Subgroups — #12014622

Order 1280.

358 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	C₅×#1726	C₂	1	1	yes	maximal, characteristic
2	C₂×#64683	C₂	1	2	yes	maximal
3	C₄×#3944	C₂	1	2	yes	maximal
4	#10598149	C₂	1	2	yes	maximal
5	C₂×C₅×#97	C₄	1	1	yes	characteristic
6	C₂×C₅×#97	C₄	1	1	yes	characteristic
7	C₂×C₅×#97	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×#376	C₄	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	4	yes	—
13	#64683	C₄	1	8	yes	complemented
14	#64681	—	2	2	no	—
15	C₄×#672	—	2	2	no	—
16	#12656	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×#97	—	2	2	no	—
25	C₅×#97	C₈	1	4	yes	—
26	C₅×#97	C₈	1	4	yes	—
27	C₄×C₅×C₈	—	2	2	no	—
28	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₄×C₅×C₈	D₄	1	2	yes	—
30	C₄×C₅×C₈	Q₈	1	2	yes	—
31	C₅×#95	—	2	2	no	—
32	#3944	—	2	4	no	—
33	#3944	C₂×C₄	1	8	yes	—
34	C₂×#672	—	2	4	no	—
35	C₂×#364	—	5	2	no	—
36	C₄×M₃₂	—	5	2	no	—
37	#1726	D₅	1	1	yes	characteristic
38	#1376	—	5	2	no	—
39	C₂²×C₄×C₅	—	2	1	no	—
40	C₂²×C₄×C₅	—	2	1	no	—
41	C₂²×C₄×C₅	M₁₆	1	2	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	—	2	1	no	—
44	C₂×C₅×C₈	—	2	1	no	—
45	C₂×C₅×C₈	C₄²	1	2	yes	—
46	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
48	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
49	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
50	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
51	C₂×C₅×C₈	M₁₆	1	2	yes	—
52	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
53	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₅×C₈	#41	1	2	yes	—
56	C₄²×C₅	—	2	1	no	—
57	C₄²×C₅	—	2	1	no	—
58	C₄²×C₅	C₄²	1	2	yes	—
59	C₄²×C₅	C₄⋊C₄	1	2	yes	—
60	C₄²×C₅	C₂×C₈	1	2	yes	—
61	C₄²×C₅	C₂×C₈	1	2	yes	—
62	#672	—	2	8	no	—
63	C₂×M₃₂	—	5	4	no	—
64	C₂×#97	Dic₅	1	1	yes	characteristic
65	C₂×#97	Dic₅	1	1	yes	characteristic
66	C₂×#97	Dic₅	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
70	#376	Dic₅	1	1	yes	characteristic
71	#364	—	5	8	no	—
72	#346	—	10	2	no	—
73	C₄×C₁₆	—	10	2	no	—
74	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	4	2	no	—
78	C₂×C₄×C₅	—	4	2	no	—
79	C₂×C₄×C₅	—	4	2	no	—
80	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅	#117	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	C₄×C₈	1	4	yes	—
89	C₅×C₈	C₄⋊C₈	1	4	yes	—
90	C₅×C₈	C₄×D₄	1	2	yes	—
91	C₅×C₈	C₄×Q₈	1	2	yes	—
92	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
98	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
99	M₃₂	—	5	8	no	—
100	M₃₂	—	10	4	no	—
101	C₂×C₁₆	—	10	4	no	—
102	#97	—	2	2	no	—
103	#97	#168	1	4	yes	—
104	#97	#168	1	4	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	Dic₁₀	1	2	yes	—
107	C₄×C₈	C₄×D₅	1	4	yes	—
108	C₄×C₈	D₂₀	1	2	yes	—
109	#95	—	2	2	no	—
110	C₂²×C₅	—	2	2	no	—
111	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
112	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂²×C₅	#431	1	1	yes	characteristic
114	C₄×C₅	—	2	4	no	—
115	C₄×C₅	—	4	2	no	—
116	C₄×C₅	—	4	2	no	—
117	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
119	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₄×C₅	#431	1	1	yes	characteristic
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	#681	1	2	yes	—
124	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	C₂×#168	1	2	yes	—
128	C₂×C₈	C₂×#168	1	2	yes	—
129	C₂×C₈	C₂×#168	1	2	yes	—
130	C₂×C₈	C₂×#168	1	2	yes	—
131	C₂×C₈	#681	1	2	yes	—
132	C₂×C₈	C₄×Dic₅	1	2	yes	—
133	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
134	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
136	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
137	C₂×C₈	#709	1	2	yes	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	C₂×#168	1	2	yes	—
141	C₄²	C₂×#168	1	2	yes	—
142	C₄²	C₄×Dic₅	1	2	yes	—
143	C₄²	C₄⋊Dic₅	1	2	yes	—
144	C₁₆	—	10	8	no	—
145	C₂×C₅	—	2	1	no	—
146	C₂×C₅	—	2	1	no	—
147	C₂×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₅	#2153	1	2	yes	—
149	C₂³	C₂×#681	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₄⋊#168	1	4	yes	—
156	C₂×C₄	C₂²×#168	1	1	yes	characteristic
157	C₂×C₄	C₂²×#168	1	1	yes	characteristic
158	C₂×C₄	C₂×#681	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
161	C₂×C₄	#4072	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	C₄×#168	1	4	yes	—
164	C₈	C₄⋊#168	1	4	yes	—
165	C₈	C₄×Dic₁₀	1	2	yes	—
166	C₈	C₄×D₂₀	1	2	yes	—
167	C₅	#12656	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	2	no	—
169	C₂²	C₄×#681	1	1	yes	characteristic
170	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
171	C₂²	#65214	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₂×C₄×#168	1	1	yes	characteristic
176	C₄	C₂×C₄⋊#168	1	1	yes	characteristic
177	C₄	#65214	1	1	yes	characteristic
178	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×C₄⋊#168	1	1	yes	characteristic
182	C₂	#10600678	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1726	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1726	D₅
hypocenter	C₅	#12656
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
13. #64683	C₄	semidirect
167. C₅	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1726	2
C₂×#64683	2
C₄×#3944	2
#10598149	2
#12656	5