Finite Group Databasebeta

Subgroups — #14512397

Order 1920.

230 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11751610	C₂	1	1	yes	maximal, characteristic
2	SL(2,𝔽₃)×#170	C₂	1	1	yes	maximal, characteristic, complemented
3	SL(2,𝔽₃)⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
4	#10603972	—	3	1	no	maximal, Hall
5	C₂×C₅⋊GL(2,𝔽₃)	C₄	1	1	yes	characteristic
6	C₂×#6740	—	4	1	no	maximal
7	Dic₅×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₅×#859	C₄	1	1	yes	characteristic
9	#73018	—	5	1	no	maximal, Hall
10	Q₈×#170	D₃	1	1	yes	characteristic, complemented
11	#65705	—	3	1	no	—
12	#65077	—	3	1	no	—
13	Q₈⋊#170	—	3	1	no	—
14	#64881	—	3	1	no	—
15	#170⋊C₈	—	3	1	no	—
16	#64871	—	3	1	no	—
17	C₂×C₅×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
18	C₂×#1200	—	4	1	no	—
19	C₂×C₃⋊#170	—	4	1	no	—
20	C₂×C₃×#170	—	4	1	no	—
21	C₅⋊GL(2,𝔽₃)	—	2	1	no	—
22	#6740	—	4	4	no	—
23	C₄×GL(2,𝔽₃)	—	5	1	no	—
24	C₈×SL(2,𝔽₃)	—	5	1	no	—
25	SL(2,𝔽₃)⋊C₈	—	5	1	no	—
26	C₂×#700	—	6	1	no	—
27	Q₈×Dic₅	C₂×D₃	1	1	yes	characteristic
28	C₂×#688	—	3	1	no	—
29	#4039	—	3	1	no	—
30	#4003	—	6	1	no	—
31	C₄⋊#170	—	3	1	no	—
32	C₄⋊#170	—	3	1	no	—
33	C₄×#170	—	3	1	no	—
34	C₄×#170	—	3	1	no	—
35	C₅×#103	—	3	1	no	—
36	#3953	—	3	1	no	—
37	C₈×Dic₅	—	3	1	no	—
38	#3940	—	3	1	no	—
39	#1583	—	15	1	no	Sylow
40	C₂²×D₁₅	—	4	1	no	—
41	C₂×C₅×Dic₃	—	4	1	no	—
42	C₂×C₃×Dic₅	—	4	1	no	—
43	C₅×SL(2,𝔽₃)	M₁₆	1	1	yes	characteristic, complemented
44	#1200	—	4	2	no	—
45	#1200	—	4	2	no	—
46	C₃⋊#170	—	4	2	no	—
47	C₃×#170	—	4	2	no	—
48	C₂×GL(2,𝔽₃)	—	5	1	no	—
49	C₂×ΓL(1,𝔽₂₅)	—	20	1	no	—
50	C₄×SL(2,𝔽₃)	—	5	1	no	—
51	#859	AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
53	C₂×D₂₀	—	3	1	no	—
54	C₂×C₄×D₅	—	6	1	no	—
55	C₂×#170	—	3	1	no	—
56	C₂×#170	—	6	1	no	—
57	C₂×#170	—	6	1	no	—
58	C₂×#170	S₄	1	1	yes	characteristic
59	#700	—	12	2	no	—
60	C₂×C₅×C₈	—	3	1	no	—
61	C₅×C₄⋊C₄	—	3	1	no	—
62	#688	—	6	1	no	—
63	#688	—	6	1	no	—
64	#685	—	6	1	no	—
65	C₄⋊Dic₅	—	3	1	no	—
66	C₄×Dic₅	—	3	1	no	—
67	C₂×#168	—	3	1	no	—
68	C₈×Q₈	—	5	1	no	—
69	C₄×SD₁₆	—	15	1	no	—
