Subgroups — #14276839

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752407	C₂	1	2	yes	maximal
2	C₂×C₃×#3947	C₂	1	1	yes	maximal, characteristic
3	#10598774	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
6	C₂²×C₃×#168	C₄	1	1	yes	characteristic
7	#91241	C₄	1	4	yes	—
8	C₄⋊(C₃⋊#170)	—	2	2	no	—
9	C₄×C₃⋊#170	—	2	2	no	—
10	C₃×#3947	—	2	2	no	—
11	#67661	—	5	1	no	maximal, Hall
12	#65748	—	3	2	no	—
13	C₂×#3947	D₃	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
16	C₂×C₃⋊#170	—	2	4	no	—
17	C₂×C₃×C₅×C₈	—	2	1	no	—
18	C₂×C₃×C₅×C₈	—	2	1	no	—
19	C₂×C₃×C₅×C₈	C₈	1	2	yes	complemented
20	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
21	C₃×C₄×Dic₅	D₄	1	2	yes	—
22	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
23	C₂×C₃×#168	—	2	1	no	—
24	C₂×C₃×#168	—	2	1	no	—
25	C₂×C₃×#168	C₈	1	2	yes	complemented
26	C₂×C₃×#97	—	5	1	no	—
27	#4925	—	5	2	no	—
28	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic
29	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
30	C₂²×#168	Dic₃	1	1	yes	characteristic
31	C₂²⋊#170	—	3	4	no	—
32	C₄⋊#170	—	6	2	no	—
33	C₄×#170	—	6	2	no	—
34	#3947	—	2	2	no	—
35	#1281	—	15	1	no	Sylow
36	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
43	C₂×C₃×Dic₅	—	4	2	no	—
44	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
45	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
46	C₃⋊#170	—	4	4	no	—
47	C₃×C₅×C₈	—	2	2	no	—
48	C₃×#168	—	2	2	no	—
49	C₂²×C₃×C₈	—	5	1	no	—
50	C₂²×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂×C₃×C₄²	—	5	1	no	—
52	C₃×#97	—	10	2	no	—
53	#830	—	5	4	no	—
54	C₄⋊(C₃⋊C₈)	—	10	2	no	—
55	C₄×C₃⋊C₈	—	10	2	no	—
56	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
57	C₂²×Dic₅	C₄×D₃	1	2	yes	—
58	C₂×#170	—	6	4	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	C₃⋊C₈	1	2	yes	complemented
62	C₄×Dic₅	Dic₆	1	1	yes	characteristic
63	C₄×Dic₅	D₁₂	1	1	yes	characteristic
64	C₄×Dic₅	#67	1	2	yes	—
65	C₂×#168	—	2	1	no	—
66	C₂×#168	—	2	1	no	—
67	C₂×#168	C₃⋊C₈	1	2	yes	complemented
68	#431	—	15	2	no	—
69	C₂×#97	—	5	1	no	—
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	#95	1	1	yes	characteristic
72	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
73	C₂²×C₃×C₅	#97	1	1	yes	characteristic
74	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
75	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
76	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
77	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
78	C₃×Dic₅	—	2	2	no	—
79	C₃×Dic₅	—	2	2	no	—
80	C₂²×C₃×C₄	—	5	2	no	—
81	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×C₃×C₈	—	2	1	no	—
83	C₂×C₃×C₈	—	2	1	no	—
84	C₂×C₃×C₈	—	5	2	no	—
85	C₂×C₃×C₈	—	10	1	no	—
86	C₂×C₃×C₈	—	10	1	no	—
87	C₂×C₃×C₈	#170	1	2	yes	complemented
88	C₃×C₄²	—	5	1	no	—
89	C₃×C₄²	—	5	1	no	—
90	C₃×C₄²	—	5	2	no	—
91	C₂×C₃⋊C₈	—	10	4	no	—
92	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	#208	1	1	yes	characteristic
96	C₂×C₄×C₅	#208	1	1	yes	characteristic
97	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
98	C₂×C₄×C₅	#217	1	1	yes	characteristic
99	C₂×Dic₅	—	4	2	no	—
100	C₂×Dic₅	Dic₃⋊C₄	1	2	yes	—
101	C₂×Dic₅	(C₂×D₃)⋊C₄	1	2	yes	—
102	#170	—	12	4	no	—
103	C₅×C₈	—	2	2	no	—
104	#168	—	2	2	no	—
105	C₂²×C₈	—	5	1	no	—
106	C₂²×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
107	C₂×C₄²	—	5	1	no	—
108	C₄⋊C₈	—	30	2	no	—
109	C₂²⋊C₈	—	15	4	no	—
110	#97	—	10	2	no	—
111	C₄×C₈	—	30	2	no	—
112	C₂×C₃×C₅	—	2	1	no	—
113	C₂×C₃×C₅	—	2	1	no	—
114	C₂×C₃×C₅	#336	1	1	yes	characteristic
115	C₂×C₃×C₅	#344	1	2	yes	—
