Subgroups — #14296921

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#91120	C₂	1	4	yes	maximal, complemented
3	#11751308	C₂	1	2	yes	maximal
4	#10602926	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×Q₈	—	2	8	no	—
6	C₂²×C₃×C₅×Q₈	—	2	4	no	—
7	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
10	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×#6710	C₂²	1	4	yes	—
12	#91120	—	2	8	no	—
13	#91114	—	2	4	no	—
14	#72133	—	5	1	no	maximal, Hall
15	C₂³×C₅×Q₈	D₃	1	1	yes	characteristic
16	C₂×#3968	—	3	4	no	—
17	#65496	—	3	2	no	—
18	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₅×Q₈	—	2	32	no	—
20	C₂×C₃×C₅×Q₈	—	2	8	no	—
21	C₂×C₃×C₅×Q₈	—	2	4	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	4	no	—
24	C₂×C₃×C₅×Q₈	D₄	1	8	yes	—
25	C₂²×C₃×C₄×C₅	—	2	4	no	—
26	C₂²×C₃×C₄×C₅	—	2	8	no	—
27	C₂²×C₃×C₄×C₅	—	2	4	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	#6710	—	2	8	no	—
33	C₂×C₃⋊#168	—	2	4	no	—
34	C₂³×C₃×Q₈	D₅	1	1	yes	characteristic
35	C₂×#836	—	5	4	no	—
36	#5207	—	5	2	no	—
37	C₂²×C₅×Q₈	—	2	8	no	—
38	C₂²×C₅×Q₈	—	2	4	no	—
39	C₂²×C₅×Q₈	Dic₃	1	2	yes	—
40	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
41	C₂³×C₄×C₅	Dic₃	1	2	yes	—
42	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×#681	—	3	4	no	—
44	#3968	—	6	8	no	—
45	C₂²⋊#168	—	6	4	no	—
46	#1784	—	15	1	no	Sylow
47	C₂³×C₃×C₅	—	2	4	no	—
48	C₂³×C₃×C₅	—	2	2	no	—
49	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₃×C₅×Q₈	—	2	16	no	—
52	C₃×C₅×Q₈	—	2	16	no	—
53	C₂×C₃×C₄×C₅	—	2	16	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	—	2	8	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	8	yes	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
62	C₃⋊#168	—	4	4	no	—
63	C₂²×C₃×Q₈	—	2	8	no	—
64	C₂²×C₃×Q₈	—	2	4	no	—
65	C₂²×C₃×Q₈	Dic₅	1	2	yes	—
66	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
67	C₂³×C₃×C₄	Dic₅	1	2	yes	—
68	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
69	C₂×#208	—	5	4	no	—
70	#836	—	10	8	no	—
71	#830	—	10	4	no	—
72	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
73	C₂×C₅×Q₈	—	2	32	no	—
74	C₂×C₅×Q₈	—	2	8	no	—
75	C₂×C₅×Q₈	—	2	4	no	—
76	C₂×C₅×Q₈	—	2	4	no	—
77	C₂×C₅×Q₈	—	2	4	no	—
78	C₂×C₅×Q₈	#67	1	8	yes	—
79	C₂²×C₄×C₅	—	2	4	no	—
80	C₂²×C₄×C₅	—	2	8	no	—
81	C₂²×C₄×C₅	—	2	4	no	—
82	C₂²×C₄×C₅	C₂×Dic₃	1	4	yes	—
83	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	#67	1	4	yes	—
85	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
86	#681	—	6	8	no	—
87	C₂×#168	—	6	4	no	—
88	C₂³×Q₈	D₁₅	1	1	yes	characteristic
89	C₂×#101	—	15	4	no	—
90	#407	—	15	2	no	—
91	C₂²×C₃×C₅	—	2	8	no	—
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	—	2	2	no	—
94	C₂²×C₃×C₅	—	2	2	no	—
95	C₂²×C₃×C₅	#101	1	4	yes	complemented
96	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₃×C₄×C₅	—	2	8	no	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	#120	1	4	yes	—
101	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	2	32	no	—
103	C₂×C₃×Q₈	—	2	8	no	—
104	C₂×C₃×Q₈	—	2	4	no	—
105	C₂×C₃×Q₈	—	2	4	no	—
106	C₂×C₃×Q₈	—	2	4	no	—
107	C₂×C₃×Q₈	C₂²⋊D₅	1	8	yes	—
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	—	2	8	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
112	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂²⋊D₅	1	4	yes	—
114	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
115	#208	—	10	8	no	—
116	C₂×C₃⋊C₈	—	10	4	no	—
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	#217	1	2	yes	—
120	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
121	C₅×Q₈	—	2	16	no	—
122	C₅×Q₈	—	2	16	no	—
123	C₂×C₄×C₅	—	2	16	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	8	no	—
