Subgroups — #14313970

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#11751207	C₂	1	1	yes	maximal, characteristic
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic
4	#11746385	C₂	1	4	yes	maximal, complemented
5	#10602854	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
9	C₂×C₃×C₅×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	#91815	—	2	1	no	—
13	#91102	C₂²	1	2	yes	—
14	#91101	—	2	1	no	—
15	#91096	—	2	2	no	—
16	C₂×C₃⋊#672	C₂²	1	2	yes	—
17	#72061	—	5	1	no	maximal, Hall
18	C₂²×C₅×Q₁₆	D₃	1	1	yes	characteristic
19	#65395	—	3	1	no	—
20	C₂×#3944	—	3	1	no	—
21	#64784	—	3	4	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	—	4	1	no	—
25	C₂×C₃×C₅×Q₈	—	4	2	no	—
26	C₂²×C₃×C₄×C₅	—	4	1	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₃×C₅×Q₁₆	—	2	4	no	—
29	C₃×C₅×Q₁₆	—	2	2	no	—
30	C₃×C₅×Q₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	#6719	—	4	1	no	—
36	C₄⋊Dic₁₅	—	2	2	no	—
37	C₄×Dic₁₅	—	4	1	no	—
38	C₃⋊#672	—	2	2	no	—
39	C₂²×C₃×Q₁₆	D₅	1	1	yes	characteristic
40	#5106	—	5	1	no	—
41	C₂×#812	—	5	1	no	—
42	#4564	—	5	4	no	—
43	C₂²×C₅×Q₈	—	2	1	no	—
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	—	2	2	no	—
46	C₂×C₅×Q₁₆	Dic₃	1	4	yes	complemented
47	C₂×C₅×Q₁₆	C₂×D₃	1	2	yes	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	#4072	—	6	1	no	—
50	#3950	—	3	2	no	—
51	#3949	—	6	1	no	—
52	#3944	—	6	2	no	—
53	C₂×#672	—	3	2	no	—
54	#2141	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	2	4	no	—
57	C₃×C₅×Q₈	—	4	4	no	—
58	C₃×C₅×Q₈	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×Dic₁₅	—	4	2	no	—
66	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
67	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×Q₈	—	2	1	no	—
70	C₂×C₃×Q₁₆	—	2	1	no	—
71	C₂×C₃×Q₁₆	—	2	2	no	—
72	C₂×C₃×Q₁₆	Dic₅	1	4	yes	complemented
73	C₂×C₃×Q₁₆	C₂×D₅	1	2	yes	—
74	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
75	#926	—	10	1	no	—
76	#818	—	5	2	no	—
77	#817	—	10	1	no	—
78	#812	—	10	2	no	—
79	C₂×C₃⋊C₁₆	—	5	2	no	—
80	C₂×C₅×Q₈	—	2	4	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×Q₈	—	4	2	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	#67	1	1	yes	characteristic
86	C₅×Q₁₆	—	2	4	no	—
87	C₅×Q₁₆	—	2	2	no	—
88	C₅×Q₁₆	C₂×Dic₃	1	4	yes	—
89	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
90	C₂×C₅×C₈	#67	1	1	yes	characteristic
91	C₂×C₅×C₈	#67	1	2	yes	—
92	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
93	C₂²⋊Dic₅	—	12	1	no	—
94	C₄⋊Dic₅	—	6	2	no	—
95	C₄×Dic₅	—	12	1	no	—
96	#672	—	6	2	no	—
97	C₂²×Q₁₆	D₁₅	1	1	yes	characteristic
98	C₂×M₃₂	—	15	1	no	—
99	#427	—	15	1	no	—
100	#358	—	15	4	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	#102	1	2	yes	—
103	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
104	C₃×C₄×C₅	—	4	4	no	—
105	C₃×C₄×C₅	#102	1	2	yes	—
106	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
108	Dic₁₅	—	8	2	no	—
109	C₂×C₃×Q₈	—	2	4	no	—
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×Q₈	—	4	1	no	—
112	C₂×C₃×Q₈	—	4	2	no	—
113	C₂²×C₃×C₄	—	4	1	no	—
114	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
115	C₃×Q₁₆	—	2	4	no	—
116	C₃×Q₁₆	—	2	2	no	—
117	C₃×Q₁₆	C₂×Dic₅	1	4	yes	—
118	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
119	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
121	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
122	#217	—	20	1	no	—
123	C₄⋊Dic₃	—	10	2	no	—
124	C₄×Dic₃	—	20	1	no	—
125	C₃⋊C₁₆	—	10	2	no	—
126	C₂³×C₅	#213	1	1	yes	characteristic
127	C₅×Q₈	—	2	4	no	—
128	C₅×Q₈	—	4	4	no	—
129	C₅×Q₈	—	4	2	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	#213	1	1	yes	characteristic
133	C₂×C₄×C₅	#214	1	2	yes	—
134	C₂×C₄×C₅	#217	1	2	yes	—
135	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
