Subgroups — #14285656

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752340	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#11746752	C₂	1	4	yes	maximal, complemented
4	#10603816	—	3	1	no	maximal, Hall
5	C₂×#6758	—	2	2	no	—
6	C₂×C₃×C₈×D₅	C₄	1	2	yes	—
7	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
8	C₄⋊(C₃⋊#170)	—	2	2	no	—
9	C₄⋊(C₃⋊#170)	C₂²	1	4	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₃×C₈×Dic₅	C₂²	1	2	yes	—
12	C₃×C₄×#168	C₄	1	1	yes	characteristic
13	#69730	—	5	1	no	maximal, Hall
14	#65681	—	3	2	no	—
15	C₄×C₈×D₅	D₃	1	1	yes	characteristic
16	#64881	—	3	4	no	—
17	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
19	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
20	C₂×C₃⋊#170	—	2	4	no	—
21	#6758	—	4	4	no	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
25	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
26	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
27	C₂×C₃×#168	C₂×C₄	1	2	yes	—
28	C₃×C₈×D₅	—	2	4	no	—
29	C₂×C₃×C₄×C₈	—	5	1	no	—
30	#4924	—	5	2	no	—
31	#4457	—	5	4	no	—
32	C₂×#700	—	6	2	no	—
33	C₂×C₈×D₅	Dic₃	1	2	yes	—
34	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
35	C₄⋊#170	—	3	4	no	—
36	C₄⋊#170	—	6	2	no	—
37	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic
38	C₈×Dic₅	C₂×D₃	1	2	yes	—
39	C₄×#168	Dic₃	1	1	yes	characteristic
40	#1566	—	15	1	no	Sylow
41	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
48	C₃×C₄×D₅	—	2	2	no	—
49	C₃×C₄×D₅	—	2	2	no	—
50	C₃×C₄×D₅	SD₁₆	1	4	yes	—
51	C₃⋊#170	—	4	4	no	—
52	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
53	C₃×#168	—	2	2	no	—
54	C₂²×C₃×C₈	—	5	2	no	—
55	C₂×C₃×C₄²	—	5	1	no	—
56	C₂×#208	—	10	2	no	—
57	C₃×C₄×C₈	—	5	2	no	—
58	C₃×C₄×C₈	—	5	1	no	—
59	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₄⋊(C₃⋊C₈)	—	5	4	no	—
61	C₄⋊(C₃⋊C₈)	—	10	2	no	—
62	C₂×C₄×D₅	Dic₆	1	1	yes	characteristic
63	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
64	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
65	C₂×#170	—	6	4	no	—
66	#700	—	12	4	no	—
67	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
68	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
69	C₄×Dic₅	Dic₆	1	1	yes	characteristic
70	C₄×Dic₅	D₁₂	1	1	yes	characteristic
71	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
72	C₂×#168	C₂×Dic₃	1	2	yes	—
73	C₈×D₅	—	2	4	no	—
74	#423	—	15	2	no	—
75	C₂×C₄×C₈	—	5	1	no	—
76	#334	—	15	4	no	—
77	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₃×D₅	—	2	1	no	—
79	C₂×C₃×D₅	—	2	1	no	—
80	C₂×C₃×D₅	#106	1	2	yes	—
81	C₃×C₄×C₅	#106	1	2	yes	—
82	C₃×C₄×C₅	#108	1	2	yes	—
83	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
84	C₃×Dic₅	—	2	1	no	—
85	C₃×Dic₅	—	2	1	no	—
86	C₃×Dic₅	C₂×SD₁₆	1	2	yes	—
87	C₂²×C₃×C₄	—	5	1	no	—
88	C₂²×C₃×C₄	—	5	1	no	—
89	C₂²×C₃×C₄	—	5	1	no	—
90	C₂×C₃×C₈	—	5	2	no	—
91	C₂×C₃×C₈	—	10	4	no	—
92	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
93	C₃×C₄²	—	5	1	no	—
94	C₃×C₄²	—	5	1	no	—
95	C₃×C₄²	—	5	1	no	—
96	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
97	#208	—	20	4	no	—
98	C₂×C₃⋊C₈	—	10	4	no	—
99	C₂²×D₅	C₄⋊Dic₃	1	1	yes	characteristic
100	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
101	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
103	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
104	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
105	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
