Subgroups — #12014614

Order 1280.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64721	C₂	1	2	yes	maximal
3	#10600542	C₂	1	1	yes	maximal, characteristic
4	C₂×#64681	C₂	1	1	yes	maximal, characteristic
5	#10598295	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3947	C₄	1	1	yes	characteristic
9	C₂×#3947	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂²×#672	C₂²	1	2	yes	—
13	#65221	C₄	1	1	yes	characteristic
14	C₂×#3935	C₄	1	1	yes	characteristic
15	C₄×C₅×C₁₆	—	2	2	no	—
16	#64721	C₄	1	8	yes	complemented
17	#64681	—	2	2	no	—
18	#12654	—	5	1	no	maximal, Sylow
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×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×#168	C₂×C₄	1	1	yes	characteristic
25	C₂²×#168	C₂×C₄	1	1	yes	characteristic
26	C₂×C₅×C₁₆	—	2	2	no	—
27	C₂×C₅×C₁₆	—	2	2	no	—
28	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
29	C₄×C₅×C₈	D₄	1	2	yes	—
30	C₄×C₅×C₈	Q₈	1	2	yes	—
31	#3963	C₈	1	4	yes	—
32	#3947	—	2	2	no	—
33	#3947	C₈	1	4	yes	—
34	C₈×Dic₅	—	2	2	no	—
35	C₈×Dic₅	C₂×C₄	1	4	yes	—
36	C₂×#672	—	2	2	no	—
37	C₂×#672	—	2	2	no	—
38	C₂×#672	C₂×C₄	1	4	yes	—
39	#3935	—	2	2	no	—
40	C₂×C₄⋊C₁₆	—	5	2	no	—
41	C₂×#346	—	5	1	no	—
42	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
43	#1726	—	5	1	no	—
44	#1374	—	5	2	no	—
45	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₅	C₂×C₈	1	2	yes	—
48	C₂²×Dic₅	C₂×C₈	1	2	yes	—
49	C₂×C₅×C₈	C₄⋊C₄	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	C₄⋊C₄	1	2	yes	—
57	C₄²×C₅	M₁₆	1	2	yes	—
58	C₄×Dic₅	—	2	1	no	—
59	C₄×Dic₅	—	2	1	no	—
60	C₄×Dic₅	C₄²	1	2	yes	—
61	C₄×Dic₅	C₂×C₈	1	2	yes	—
62	C₂×#168	—	2	1	no	—
63	C₂×#168	—	2	1	no	—
64	C₂×#168	—	2	1	no	—
65	C₂×#168	—	2	1	no	—
66	C₂×#168	C₄²	1	2	yes	—
67	C₂×#168	C₂×C₈	1	2	yes	—
68	C₅×C₁₆	—	2	4	no	—
69	#672	—	2	4	no	—
70	C₂²×C₁₆	—	5	2	no	—
71	C₂²×C₁₆	C₂×D₅	1	2	yes	—
72	C₂×#97	—	5	1	no	—
73	C₂×#97	—	5	1	no	—
74	C₂×#97	—	5	1	no	—
75	C₂×C₄×C₈	—	5	1	no	—
76	C₂×C₄×C₈	—	5	1	no	—
77	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
78	#376	—	5	1	no	—
79	C₄⋊C₁₆	—	5	8	no	—
80	#346	—	10	2	no	—
81	C₄×C₁₆	—	2	2	no	—
82	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
85	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅	#117	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₂×Dic₅	—	2	4	no	—
92	C₂×Dic₅	—	2	2	no	—
93	C₂×Dic₅	—	2	4	no	—
94	C₂×Dic₅	C₄×C₈	1	4	yes	—
95	C₅×C₈	C₄⋊C₈	1	4	yes	—
96	C₅×C₈	C₄×D₄	1	2	yes	—
97	C₅×C₈	C₄×Q₈	1	2	yes	—
98	#168	—	2	2	no	—
99	#168	—	2	2	no	—
100	C₂²×C₈	—	5	1	no	—
101	C₂²×C₈	—	5	1	no	—
102	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
104	C₂×C₄²	—	5	1	no	—
105	C₂×C₄²	—	5	1	no	—
106	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	5	4	no	—
110	C₂×C₁₆	—	10	2	no	—
111	C₂×C₁₆	—	10	2	no	—
112	C₂×C₁₆	C₄×D₅	1	4	yes	—
113	#97	—	5	4	no	—
