Subgroups — #12014626

Order 1280.

334 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×#1726	C₂	1	1	yes	maximal, characteristic
2	C₂×#64686	C₂	1	2	yes	maximal
3	#10600536	C₂	1	2	yes	maximal
4	#10598154	C₂	1	2	yes	maximal
5	C₂×C₅×#97	C₄	1	1	yes	characteristic
6	C₂×C₅×#97	C₂²	1	1	yes	characteristic
7	C₂×C₅×#97	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₄	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	#64686	—	2	4	no	—
14	C₄⋊#672	C₄	1	8	yes	complemented
15	#12657	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	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	C₅×#97	—	2	2	no	—
24	C₅×#97	—	2	2	no	—
25	C₅×#97	D₄	1	2	yes	—
26	C₅×#97	Q₈	1	2	yes	—
27	C₄×C₅×C₈	—	2	2	no	—
28	C₄×C₅×C₈	—	2	2	no	—
29	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
30	C₅×#95	—	2	2	no	—
31	#3944	—	2	4	no	—
32	#3944	—	2	4	no	—
33	C₂×#672	C₂×C₄	1	8	yes	—
34	#2150	—	5	2	no	—
35	C₂×#347	—	5	2	no	—
36	#1726	D₅	1	1	yes	characteristic
37	#1375	—	5	2	no	—
38	C₂²×C₄×C₅	—	2	1	no	—
39	C₂²×C₄×C₅	—	2	1	no	—
40	C₂²×C₄×C₅	M₁₆	1	2	yes	—
41	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	—	2	1	no	—
43	C₂×C₅×C₈	—	2	1	no	—
44	C₂×C₅×C₈	—	2	1	no	—
45	C₂×C₅×C₈	—	2	1	no	—
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	C₄²	1	2	yes	—
49	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₅×C₈	M₁₆	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
52	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₅×C₈	#41	1	2	yes	—
55	C₄²×C₅	—	2	1	no	—
56	C₄²×C₅	—	2	1	no	—
57	C₄²×C₅	C₄²	1	2	yes	—
58	C₄²×C₅	C₄⋊C₄	1	2	yes	—
59	C₄²×C₅	M₁₆	1	2	yes	—
60	C₄²×C₅	M₁₆	1	2	yes	—
61	#672	—	2	8	no	—
62	C₂×M₃₂	—	5	4	no	—
63	C₂×#97	Dic₅	1	1	yes	characteristic
64	C₂×#97	C₂×D₅	1	1	yes	characteristic
65	C₂×#97	C₂×D₅	1	1	yes	characteristic
66	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
67	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
69	#376	Dic₅	1	1	yes	characteristic
70	C₄⋊C₁₆	—	5	8	no	—
71	#347	—	10	4	no	—
72	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	4	2	no	—
76	C₂×C₄×C₅	—	4	2	no	—
77	C₂×C₄×C₅	—	4	2	no	—
78	C₂×C₄×C₅	#97	1	4	yes	—
79	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅	#117	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
85	C₅×C₈	—	2	2	no	—
86	C₅×C₈	—	2	2	no	—
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	C₄×D₄	1	2	yes	—
89	C₅×C₈	C₄×Q₈	1	2	yes	—
90	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
92	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
97	M₃₂	—	10	4	no	—
98	M₃₂	—	10	4	no	—
99	C₂×C₁₆	—	5	8	no	—
100	#97	—	2	2	no	—
101	#97	—	2	2	no	—
102	#97	Dic₁₀	1	2	yes	—
103	#97	D₂₀	1	2	yes	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	C₄×D₅	1	4	yes	—
107	#95	—	2	2	no	—
108	C₂²×C₅	—	2	2	no	—
109	C₂²×C₅	C₂×#97	1	1	yes	characteristic
110	C₂²×C₅	#423	1	1	yes	characteristic
111	C₂²×C₅	#432	1	1	yes	characteristic
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	—	4	2	no	—
114	C₄×C₅	—	4	2	no	—
115	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
117	C₄×C₅	#423	1	1	yes	characteristic
118	C₄×C₅	#432	1	1	yes	characteristic
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	#681	1	2	yes	—
122	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	#681	1	2	yes	—
130	C₂×C₈	C₄×Dic₅	1	2	yes	—
131	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
132	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
134	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
135	C₂×C₈	#709	1	2	yes	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	#681	1	2	yes	—
139	C₄²	#681	1	2	yes	—
140	C₄²	C₄×Dic₅	1	2	yes	—
141	C₄²	C₄⋊Dic₅	1	2	yes	—
142	C₁₆	—	10	8	no	—
143	C₂×C₅	—	2	1	no	—
144	C₂×C₅	—	2	1	no	—
145	C₂×C₅	#1767	1	1	yes	characteristic
146	C₂×C₅	#2110	1	2	yes	—
147	C₂³	C₂×#681	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	#3935	1	4	yes	—
154	C₂×C₄	C₂×#681	1	1	yes	characteristic
155	C₂×C₄	C₂×#681	1	1	yes	characteristic
156	C₂×C₄	C₂×#681	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
159	C₂×C₄	#4072	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	C₄×Dic₁₀	1	2	yes	—
164	C₈	C₄×D₂₀	1	2	yes	—
165	C₅	#12657	1	1	yes	characteristic, complemented, Sylow
166	C₂²	—	2	2	no	—
167	C₂²	C₂×#3935	1	1	yes	characteristic
168	C₂²	#65213	1	1	yes	characteristic
169	C₂²	#65215	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	C₄×#681	1	1	yes	characteristic
174	C₄	#65213	1	1	yes	characteristic
175	C₄	#65215	1	1	yes	characteristic
176	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#10600184	1	1	yes	characteristic
180	C₂	#10600677	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1726	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1726	D₅
hypocenter	C₅	#12657
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
14. C₄⋊#672	C₄	semidirect
165. C₅	#12657	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×#64686	2
#10600536	2
#10598154	2
#12657	5