Subgroups — #12015711

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64721	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₅	C₄	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₂²×#672	C₂²	1	2	yes	—
10	C₄²×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₄×#168	C₄	1	1	yes	characteristic
12	C₅×C₄⋊C₁₆	—	2	2	no	—
13	#64721	—	2	8	no	—
14	C₄⋊#672	—	2	2	no	—
15	#12898	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₈	1	2	yes	—
21	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
23	C₂²×#168	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	—	2	2	no	—
25	C₂×C₅×C₁₆	—	2	4	no	—
26	C₄×C₅×C₈	D₄	1	2	yes	—
27	C₄×C₅×C₈	Q₈	1	2	yes	—
28	C₈×Dic₅	—	2	4	no	—
29	C₈×Dic₅	D₄	1	4	yes	—
30	C₈×Dic₅	Q₈	1	4	yes	—
31	C₂×#672	—	2	2	no	—
32	C₂×#672	—	2	4	no	—
33	C₄×#168	—	2	2	no	—
34	C₂×C₄⋊C₁₆	—	5	4	no	—
35	C₂×C₄⋊C₁₆	—	5	1	no	—
36	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic
37	C₄²×C₈	—	5	1	no	—
38	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅	C₂×C₈	1	2	yes	—
42	C₂²×Dic₅	C₂×C₈	1	2	yes	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
46	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
49	C₄²×C₅	C₄⋊C₄	1	2	yes	—
50	C₄²×C₅	M₁₆	1	2	yes	—
51	C₄×Dic₅	—	2	2	no	—
52	C₄×Dic₅	—	2	4	no	—
53	C₄×Dic₅	C₁₆	1	4	yes	complemented
54	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
55	C₄×Dic₅	M₁₆	1	4	yes	—
56	C₂×#168	—	2	2	no	—
57	C₂×#168	—	2	2	no	—
58	C₂×#168	C₄⋊C₄	1	4	yes	—
59	C₅×C₁₆	—	4	2	no	—
60	#672	—	4	2	no	—
61	C₂²×C₁₆	—	5	2	no	—
62	C₂²×C₁₆	C₂×D₅	1	2	yes	—
63	C₂×C₄×C₈	—	5	2	no	—
64	C₂×C₄×C₈	—	5	2	no	—
65	C₂×C₄×C₈	—	5	1	no	—
66	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	C₄³	—	5	1	no	—
68	C₄⋊C₁₆	—	2	2	no	—
69	C₄⋊C₁₆	—	10	8	no	—
70	C₄⋊C₁₆	—	10	2	no	—
71	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₅	M₃₂	1	2	yes	—
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
81	C₂×Dic₅	—	2	2	no	—
82	C₂×Dic₅	—	2	2	no	—
83	C₂×Dic₅	C₄⋊C₈	1	4	yes	—
84	C₂×Dic₅	C₄⋊C₈	1	4	yes	—
85	C₂×Dic₅	C₂×C₁₆	1	4	yes	—
86	C₂×Dic₅	M₃₂	1	4	yes	—
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	#127	1	1	yes	characteristic
89	C₅×C₈	C₄⋊Q₈	1	2	yes	—
90	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
91	#168	—	2	4	no	—
92	C₂²×C₈	—	5	2	no	—
93	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
95	C₂×C₄²	—	5	1	no	—
96	C₂×C₄²	—	5	2	no	—
97	C₂×C₄²	—	5	2	no	—
98	C₂×C₄²	—	5	1	no	—
99	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
100	C₂×C₁₆	—	2	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	10	4	no	—
103	C₂×C₁₆	—	10	2	no	—
104	C₄×C₈	—	5	4	no	—
105	C₄×C₈	—	5	4	no	—
106	C₄×C₈	—	10	4	no	—
107	C₄×C₈	—	10	2	no	—
108	C₄×C₈	Dic₁₀	1	2	yes	—
109	C₄×C₈	C₂²⋊D₅	1	2	yes	—
110	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