136	C₂×Dic₅	—	12	2	no	—
137	C₅×C₈	#217	1	2	yes	—
138	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
139	C₅×C₈	C₂×#67	1	1	yes	characteristic
140	C₂²×Q₈	—	2	1	no	—
141	C₂×Q₁₆	—	2	1	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	Dic₁₅	1	4	yes	complemented
144	C₂×Q₁₆	C₂×D₁₅	1	2	yes	—
145	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
146	#117	—	30	1	no	—
147	M₃₂	—	30	2	no	—
148	C₂×C₁₆	—	15	2	no	—
149	#107	—	15	2	no	—
150	#106	—	30	1	no	—
151	C₂×C₃×C₅	—	2	2	no	—
152	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
153	C₂×C₃×C₅	#510	1	2	yes	—
154	C₂³×C₃	#686	1	1	yes	characteristic
155	C₃×Q₈	—	2	4	no	—
156	C₃×Q₈	—	4	4	no	—
157	C₃×Q₈	—	4	2	no	—
158	C₂×C₃×C₄	—	4	4	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	#686	1	1	yes	characteristic
161	C₂×C₃×C₄	#687	1	2	yes	—
162	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
164	C₂×Dic₃	—	20	2	no	—
165	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
166	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
167	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	#832	1	2	yes	—
170	C₂²×C₅	C₂×#213	1	1	yes	characteristic
171	C₄×C₅	—	4	4	no	—
172	C₄×C₅	#832	1	2	yes	—
173	C₄×C₅	C₂×#214	1	1	yes	characteristic
174	C₄×C₅	C₂×#217	1	1	yes	characteristic
175	Dic₅	—	24	2	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	#1220	1	1	yes	characteristic
182	Q₁₆	—	2	4	no	—
183	Q₁₆	—	2	2	no	—
184	Q₁₆	C₂×Dic₁₅	1	4	yes	—
185	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
186	C₂×C₈	#1220	1	1	yes	characteristic
187	C₂×C₈	#1220	1	2	yes	—
188	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
189	C₄⋊C₄	—	30	2	no	—
190	C₂²⋊C₄	—	60	1	no	—
191	C₄²	—	60	1	no	—
192	C₁₆	—	30	2	no	—
193	C₃×C₅	#2141	1	1	yes	characteristic, complemented, Hall
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	#3964	1	2	yes	—
196	C₂²×C₃	C₂×#686	1	1	yes	characteristic
197	C₃×C₄	—	4	4	no	—
198	C₃×C₄	#3964	1	2	yes	—
199	C₃×C₄	C₂×#687	1	1	yes	characteristic
200	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
201	Dic₃	—	40	2	no	—
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	#5179	1	2	yes	—
204	C₂×C₅	C₂×#832	1	1	yes	characteristic
205	C₂³	#6715	1	1	yes	characteristic
206	Q₈	—	2	4	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	60	2	no	—
212	C₂×C₄	#6715	1	1	yes	characteristic
213	C₂×C₄	#6716	1	2	yes	—
214	C₂×C₄	#6719	1	2	yes	—
215	C₂×C₄	C₂×#1220	1	1	yes	characteristic
216	C₈	#6719	1	2	yes	—
217	C₈	C₂²×Dic₁₅	1	1	yes	characteristic
218	C₈	C₂×#1220	1	1	yes	characteristic
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	#65468	1	2	yes	—
221	C₂×C₃	C₂×#3964	1	1	yes	characteristic
222	C₅	#72061	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	2	2	no	—
224	C₂²	#91116	1	2	yes	—
225	C₂²	C₂×#6715	1	1	yes	characteristic
226	C₄	—	4	4	no	—
227	C₄	—	120	2	no	—
228	C₄	#91116	1	2	yes	—
229	C₄	C₂×#6716	1	1	yes	characteristic
230	C₄	C₂×#6719	1	1	yes	characteristic
231	C₃	#10602854	1	1	yes	characteristic, complemented, Sylow
232	C₂	—	2	2	no	—
233	C₂	#11751280	1	2	yes	—
234	C₂	C₂×#91116	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6715
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂²×C₃×C₅×Q₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂³×C₃×C₅	D₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×Q₁₆	D₁₅
hypocenter	C₃×C₅	#2141
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #11746385	C₂	semidirect
9. C₂×C₃×C₅×Q₁₆	C₄	semidirect
46. C₂×C₅×Q₁₆	Dic₃	semidirect
72. C₂×C₃×Q₁₆	Dic₅	semidirect
143. C₂×Q₁₆	Dic₁₅	semidirect
193. C₃×C₅	#2141	semidirect
222. C₅	#72061	semidirect
231. C₃	#10602854	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×Q₁₆	2
#11751207	2
C₂×#91096	2
#11746385	2
#10602854	3
#72061	5