Subgroups — #12015713

Order 1280.

370 conjugacy classes of subgroups.

The subgroup lattice has 166 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	#10600542	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×#3947	C₄	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₂²×#672	C₂²	1	2	yes	—
10	#65221	C₄	1	1	yes	characteristic
11	C₂×#3935	C₄	1	1	yes	characteristic
12	C₅×C₄⋊C₁₆	—	2	2	no	—
13	#64721	—	2	8	no	—
14	C₄⋊#672	—	2	2	no	—
15	#12899	—	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	C₂×C₄	1	2	yes	—
20	C₂²×#168	C₂×C₄	1	2	yes	—
21	C₂×C₅×C₁₆	—	2	2	no	—
22	C₂×C₅×C₁₆	—	2	4	no	—
23	C₄×C₅×C₈	D₄	1	2	yes	—
24	C₄×C₅×C₈	Q₈	1	2	yes	—
25	#3963	C₈	1	2	yes	—
26	#3963	C₈	1	2	yes	—
27	#3947	—	2	4	no	—
28	C₈×Dic₅	D₄	1	4	yes	—
29	C₈×Dic₅	Q₈	1	4	yes	—
30	C₂×#672	—	2	2	no	—
31	C₂×#672	—	2	4	no	—
32	#3935	—	2	2	no	—
33	C₂×C₄⋊C₁₆	—	5	4	no	—
34	C₂×C₄⋊C₁₆	—	5	1	no	—
35	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic
36	#1726	—	5	1	no	—
37	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅	C₂×C₈	1	4	yes	—
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
44	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
47	C₄²×C₅	C₄⋊C₄	1	2	yes	—
48	C₄²×C₅	M₁₆	1	2	yes	—
49	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
50	C₄×Dic₅	M₁₆	1	4	yes	—
51	C₂×#168	—	2	2	no	—
52	C₂×#168	—	2	2	no	—
53	C₂×#168	C₄⋊C₄	1	4	yes	—
54	C₅×C₁₆	—	4	2	no	—
55	#672	—	4	2	no	—
56	C₂²×C₁₆	—	5	2	no	—
57	C₂²×C₁₆	C₂×D₅	1	2	yes	—
58	C₂×#97	—	5	2	no	—
59	C₂×#97	—	5	1	no	—
60	C₂×C₄×C₈	—	5	2	no	—
61	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
62	#376	—	5	1	no	—
63	C₄⋊C₁₆	—	2	2	no	—
64	C₄⋊C₁₆	—	10	8	no	—
65	C₄⋊C₁₆	—	10	2	no	—
66	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
70	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
71	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
75	C₂×Dic₅	—	2	4	no	—
76	C₂×Dic₅	—	2	4	no	—
77	C₂×Dic₅	C₄⋊C₈	1	8	yes	—
78	C₅×C₈	—	2	2	no	—
79	C₅×C₈	#127	1	1	yes	characteristic
80	C₅×C₈	C₄⋊Q₈	1	2	yes	—
81	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
82	#168	—	2	4	no	—
83	C₂²×C₈	—	5	2	no	—
84	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
85	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
86	C₂×C₄²	—	5	2	no	—
87	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
88	C₂×C₁₆	—	2	4	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	10	4	no	—
91	C₂×C₁₆	—	10	2	no	—
92	#97	—	10	4	no	—
93	#97	—	10	2	no	—
94	C₄×C₈	—	5	4	no	—
95	C₄×C₈	—	5	4	no	—
96	C₄×C₈	Dic₁₀	1	2	yes	—
97	C₄×C₈	C₂²⋊D₅	1	2	yes	—
98	#95	—	5	2	no	—
99	#95	—	5	2	no	—
100	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
101	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₅	#504	1	2	yes	—
103	C₂²×C₅	#504	1	2	yes	—
104	C₄×C₅	—	2	2	no	—
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	#384	1	1	yes	characteristic
107	C₄×C₅	#423	1	2	yes	—
108	C₄×C₅	#423	1	1	yes	characteristic
109	Dic₅	—	2	8	no	—
110	C₂²×C₄	—	5	4	no	—
111	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
112	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	5	4	no	—
116	C₂×C₈	—	10	2	no	—
117	C₂×C₈	—	10	2	no	—
118	C₂×C₈	#683	1	2	yes	—
119	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
120	C₂×C₈	D₄×D₅	1	2	yes	—
121	C₂×C₈	Q₈×D₅	1	2	yes	—
122	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
123	C₄²	—	5	4	no	—
124	C₄²	—	5	4	no	—
125	C₄²	#676	1	2	yes	—
126	C₄²	#683	1	2	yes	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	20	2	no	—
129	C₂×C₅	#1842	1	1	yes	characteristic
130	C₂×C₅	#2151	1	4	yes	—
131	C₂×C₅	#2151	1	1	yes	characteristic
132	C₂×C₅	#2151	1	1	yes	characteristic
133	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	5	8	no	—
137	C₂×C₄	—	10	4	no	—
138	C₂×C₄	—	10	4	no	—
139	C₂×C₄	#3946	1	2	yes	—
140	C₂×C₄	#3946	1	2	yes	—
141	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
142	C₂×C₄	C₂×#676	1	1	yes	characteristic
143	C₂×C₄	D₅×M₁₆	1	2	yes	—
144	C₂×C₄	C₂×#683	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	10	4	no	—
147	C₈	#4034	1	2	yes	—
148	C₈	#4086	1	1	yes	characteristic
149	C₈	#4090	1	1	yes	characteristic
150	C₅	#12899	1	1	yes	characteristic, complemented, Sylow
151	C₂²	D₅×C₄⋊C₈	1	2	yes	—
152	C₂²	#65190	1	2	yes	—
153	C₂²	#65196	1	2	yes	—
154	C₂²	C₂×#3946	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	10	8	no	—
158	C₄	#65126	1	2	yes	—
159	C₄	#65265	1	1	yes	characteristic
160	C₄	#65407	1	1	yes	characteristic
161	C₂	#10600078	1	4	yes	—
162	C₂	#10601238	1	1	yes	characteristic
163	C₂	#10602194	1	1	yes	characteristic
164	C₂	#10602214	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator, socle, prime radical	C₂³×C₅	C₂²×C₈
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	#12899
LCS	G ▷ C₂³×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
150. C₅	#12899	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
#10600542	2
C₂×C₄⋊#672	2
#12899	5