111	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
113	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
114	C₂²×C₅	C₂×M₃₂	1	2	yes	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	C₄⋊C₁₆	1	4	yes	complemented
117	C₄×C₅	#384	1	1	yes	characteristic
118	C₄×C₅	#423	1	2	yes	—
119	C₄×C₅	#423	1	1	yes	characteristic
120	Dic₅	—	2	4	no	—
121	Dic₅	C₄⋊C₁₆	1	8	yes	complemented
122	C₂²×C₄	—	5	2	no	—
123	C₂²×C₄	—	5	2	no	—
124	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
125	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	5	4	no	—
129	C₂×C₈	—	10	2	no	—
130	C₂×C₈	—	10	2	no	—
131	C₂×C₈	#683	1	2	yes	—
132	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
133	C₂×C₈	D₄×D₅	1	2	yes	—
134	C₂×C₈	Q₈×D₅	1	2	yes	—
135	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
136	C₄²	—	5	4	no	—
137	C₄²	—	5	4	no	—
138	C₄²	—	5	4	no	—
139	C₄²	—	10	4	no	—
140	C₄²	—	10	2	no	—
141	C₄²	#676	1	2	yes	—
142	C₄²	#683	1	2	yes	—
143	C₁₆	—	4	2	no	—
144	C₁₆	—	20	2	no	—
145	C₂×C₅	#1842	1	1	yes	characteristic
146	C₂×C₅	C₂×C₄⋊C₁₆	1	2	yes	—
147	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
148	C₂×C₅	#2150	1	2	yes	—
149	C₂×C₅	#2150	1	1	yes	characteristic
150	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	5	4	no	—
153	C₂×C₄	—	5	4	no	—
154	C₂×C₄	—	5	4	no	—
155	C₂×C₄	—	5	4	no	—
156	C₂×C₄	—	10	2	no	—
157	C₂×C₄	—	10	2	no	—
158	C₂×C₄	D₅×C₁₆	1	2	yes	—
159	C₂×C₄	#3930	1	2	yes	—
160	C₂×C₄	#3946	1	2	yes	—
161	C₂×C₄	#3946	1	2	yes	—
162	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
163	C₂×C₄	C₂×#676	1	1	yes	characteristic
164	C₂×C₄	D₅×M₁₆	1	2	yes	—
165	C₂×C₄	C₂×#683	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	10	4	no	—
168	C₈	#4034	1	2	yes	—
169	C₈	#4086	1	1	yes	characteristic
170	C₈	#4090	1	1	yes	characteristic
171	C₅	#12898	1	1	yes	characteristic, complemented, Sylow
172	C₂²	D₅×C₄⋊C₈	1	2	yes	—
173	C₂²	C₂×D₅×C₁₆	1	1	yes	characteristic
174	C₂²	C₂×#3930	1	1	yes	characteristic
175	C₂²	D₅×M₃₂	1	2	yes	—
176	C₂²	C₂×#3946	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	5	8	no	—
179	C₄	—	10	4	no	—
180	C₄	#64721	1	4	yes	complemented
181	C₄	#65126	1	2	yes	—
182	C₄	#65265	1	1	yes	characteristic
183	C₄	#65407	1	1	yes	characteristic
184	C₂	D₅×C₄⋊C₁₆	1	2	yes	—
185	C₂	#10600077	1	2	yes	—
186	C₂	#10601238	1	1	yes	characteristic
187	C₂	C₂×#64721	1	1	yes	characteristic
188	C₂	#10602213	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₁₆
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	#12898
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
53. C₄×Dic₅	C₁₆	semidirect
116. C₄×C₅	C₄⋊C₁₆	semidirect
121. Dic₅	C₄⋊C₁₆	semidirect
171. C₅	#12898	semidirect
180. C₄	#64721	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₄⋊C₁₆	2
C₂×#64721	2
C₄×C₈×Dic₅	2
C₂×C₄⋊#672	2
#12898	5