116	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₃×C₄	—	5	2	no	—
118	C₂×C₃×C₄	—	5	2	no	—
119	C₂×C₃×C₄	—	20	2	no	—
120	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
121	C₂×C₃×C₄	#700	1	1	yes	characteristic
122	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
124	C₂×C₃×C₄	#704	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₃×C₈	—	2	2	no	—
127	C₃×C₈	—	10	2	no	—
128	C₃⋊C₈	—	20	4	no	—
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	C₄×C₃⋊C₈	1	1	yes	characteristic
131	C₂²×C₅	#803	1	1	yes	characteristic
132	C₂²×C₅	C₃⋊#95	1	1	yes	characteristic
133	C₄×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
134	C₄×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
135	C₄×C₅	#830	1	1	yes	characteristic
136	C₄×C₅	#830	1	1	yes	characteristic
137	Dic₅	—	2	2	no	—
138	Dic₅	—	2	2	no	—
139	C₂²×C₄	—	5	2	no	—
140	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	5	2	no	—
144	C₂×C₈	—	10	1	no	—
145	C₂×C₈	—	10	1	no	—
146	C₂×C₈	—	30	4	no	—
147	C₂×C₈	C₃⋊#170	1	2	yes	complemented
148	C₄²	—	5	1	no	—
149	C₄²	—	5	1	no	—
150	C₄²	—	5	2	no	—
151	C₃×C₅	#1281	1	1	yes	characteristic, complemented, Hall
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	C₄×#170	1	1	yes	characteristic
154	C₂²×C₃	#4002	1	1	yes	characteristic
155	C₂²×C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
156	C₃×C₄	—	10	2	no	—
157	C₃×C₄	—	10	2	no	—
158	C₃×C₄	C₄⋊#170	1	1	yes	characteristic
159	C₃×C₄	#4003	1	1	yes	characteristic
160	C₃×C₄	#4004	1	1	yes	characteristic
161	C₃×C₄	C₂²⋊#170	1	1	yes	characteristic
162	C₂×C₅	—	2	1	no	—
163	C₂×C₅	—	2	1	no	—
164	C₂×C₅	#4524	1	1	yes	characteristic
165	C₂×C₅	#4556	1	2	yes	—
166	C₂³	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
167	C₂×C₄	—	5	2	no	—
168	C₂×C₄	—	5	2	no	—
169	C₂×C₄	—	20	2	no	—
170	C₂×C₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
171	C₂×C₄	#6758	1	1	yes	characteristic
172	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
173	C₂×C₄	C₂×C₃⋊#170	1	1	yes	characteristic
174	C₂×C₄	#6762	1	1	yes	characteristic
175	C₂×C₄	#6763	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	10	2	no	—
178	C₈	—	60	4	no	—
179	C₂×C₃	—	2	1	no	—
180	C₂×C₃	—	2	1	no	—
181	C₂×C₃	#64894	1	2	yes	—
182	C₂×C₃	#64918	1	1	yes	characteristic
183	C₅	#67661	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	C₄×C₃⋊#170	1	1	yes	characteristic
186	C₂²	#91231	1	1	yes	characteristic
187	C₂²	#91235	1	1	yes	characteristic
188	C₄	—	10	2	no	—
189	C₄	—	10	2	no	—
190	C₄	C₄⋊(C₃⋊#170)	1	1	yes	characteristic
191	C₄	#91232	1	1	yes	characteristic
192	C₄	#91233	1	1	yes	characteristic
193	C₄	#91241	1	1	yes	characteristic
194	C₃	#10598774	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#11746765	1	2	yes	—
198	C₂	#11746789	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6763
commutator	C₂²×C₃×C₅	C₄×C₈
Frattini	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)
Fitting	C₂²×C₃×C₅×C₈	C₄
socle	C₂²×C₃×C₅	#95
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₂×C₃×C₄×Dic₅	C₂²
hypercenter	C₂²×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1281
LCS	G ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
19. C₂×C₃×C₅×C₈	C₈	semidirect
25. C₂×C₃×#168	C₈	semidirect
61. C₂×C₅×C₈	C₃⋊C₈	semidirect
67. C₂×#168	C₃⋊C₈	semidirect
87. C₂×C₃×C₈	#170	semidirect
147. C₂×C₈	C₃⋊#170	semidirect
151. C₃×C₅	#1281	semidirect
183. C₅	#67661	semidirect
194. C₃	#10598774	semidirect

Maximal subgroups

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

subgroup	index
#11752407	2
C₂×C₃×#3947	2
#10598774	3
#67661	5