100	C₂×C₁₆	C₄×D₅	1	2	yes	—
101	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
103	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
104	#108	—	2	2	no	—
105	#108	C₂×Dic₅	1	4	yes	—
106	#107	—	5	4	no	—
107	#106	—	10	2	no	—
108	#97	—	20	2	no	—
109	C₄×C₈	—	5	2	no	—
110	C₄×C₈	—	5	2	no	—
111	C₄×C₈	—	20	2	no	—
112	C₂²×C₅	—	2	2	no	—
113	C₂²×C₅	#427	1	1	yes	characteristic
114	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
115	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
116	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₅	C₂×#107	1	1	yes	characteristic
118	C₄×C₅	C₄×D₈	1	1	yes	characteristic
119	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
120	Dic₅	—	4	2	no	—
121	Dic₅	—	8	4	no	—
122	C₂²×C₄	—	10	1	no	—
123	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	5	2	no	—
128	C₂×C₈	—	5	2	no	—
129	C₂×C₈	—	5	2	no	—
130	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
132	C₂×C₈	#711	1	1	yes	characteristic
133	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
134	C₂×C₈	#713	1	1	yes	characteristic
135	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
136	C₄⋊C₄	—	10	4	no	—
137	C₂²⋊C₄	—	20	2	no	—
138	C₄²	—	5	2	no	—
139	C₄²	—	10	1	no	—
140	C₄²	—	20	2	no	—
141	C₁₆	—	10	2	no	—
142	C₁₆	C₄×Dic₅	1	2	yes	—
143	C₁₆	C₄×Dic₅	1	2	yes	—
144	C₂×C₅	—	2	1	no	—
145	C₂×C₅	—	2	1	no	—
146	C₂×C₅	C₄×#107	1	1	yes	characteristic
147	C₂×C₅	#2179	1	2	yes	—
148	C₂³	#4065	1	1	yes	characteristic
149	C₂×C₄	—	10	2	no	—
150	C₂×C₄	—	20	2	no	—
151	C₂×C₄	—	20	4	no	—
152	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	#4039	1	1	yes	characteristic
154	C₂×C₄	D₅×D₈	1	1	yes	characteristic
155	C₂×C₄	#4058	1	1	yes	characteristic
156	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
157	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
158	C₈	—	4	4	no	—
159	C₈	—	5	4	no	—
160	C₈	#4033	1	1	yes	characteristic
161	C₈	#4038	1	1	yes	characteristic
162	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
163	C₈	Q₈×Dic₅	1	1	yes	characteristic
164	C₅	#12671	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	2	2	no	—
166	C₂²	#65161	1	1	yes	characteristic
167	C₂²	#65169	1	1	yes	characteristic
168	C₂²	Q₁₆×Dic₅	1	1	yes	characteristic
169	C₄	—	20	2	no	—
170	C₄	—	40	4	no	—
171	C₄	#65162	1	1	yes	characteristic
172	C₄	D₅×#107	1	1	yes	characteristic
173	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
174	C₄	D₈×Dic₅	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#10600151	1	2	yes	—
178	C₂	Dic₅×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#368	C₂
socle	C₂²×C₅	C₄×D₈
prime radical	C₂³×C₅	#135
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#368	D₅
hypocenter	C₅	#12671
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
12. C₅×#368	C₄	semidirect
14. #64725	C₄	semidirect
15. #64724	C₄	semidirect
16. #64722	C₄	semidirect
71. #368	Dic₅	semidirect
164. C₅	#12671	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#368	2
#10602197	2
C₂×#64722	2
#10601495	2
#10598220	2
#12671	5