128	C₂×C₄×C₅	#217	1	8	yes	—
129	C₂×C₄×C₅	#217	1	2	yes	—
130	C₂×C₄×C₅	C₂×#67	1	2	yes	—
131	C₂×C₄×C₅	C₂×#67	1	4	yes	—
132	#168	—	12	4	no	—
133	C₂²×Q₈	—	2	8	no	—
134	C₂²×Q₈	—	2	4	no	—
135	C₂²×Q₈	Dic₁₅	1	2	yes	—
136	C₂²×Q₈	C₂×D₁₅	1	2	yes	—
137	C₂³×C₄	Dic₁₅	1	2	yes	—
138	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
139	C₂×M₁₆	—	15	4	no	—
140	#101	—	30	8	no	—
141	C₂²⋊C₈	—	30	4	no	—
142	C₂×C₃×C₅	—	2	4	no	—
143	C₂×C₃×C₅	—	2	2	no	—
144	C₂×C₃×C₅	#379	1	1	yes	characteristic
145	C₂×C₃×C₅	C₂×#101	1	2	yes	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	—	2	4	no	—
148	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
149	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
150	C₃×Q₈	—	2	16	no	—
151	C₃×Q₈	—	2	16	no	—
152	C₂×C₃×C₄	—	2	16	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	8	no	—
157	C₂×C₃×C₄	C₂²⋊Dic₅	1	8	yes	—
158	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
159	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
161	C₃⋊C₈	—	20	4	no	—
162	C₂²×C₅	—	2	8	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	#836	1	4	yes	complemented
167	C₂²×C₅	C₂×#217	1	2	yes	—
168	C₂²×C₅	C₂×#217	1	1	yes	characteristic
169	C₄×C₅	—	2	8	no	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	#953	1	4	yes	—
172	C₂⁴	C₂×Dic₁₅	1	1	yes	characteristic
173	C₂×Q₈	—	2	32	no	—
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	#1220	1	8	yes	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	C₂×Dic₁₅	1	4	yes	—
183	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
184	C₂²×C₄	#1220	1	4	yes	—
185	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
186	M₁₆	—	30	8	no	—
187	C₂×C₈	—	30	4	no	—
188	C₃×C₅	#1784	1	1	yes	characteristic, complemented, Hall
189	C₂²×C₃	—	2	8	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	#3968	1	4	yes	complemented
194	C₂²×C₃	C₂×C₂²⋊Dic₅	1	2	yes	—
195	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
196	C₃×C₄	—	2	8	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	#4099	1	4	yes	—
199	C₂×C₅	—	2	4	no	—
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	C₂×#836	1	2	yes	—
202	C₂×C₅	#5247	1	1	yes	characteristic
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	#6719	1	2	yes	—
206	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
207	Q₈	—	2	16	no	—
208	Q₈	—	2	16	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	#6719	1	8	yes	—
215	C₂×C₄	#6719	1	2	yes	—
216	C₂×C₄	C₂×#1220	1	4	yes	—
217	C₂×C₄	C₂×#1220	1	2	yes	—
218	C₈	—	60	4	no	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	C₂×#3968	1	2	yes	—
222	C₂×C₃	#65536	1	1	yes	characteristic
223	C₅	#72133	1	1	yes	characteristic, complemented, Sylow
224	C₂²	—	2	8	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	#91120	1	4	yes	complemented
229	C₂²	C₂×#6719	1	2	yes	—
230	C₂²	C₂×#6719	1	1	yes	characteristic
231	C₄	—	2	8	no	—
232	C₄	—	2	2	no	—
233	C₄	#91842	1	4	yes	—
234	C₃	#10602926	1	1	yes	characteristic, complemented, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	2	4	no	—
237	C₂	C₂×#91120	1	2	yes	—
238	C₂	#11751348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6719
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂³×C₃×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×Q₈	D₁₅
hypocenter	C₃×C₅	#1784
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#91120	C₂	semidirect
95. C₂²×C₃×C₅	#101	semidirect
166. C₂²×C₅	#836	semidirect
188. C₃×C₅	#1784	semidirect
193. C₂²×C₃	#3968	semidirect
223. C₅	#72133	semidirect
228. C₂²	#91120	semidirect
234. C₃	#10602926	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×Q₈	2
C₂×#91120	2
#11751308	2
#10602926	3
#72133	5