Subgroups — #12252302

Order 1280.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3407	C₂	1	1	yes	maximal, characteristic
2	Dic₅×#131	C₂	1	1	yes	maximal, characteristic
3	#10615511	C₂	1	1	yes	maximal, characteristic
4	#10602216	C₂	1	6	yes	maximal, complemented
5	C₂×#64731	C₂	1	6	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	#41×Dic₅	C₄	1	1	yes	characteristic
8	C₂×#4094	C₄	1	1	yes	characteristic
9	C₅×#504	—	2	4	no	—
10	C₂×C₅×M₃₂	C₂²	1	6	yes	—
11	#65429	—	2	4	no	—
12	#65411	C₄	1	3	yes	—
13	M₁₆×Dic₅	C₄	1	3	yes	—
14	#65390	C₂²	1	3	yes	—
15	C₂×C₈×Dic₅	C₂²	1	3	yes	—
16	C₂×#3944	C₂²	1	6	yes	—
17	#64731	C₂²	1	4	yes	—
18	#64731	C₂²	1	12	yes	—
19	#33401	—	5	1	no	maximal, Sylow
20	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
21	C₅×#131	D₄	1	4	yes	—
22	C₅×#131	Q₈	1	4	yes	—
23	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
24	C₂²×C₅×C₈	C₂³	1	3	yes	—
25	#4094	C₈	1	8	yes	—
26	Q₈×Dic₅	—	2	1	no	—
27	D₄×Dic₅	—	2	3	no	—
28	#4072	C₂×C₄	1	3	yes	—
29	C₂×C₄×Dic₅	C₂×C₄	1	3	yes	—
30	C₂×#681	C₂×C₄	1	3	yes	—
31	C₂²×#168	C₂×C₄	1	3	yes	—
32	C₅×M₃₂	—	2	6	no	—
33	C₅×M₃₂	C₂³	1	4	yes	—
34	C₂×C₅×C₁₆	—	2	6	no	—
35	#3947	C₂×C₄	1	6	yes	—
36	C₈×Dic₅	C₂×C₄	1	3	yes	—
37	C₈×Dic₅	C₂×C₄	1	3	yes	—
38	C₈×Dic₅	C₂³	1	1	yes	characteristic
39	C₈×Dic₅	C₂³	1	3	yes	—
40	#3944	—	2	6	no	—
41	#3944	C₂³	1	4	yes	—
42	C₂×#672	—	2	6	no	—
43	#3407	—	5	1	no	—
44	#3407	D₅	1	1	yes	characteristic
45	C₄×#131	—	5	1	no	—
46	#2153	—	5	6	no	—
47	C₂×#364	—	5	6	no	—
48	C₅×#41	C₄⋊C₄	1	4	yes	—
49	C₅×#41	C₂×C₈	1	4	yes	—
50	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
51	C₂×C₅×D₄	M₁₆	1	3	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
53	C₂²×Dic₅	—	2	3	no	—
54	C₅×M₁₆	C₄⋊C₄	1	12	yes	—
55	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
56	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
57	C₂×C₅×C₈	C₂×Q₈	1	6	yes	—
58	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
59	C₂²⋊Dic₅	—	2	3	no	—
60	C₄⋊Dic₅	—	2	3	no	—
61	C₄×Dic₅	—	2	3	no	—
62	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₅	C₂²×C₄	1	3	yes	—
64	#681	C₂×C₈	1	12	yes	—
65	C₂×#168	C₂×C₈	1	12	yes	—
66	C₂×#168	C₂²×C₄	1	1	yes	characteristic
67	C₂×#168	C₂²×C₄	1	3	yes	—
68	C₅×C₁₆	—	2	4	no	—
69	#672	—	2	4	no	—
70	C₂×#131	—	5	1	no	—
71	C₂×#131	C₂×D₅	1	1	yes	characteristic
72	C₄×#41	—	5	1	no	—
73	#504	—	2	4	no	—
74	#504	—	10	4	no	—
75	C₂×M₃₂	—	5	6	no	—
76	C₂×M₃₂	C₂×D₅	1	6	yes	—
77	#405	—	5	3	no	—
78	#405	—	5	3	no	—
79	C₄×M₁₆	—	5	3	no	—
80	C₂×C₄×C₈	—	5	3	no	—
81	#364	—	5	4	no	—
82	#364	—	5	12	no	—
83	C₂³×C₅	C₂×M₁₆	1	3	yes	—
84	C₅×Q₈	C₄⋊C₈	1	4	yes	—
85	C₅×D₄	C₄⋊C₈	1	12	yes	—
86	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
87	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
88	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
89	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₅	—	2	1	no	—
91	C₂×Dic₅	—	2	3	no	—
92	C₂×Dic₅	—	4	6	no	—
93	C₅×C₈	C₂×C₄⋊C₄	1	6	yes	—
94	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