114	#97	—	10	2	no	—
115	#97	—	10	2	no	—
116	C₄×C₈	—	5	4	no	—
117	C₄×C₈	—	10	2	no	—
118	C₄×C₈	Dic₁₀	1	2	yes	—
119	C₄×C₈	C₂²⋊D₅	1	2	yes	—
120	#95	—	5	4	no	—
121	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
122	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂²×C₅	#431	1	1	yes	characteristic
124	C₂²×C₅	#504	1	4	yes	—
125	C₄×C₅	—	2	4	no	—
126	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
128	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₄×C₅	#431	1	1	yes	characteristic
130	Dic₅	—	2	4	no	—
131	Dic₅	—	4	2	no	—
132	C₂²×C₄	—	5	2	no	—
133	C₂²×C₄	—	5	2	no	—
134	C₂²×C₄	C₈×D₅	1	2	yes	—
135	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
136	C₂×C₈	—	5	2	no	—
137	C₂×C₈	—	5	2	no	—
138	C₂×C₈	—	10	1	no	—
139	C₂×C₈	—	10	1	no	—
140	C₂×C₈	—	10	1	no	—
141	C₂×C₈	—	10	1	no	—
142	C₂×C₈	C₈×D₅	1	2	yes	—
143	C₂×C₈	#676	1	2	yes	—
144	C₂×C₈	#683	1	2	yes	—
145	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
146	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
147	C₂×C₈	#709	1	2	yes	—
148	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
149	C₄²	—	5	2	no	—
150	C₄²	—	5	2	no	—
151	C₄²	—	10	1	no	—
152	C₄²	—	10	1	no	—
153	C₄²	#676	1	2	yes	—
154	C₄²	#683	1	2	yes	—
155	C₁₆	—	2	4	no	—
156	C₁₆	—	10	4	no	—
157	C₂×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic
158	C₂×C₅	#2108	1	2	yes	—
159	C₂×C₅	#2151	1	2	yes	—
160	C₂×C₅	#2164	1	2	yes	—
161	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	5	4	no	—
164	C₂×C₄	—	10	4	no	—
165	C₂×C₄	—	10	4	no	—
166	C₂×C₄	—	10	2	no	—
167	C₂×C₄	#3946	1	4	yes	—
168	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
169	C₂×C₄	#4018	1	1	yes	characteristic
170	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
171	C₂×C₄	C₂×#676	1	1	yes	characteristic
172	C₂×C₄	C₂×#676	1	1	yes	characteristic
173	C₂×C₄	C₂×#683	1	1	yes	characteristic
174	C₈	—	10	2	no	—
175	C₈	—	10	2	no	—
176	C₈	#3946	1	4	yes	—
177	C₈	C₄×Dic₁₀	1	2	yes	—
178	C₈	C₄×C₂²⋊D₅	1	2	yes	—
179	C₅	#12654	1	1	yes	characteristic, complemented, Sylow
180	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
181	C₂²	#64974	1	1	yes	characteristic
182	C₂²	#65190	1	4	yes	—
183	C₂²	C₂×#3946	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	10	4	no	—
186	C₄	—	20	2	no	—
187	C₄	#64974	1	1	yes	characteristic
188	C₄	C₄×#676	1	1	yes	characteristic
189	C₄	C₄×#683	1	1	yes	characteristic
190	C₄	C₂×#3946	1	1	yes	characteristic
191	C₂	#10599844	1	2	yes	—
192	C₂	#10599846	1	2	yes	—
193	C₂	C₄×#3946	1	1	yes	characteristic
194	C₂	#10602194	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	#12654
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
16. #64721	C₄	semidirect
179. C₅	#12654	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₆	2
C₂×#64721	2
#10600542	2
C₂×#64681	2
#10598295	2
#12654	5