70	#436	—	15	1	no	—
71	#334	—	15	1	no	—
72	Q₈⋊C₈	—	15	1	no	—
73	D₄⋊C₈	—	15	1	no	—
74	#322	—	15	1	no	—
75	C₂²×C₃×C₅	—	4	1	no	—
76	C₂×D₁₅	—	4	2	no	—
77	C₂×D₁₅	—	8	1	no	—
78	C₂×D₁₅	—	8	1	no	—
79	C₃×Dic₅	—	4	1	no	—
80	C₃×Dic₅	—	4	1	no	—
81	C₅×Dic₃	—	4	2	no	—
82	C₂×C₄×D₃	—	20	1	no	—
83	C₂×SL(2,𝔽₃)	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
84	GL(2,𝔽₃)	—	10	1	no	—
85	C₂×C₃×C₈	—	20	1	no	—
86	C₂×C₃⋊C₈	—	20	1	no	—
87	ΓL(1,𝔽₂₅)	—	20	4	no	—
88	C₂²×D₅	—	6	1	no	—
89	C₅×Q₈	—	3	1	no	—
90	C₅×Q₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
91	C₂×C₄×C₅	—	3	1	no	—
92	C₂×C₄×C₅	—	6	1	no	—
93	C₂×Dic₅	—	3	1	no	—
94	C₂×Dic₅	C₂×S₄	1	1	yes	characteristic
95	D₂₀	—	6	1	no	—
96	D₂₀	—	6	1	no	—
97	C₄×D₅	—	12	1	no	—
98	C₄×D₅	—	12	1	no	—
99	#170	—	6	1	no	—
100	#170	—	12	1	no	—
101	#170	—	12	1	no	—
102	#170	GL(2,𝔽₃)	1	2	yes	complemented
103	C₅×C₈	—	6	1	no	—
104	#168	—	6	1	no	—
105	C₂×SD₁₆	—	15	1	no	—
106	C₂×M₁₆	—	30	1	no	—
107	C₄×Q₈	—	5	1	no	—
108	C₄×D₄	—	15	1	no	—
109	#106	—	15	1	no	—
110	C₄⋊C₈	—	15	1	no	—
111	C₄⋊C₈	—	15	1	no	—
112	#103	—	3	1	no	—
113	#102	—	15	1	no	—
114	C₂²⋊C₈	—	30	1	no	—
115	C₄×C₈	—	15	1	no	—
116	C₄×C₈	—	15	1	no	—
117	C₄×C₈	—	15	1	no	—
118	C₂×C₃×C₅	—	4	1	no	—
119	C₂×C₃×C₅	—	4	1	no	—
120	C₂×C₃×C₅	—	4	1	no	—
121	D₁₅	—	8	2	no	—
122	C₂²×D₃	—	20	1	no	—
123	C₂×C₃×C₄	—	20	1	no	—
124	C₂×Dic₃	—	4	1	no	—
125	C₄×D₃	—	20	2	no	—
126	C₄×D₃	—	20	2	no	—
127	SL(2,𝔽₃)	#700	1	1	yes	characteristic, complemented
128	C₃×C₈	—	20	2	no	—
129	C₃⋊C₈	—	20	2	no	—
130	C₂²×C₅	C₄×S₄	1	1	yes	characteristic
131	C₂×D₅	—	12	1	no	—
132	C₂×D₅	—	12	1	no	—
133	C₂×D₅	—	12	1	no	—
134	C₄×C₅	—	3	1	no	—
135	C₄×C₅	—	3	1	no	—
136	C₄×C₅	—	12	1	no	—
137	Dic₅	—	6	1	no	—
138	Dic₅	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
139	Dic₅	#985	1	1	yes	characteristic
140	C₂×Q₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₂×D₄	—	15	1	no	—
142	C₂²×C₄	—	30	1	no	—
143	SD₁₆	—	30	1	no	—
144	SD₁₆	—	30	1	no	—
145	M₁₆	—	60	2	no	—
146	C₂×C₈	—	3	1	no	—
147	C₂×C₈	—	5	1	no	—
148	C₂×C₈	—	15	1	no	—
149	C₂×C₈	—	15	1	no	—
150	C₂×C₈	—	30	1	no	—