106	C₄×D₅	—	2	2	no	—
107	C₄×D₅	—	2	2	no	—
108	C₄×D₅	#204	1	4	yes	—
109	#170	—	12	4	no	—
110	C₅×C₈	#208	1	4	yes	complemented
111	#168	—	2	2	no	—
112	C₂×M₁₆	—	30	2	no	—
113	C₂²×C₈	—	5	2	no	—
114	C₂×C₄²	—	5	1	no	—
115	C₄⋊C₈	—	15	4	no	—
116	C₄⋊C₈	—	30	2	no	—
117	C₄×C₈	—	5	2	no	—
118	C₄×C₈	—	5	1	no	—
119	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
120	C₂×C₃×C₅	#423	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
123	C₃×D₅	—	2	2	no	—
124	C₂³×C₃	—	5	1	no	—
125	C₂×C₃×C₄	—	5	1	no	—
126	C₂×C₃×C₄	—	5	4	no	—
127	C₂×C₃×C₄	—	5	1	no	—
128	C₂×C₃×C₄	—	5	1	no	—
129	C₂×C₃×C₄	—	10	2	no	—
130	C₂×C₃×C₄	—	10	2	no	—
131	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₃×C₈	—	10	2	no	—
135	C₃×C₈	#700	1	4	yes	complemented
136	C₃⋊C₈	—	20	4	no	—
137	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
138	C₂×D₅	—	2	1	no	—
139	C₂×D₅	—	2	1	no	—
140	C₂×D₅	#817	1	2	yes	—
141	C₄×C₅	#817	1	2	yes	—
142	C₄×C₅	#819	1	2	yes	—
143	C₄×C₅	C₂×#208	1	2	yes	—
144	Dic₅	—	2	1	no	—
145	Dic₅	—	2	1	no	—
146	Dic₅	C₂×#204	1	2	yes	—
147	C₂²×C₄	—	5	1	no	—
148	C₂²×C₄	—	5	1	no	—
149	C₂²×C₄	—	5	1	no	—
150	M₁₆	—	60	4	no	—
151	C₂×C₈	—	5	2	no	—
152	C₂×C₈	—	10	4	no	—
153	C₂×C₈	—	30	4	no	—
154	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
155	C₄²	—	5	1	no	—
156	C₄²	—	5	1	no	—
157	C₄²	—	5	1	no	—
158	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
159	C₃×C₅	#1566	1	1	yes	characteristic, complemented, Hall
160	C₂²×C₃	—	5	2	no	—
161	C₂²×C₃	—	10	1	no	—
162	C₂²×C₃	—	10	1	no	—
163	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₃×C₄	—	5	2	no	—
165	C₃×C₄	—	10	1	no	—
166	C₃×C₄	—	10	1	no	—
167	C₃×C₄	#3993	1	2	yes	—
168	C₃×C₄	#3995	1	2	yes	—
169	C₃×C₄	C₂×#700	1	2	yes	—
170	C₂×C₅	#4924	1	1	yes	characteristic
171	C₂×C₅	C₂×#817	1	1	yes	characteristic
172	C₂×C₅	C₂×#819	1	1	yes	characteristic
173	D₅	—	2	2	no	—
174	C₂³	—	5	1	no	—
175	C₂×C₄	—	5	1	no	—
176	C₂×C₄	—	5	4	no	—
177	C₂×C₄	—	5	1	no	—
178	C₂×C₄	—	5	1	no	—
179	C₂×C₄	—	10	2	no	—
180	C₂×C₄	—	10	2	no	—
181	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
182	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
183	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
184	C₈	—	10	2	no	—
185	C₈	—	60	4	no	—
186	C₈	#6758	1	4	yes	complemented
187	C₂×C₃	—	10	2	no	—
188	C₂×C₃	#65681	1	1	yes	characteristic
189	C₂×C₃	C₂×#3993	1	1	yes	characteristic
190	C₂×C₃	C₂×#3995	1	1	yes	characteristic
191	C₅	#69730	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	5	2	no	—
193	C₂²	—	10	1	no	—
194	C₂²	—	10	1	no	—
195	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
196	C₄	—	5	2	no	—
197	C₄	—	10	1	no	—
198	C₄	—	10	1	no	—
199	C₄	#91222	1	2	yes	—
200	C₄	#91224	1	2	yes	—
201	C₄	C₂×#6758	1	2	yes	—
202	C₃	#10603816	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	10	2	no	—
204	C₂	#11752340	1	1	yes	characteristic
205	C₂	C₂×#91222	1	1	yes	characteristic
206	C₂	C₂×#91224	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1566
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11746752	C₂	semidirect
52. C₃×C₅×C₈	M₁₆	semidirect
110. C₅×C₈	#208	semidirect
135. C₃×C₈	#700	semidirect
159. C₃×C₅	#1566	semidirect
186. C₈	#6758	semidirect
191. C₅	#69730	semidirect
202. C₃	#10603816	semidirect

Maximal subgroups

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

subgroup	index
#11752340	2
C₃×C₄×C₈×D₅	2
#11746752	2
#10603816	3
#69730	5