Subgroups — #12014692

Order 1280.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 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×C₁₆×Dic₅	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	C₁₆×Dic₅	C₄	1	4	yes	complemented
17	#12670	—	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×C₄×C₁₆	—	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	1	yes	characteristic
61	C₂×#168	D₈	1	1	yes	characteristic
62	C₂×#168	Q₁₆	1	2	yes	—
63	C₅×C₁₆	C₄²	1	2	yes	—
64	C₅×C₁₆	C₄²	1	2	yes	—
65	#672	—	2	2	no	—
66	C₂²×C₁₆	—	5	1	no	—
67	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
68	C₂×#108	C₂×D₅	1	2	yes	—
69	#427	—	5	2	no	—
70	#405	—	10	2	no	—
71	C₂×C₄×C₈	—	5	1	no	—
72	#368	Dic₅	1	4	yes	complemented
73	#367	—	5	2	no	—
74	#366	—	5	2	no	—
75	C₄×C₁₆	—	5	4	no	—
76	#339	—	10	2	no	—
77	C₂³×C₅	#135	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅	#135	1	1	yes	characteristic
84	C₂×Dic₅	—	2	4	no	—
85	C₂×Dic₅	—	2	2	no	—
86	C₂×Dic₅	—	4	4	no	—
87	C₅×C₈	—	4	4	no	—
88	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
89	C₅×C₈	#117	1	1	yes	characteristic
90	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
91	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
92	#168	#107	1	2	yes	—
93	#168	#107	1	2	yes	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂²×C₈	—	5	1	no	—
96	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
97	#117	—	10	2	no	—
98	C₂×C₄²	—	5	1	no	—
99	C₂×C₁₆	—	5	2	no	—
100	C₂×C₁₆	—	10	1	no	—
101	C₂×C₁₆	—	10	1	no	—
102	C₂×C₁₆	C₄×D₅	1	2	yes	—
103	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
104	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
105	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
106	#108	—	2	2	no	—
107	#108	C₂×Dic₅	1	4	yes	—
108	#107	—	5	4	no	—
109	#106	—	10	2	no	—
110	#97	—	20	2	no	—
111	C₄×C₈	—	5	2	no	—
112	C₄×C₈	—	5	2	no	—
113	C₄×C₈	—	20	2	no	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	#427	1	1	yes	characteristic
116	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
117	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₅	C₂×#107	1	1	yes	characteristic
120	C₄×C₅	C₄×D₈	1	1	yes	characteristic
121	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
122	Dic₅	—	2	4	no	—
123	Dic₅	—	8	4	no	—
124	C₂²×C₄	—	10	1	no	—
125	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	5	2	no	—
130	C₂×C₈	—	5	1	no	—
131	C₂×C₈	—	5	1	no	—
132	C₂×C₈	—	5	2	no	—
133	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
134	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
135	C₂×C₈	#711	1	1	yes	characteristic
136	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
137	C₂×C₈	#713	1	1	yes	characteristic
138	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
139	C₄⋊C₄	—	10	4	no	—
140	C₂²⋊C₄	—	20	2	no	—
141	C₄²	—	5	2	no	—
142	C₄²	—	10	1	no	—
143	C₄²	—	20	2	no	—
144	C₁₆	—	10	2	no	—
145	C₁₆	C₄×Dic₅	1	2	yes	—
146	C₁₆	C₄×Dic₅	1	2	yes	—
147	C₂×C₅	—	2	1	no	—
148	C₂×C₅	—	2	1	no	—
149	C₂×C₅	C₄×#107	1	1	yes	characteristic
150	C₂×C₅	#2178	1	2	yes	—
151	C₂³	#4065	1	1	yes	characteristic
152	C₂×C₄	—	10	4	no	—
153	C₂×C₄	—	10	2	no	—
154	C₂×C₄	—	20	4	no	—
155	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄	#4039	1	1	yes	characteristic
157	C₂×C₄	D₅×D₈	1	1	yes	characteristic
158	C₂×C₄	#4058	1	1	yes	characteristic
159	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
160	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
161	C₈	—	4	4	no	—
162	C₈	—	5	2	no	—
163	C₈	—	5	2	no	—
164	C₈	#4033	1	1	yes	characteristic
165	C₈	#4038	1	1	yes	characteristic
166	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
167	C₈	Q₈×Dic₅	1	1	yes	characteristic
168	C₅	#12670	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	2	2	no	—
170	C₂²	#65161	1	1	yes	characteristic
171	C₂²	#65169	1	1	yes	characteristic
172	C₂²	Q₁₆×Dic₅	1	1	yes	characteristic
173	C₄	—	10	4	no	—
174	C₄	—	40	4	no	—
175	C₄	#65162	1	1	yes	characteristic
176	C₄	D₅×#107	1	1	yes	characteristic
177	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
178	C₄	D₈×Dic₅	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#10600150	1	2	yes	—
182	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	#12670
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. C₁₆×Dic₅	C₄	semidirect
72. #368	Dic₅	semidirect
168. C₅	#12670	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×C₁₆×Dic₅	2
#10601495	2
#10598220	2
#12670	5