95	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
96	#168	C₂²×C₈	1	2	yes	—
97	#168	C₂²×C₈	1	6	yes	—
98	C₂×#41	C₄×D₅	1	1	yes	characteristic
99	#131	—	5	8	no	—
100	#131	Dic₁₀	1	4	yes	—
101	#131	C₂²⋊D₅	1	4	yes	—
102	C₂×M₁₆	—	5	3	no	—
103	C₂×M₁₆	C₄×D₅	1	3	yes	—
104	C₂²×C₈	—	5	3	no	—
105	C₂²×C₈	C₂²×D₅	1	3	yes	—
106	C₄×Q₈	—	10	1	no	—
107	C₄×D₄	—	10	3	no	—
108	#117	—	5	3	no	—
109	C₂×C₄²	—	5	3	no	—
110	M₃₂	—	2	6	no	—
111	M₃₂	—	5	4	no	—
112	M₃₂	—	10	6	no	—
113	M₃₂	C₂²×D₅	1	4	yes	—
114	C₂×C₁₆	—	2	6	no	—
115	C₂×C₁₆	—	10	6	no	—
116	#97	—	5	6	no	—
117	C₄×C₈	—	5	3	no	—
118	C₄×C₈	—	5	3	no	—
119	C₄×C₈	—	5	1	no	—
120	C₄×C₈	—	5	3	no	—
121	C₂²×C₅	—	4	3	no	—
122	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
123	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
124	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
125	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
127	Dic₅	—	2	2	no	—
128	Dic₅	—	4	3	no	—
129	#41	C₈×D₅	1	4	yes	—
130	#41	#683	1	4	yes	—
131	C₂×Q₈	#676	1	1	yes	characteristic
132	C₂×D₄	#676	1	3	yes	—
133	C₂²×C₄	—	10	3	no	—
134	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
135	M₁₆	—	5	12	no	—
136	M₁₆	#683	1	12	yes	—
137	C₂×C₈	—	5	1	no	—
138	C₂×C₈	—	5	3	no	—
139	C₂×C₈	—	5	12	no	—
140	C₂×C₈	C₂×Dic₁₀	1	6	yes	—
141	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
142	C₂×C₈	C₂×C₂²⋊D₅	1	6	yes	—
143	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
144	C₄⋊C₄	—	10	3	no	—
145	C₂²⋊C₄	—	10	3	no	—
146	C₄²	—	5	1	no	—
147	C₄²	—	5	3	no	—
148	C₄²	—	10	3	no	—
149	C₁₆	—	2	4	no	—
150	C₁₆	—	10	4	no	—
151	C₂×C₅	—	2	1	no	—
152	C₂×C₅	—	2	6	no	—
153	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
154	C₂³	C₂×#676	1	3	yes	—
155	Q₈	#3946	1	4	yes	—
156	D₄	#3946	1	12	yes	—
157	C₂×C₄	—	10	1	no	—
158	C₂×C₄	—	10	3	no	—
159	C₂×C₄	—	20	6	no	—
160	C₂×C₄	C₂×C₈×D₅	1	6	yes	—
161	C₂×C₄	C₂×#676	1	3	yes	—
162	C₂×C₄	C₂×#683	1	6	yes	—
163	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
164	C₈	—	5	2	no	—
165	C₈	—	5	6	no	—
166	C₈	C₂×#683	1	6	yes	—
167	C₈	C₂²×Dic₁₀	1	1	yes	characteristic
168	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
169	C₅	#33401	1	1	yes	characteristic, complemented, Sylow
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#3946	1	6	yes	—
172	C₂²	C₂²×#676	1	1	yes	characteristic
173	C₄	—	10	2	no	—
174	C₄	—	20	3	no	—
175	C₄	C₂×#3946	1	6	yes	—
176	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
177	C₄	C₂²×#683	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#3946	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#3407	C₂
socle	C₂×C₅	C₂²×C₄⋊C₈
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#3407	D₅
hypocenter	C₅	#33401
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ #3407

Complements

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

N	H ≅ G/N	type
4. #10602216	C₂	semidirect
5. C₂×#64731	C₂	semidirect
169. C₅	#33401	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3407	2
Dic₅×#131	2
#10615511	2
#10602216	2
C₂×#64731	2
#33401	5