151	C₂×C₈	—	30	1	no	—
152	C₄⋊C₄	—	3	1	no	—
153	C₄⋊C₄	—	15	1	no	—
154	C₂²⋊C₄	—	30	1	no	—
155	C₄²	—	15	1	no	—
156	C₃×C₅	—	4	1	no	Hall
157	C₂²×C₃	—	4	1	no	—
158	C₂×D₃	—	20	2	no	—
159	C₂×D₃	—	40	1	no	—
160	C₂×D₃	—	40	1	no	—
161	C₃×C₄	—	20	1	no	—
162	C₃×C₄	—	20	1	no	—
163	Dic₃	—	4	2	no	—
164	C₂×C₅	C₄×GL(2,𝔽₃)	1	1	yes	characteristic
165	C₂×C₅	#5400	1	1	yes	characteristic
166	C₂×C₅	#5405	1	1	yes	characteristic
167	D₅	—	24	1	no	—
168	C₂³	—	30	1	no	—
169	Q₈	—	3	1	no	—
170	Q₈	#6740	1	1	yes	characteristic, complemented
171	D₄	—	30	1	no	—
172	D₄	—	30	1	no	—
173	C₂×C₄	—	3	1	no	—
174	C₂×C₄	—	5	1	no	—
175	C₂×C₄	—	6	1	no	—
176	C₂×C₄	—	15	1	no	—
177	C₂×C₄	—	60	1	no	—
178	C₂×C₄	—	60	1	no	—
179	C₈	—	5	2	no	—
180	C₈	—	6	1	no	—
181	C₈	—	30	1	no	—
182	C₈	—	30	1	no	—
183	C₈	—	60	1	no	—
184	C₈	—	60	1	no	—
185	C₂×C₃	—	4	1	no	—
186	C₂×C₃	—	4	1	no	—
187	C₂×C₃	—	4	1	no	—
188	D₃	—	40	2	no	—
189	C₅	#73018	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	60	1	no	—
191	C₂²	—	60	1	no	—
192	C₂²	—	60	1	no	—
193	C₂²	AGL(1,𝔽₅)×S₄	1	1	yes	characteristic
194	C₄	—	3	1	no	—
195	C₄	—	3	1	no	—
196	C₄	—	5	1	no	—
197	C₄	—	5	1	no	—
198	C₄	—	12	1	no	—
199	C₄	—	30	1	no	—
200	C₃	—	4	1	no	Sylow
201	C₂	—	120	1	no	—
202	C₂	#11756702	1	1	yes	characteristic
203	C₂	AGL(1,𝔽₅)×GL(2,𝔽₃)	1	1	yes	characteristic
204	C₂	#11756710	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	AGL(1,𝔽₅)×S₄
commutator	C₂×C₅×SL(2,𝔽₃)	C₂×C₄
Fitting	C₂×C₅×Q₈	C₄×D₃
socle	C₂²×C₅	C₄×S₄
prime radical	C₂×C₅⋊GL(2,𝔽₃)	C₄
cosocle	Dic₅×SL(2,𝔽₃)	C₂²
hypocenter	C₅×SL(2,𝔽₃)	M₁₆
LCS	G ▷ C₂×C₅×SL(2,𝔽₃) ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. SL(2,𝔽₃)×#170	C₂	semidirect
3. SL(2,𝔽₃)⋊#170	C₂	semidirect
10. Q₈×#170	D₃	semidirect
43. C₅×SL(2,𝔽₃)	M₁₆	semidirect
90. C₅×Q₈	ΓL(1,𝔽₂₅)	semidirect
102. #170	GL(2,𝔽₃)	semidirect
127. SL(2,𝔽₃)	#700	semidirect
170. Q₈	#6740	semidirect
189. C₅	#73018	semidirect

Maximal subgroups

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

subgroup	index
#11751610	2
SL(2,𝔽₃)×#170	2
SL(2,𝔽₃)⋊#170	2
#10603972	3
C₂×#6740	4
#73018	5