Subgroups — #14305945

Order 1920.

616 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	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic
2	#11751037	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic
4	C₂×#90973	C₂	1	4	yes	maximal
5	#10602579	—	3	1	no	maximal, Hall
6	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
10	C₅×#869	—	2	2	no	—
11	C₅×#869	C₄	1	4	yes	complemented
12	C₂²×C₃×#168	C₂²	1	2	yes	—
13	#91115	—	2	2	no	—
14	C₃×#3935	—	2	2	no	—
15	#90973	—	2	8	no	—
16	#69822	—	5	1	no	maximal, Hall
17	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
18	C₂×#3967	—	3	4	no	—
19	#65225	—	3	1	no	—
20	C₂×#3935	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₅	—	2	2	no	—
24	C₂²×C₅×Dic₃	—	2	2	no	—
25	C₂×C₅×Dic₆	—	2	4	no	—
26	C₂×C₅×Dic₆	—	2	4	no	—
27	C₂×C₅×Dic₆	—	2	2	no	—
28	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₄⋊Dic₁₅	—	2	4	no	—
32	C₅×C₄⋊Dic₃	—	2	8	no	—
33	C₂×C₃×#168	—	2	2	no	—
34	C₂×C₃×#168	—	2	2	no	—
35	C₂×C₃×#168	D₄	1	4	yes	—
36	C₂×#869	D₅	1	1	yes	characteristic
37	C₂×C₃×#97	—	5	1	no	—
38	C₂×#816	—	5	4	no	—
39	#4936	—	5	1	no	—
40	C₂²×C₅×Q₈	—	3	2	no	—
41	C₅×C₄⋊Q₈	—	3	4	no	—
42	C₅×C₄⋊Q₈	—	6	2	no	—
43	C₂×C₅×C₄⋊C₄	—	6	2	no	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅	—	3	2	no	—
46	C₂²×#168	C₂×D₃	1	2	yes	—
47	#3967	—	6	8	no	—
48	#3963	—	6	2	no	—
49	#3935	—	2	2	no	—
50	#1960	—	15	1	no	Sylow
51	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	#41	1	2	yes	—
58	C₂×Dic₁₅	—	4	2	no	—
59	C₂×Dic₁₅	—	4	4	no	—
60	C₂×C₅×Dic₃	—	2	4	no	—
61	C₂×C₅×Dic₃	—	4	2	no	—
62	C₂×C₅×Dic₃	—	4	2	no	—
63	C₅×Dic₆	—	2	8	no	—
64	C₅×Dic₆	—	4	4	no	—
65	C₃×#168	—	2	4	no	—
66	C₂²×Dic₆	C₂×D₅	1	2	yes	—
67	C₂²×C₃×C₈	—	5	2	no	—
68	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
69	C₂×C₄⋊Dic₃	—	2	2	no	—
70	C₂×C₄⋊Dic₃	—	5	2	no	—
71	#869	—	2	2	no	—
72	#869	Dic₅	1	4	yes	complemented
73	C₃×#97	—	10	2	no	—
74	C₃⋊#95	—	10	2	no	—
75	#816	—	10	8	no	—
76	C₂×C₅×Q₈	—	3	4	no	—
77	C₂×C₅×Q₈	—	6	2	no	—
78	C₂×C₅×Q₈	—	6	4	no	—
79	C₂×C₅×Q₈	—	6	4	no	—
80	C₂²×C₄×C₅	—	6	2	no	—
81	C₂²×C₄×C₅	D₁₂	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
83	C₂²×Dic₅	—	6	2	no	—
84	C₅×C₄⋊C₄	—	6	8	no	—
85	C₄²×C₅	C₄×D₃	1	2	yes	—
86	C₄²×C₅	#67	1	2	yes	—
87	C₄⋊Dic₅	—	6	4	no	—
88	C₂×#168	—	2	2	no	—
89	C₂×#168	—	2	2	no	—
90	C₂×#168	D₁₂	1	4	yes	—
91	C₂×C₄⋊Q₈	—	3	1	no	—
92	C₂×#103	—	15	4	no	—
93	C₂×#97	—	5	1	no	—
94	#382	—	15	1	no	—
95	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	#124	1	1	yes	characteristic
97	C₂²×C₃×C₅	#127	1	1	yes	characteristic
98	C₂²×C₃×C₅	#137	1	4	yes	—
99	C₃×C₄×C₅	—	2	4	no	—
100	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
101	C₃×C₄×C₅	#121	1	2	yes	—
102	Dic₁₅	—	8	2	no	—
103	C₅×Dic₃	—	4	4	no	—
104	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
105	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
106	C₂²×Dic₃	—	2	2	no	—
107	C₂²×Dic₃	—	10	2	no	—
108	C₂×Dic₆	—	2	4	no	—
109	C₂×Dic₆	—	2	4	no	—
110	C₂×Dic₆	—	2	2	no	—
111	C₂×Dic₆	C₂×Dic₅	1	4	yes	—
112	C₂×C₃×C₈	—	5	4	no	—
113	C₂×C₃×C₈	—	10	2	no	—
114	C₂×C₃×C₈	—	10	2	no	—
115	C₃×C₄²	C₂×Dic₅	1	2	yes	—
116	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
117	C₄⋊Dic₃	—	2	8	no	—
118	C₄⋊Dic₃	—	10	4	no	—
119	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
120	C₅×Q₈	—	6	8	no	—
121	C₅×Q₈	—	12	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	6	4	no	—
124	C₂×C₄×C₅	—	12	2	no	—
125	C₂×C₄×C₅	—	12	2	no	—
126	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
127	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
129	C₂×C₄×C₅	#235	1	2	yes	—
130	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
131	C₂×Dic₅	—	12	2	no	—
132	C₂×Dic₅	—	12	4	no	—
133	#168	—	2	4	no	—
134	C₂²×Q₈	—	3	2	no	—
135	C₂²×C₈	—	5	2	no	—
136	C₄⋊Q₈	—	3	4	no	—
137	C₄⋊Q₈	—	6	2	no	—
138	C₂×C₄⋊C₄	—	6	2	no	—
139	C₂×C₄⋊C₄	—	15	2	no	—
140	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
141	#103	—	30	8	no	—
142	#97	—	10	2	no	—
143	#95	—	30	2	no	—
144	C₂×C₃×C₅	#390	1	1	yes	characteristic
145	C₂×C₃×C₅	#419	1	2	yes	—
146	C₂×C₃×C₅	#491	1	2	yes	—
147	C₂×C₃×C₅	#497	1	2	yes	—
148	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
151	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
152	C₂×C₃×C₄	#711	1	2	yes	—
153	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
155	C₂×Dic₃	—	2	4	no	—
156	C₂×Dic₃	—	4	2	no	—
157	C₂×Dic₃	—	4	2	no	—
158	C₂×Dic₃	—	20	2	no	—
159	C₂×Dic₃	—	20	4	no	—
160	Dic₆	—	2	8	no	—
161	Dic₆	—	4	4	no	—
162	C₃×C₈	—	10	4	no	—
163	C₂²×C₅	#874	1	1	yes	characteristic
164	C₂²×C₅	#875	1	1	yes	characteristic
165	C₂²×C₅	#909	1	4	yes	—
166	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₄×C₅	—	2	4	no	—
168	C₄×C₅	—	12	4	no	—
169	C₄×C₅	C₄×D₁₂	1	2	yes	—
170	C₄×C₅	#930	1	2	yes	—
171	Dic₅	—	24	2	no	—
172	C₂×Q₈	—	3	4	no	—
173	C₂×Q₈	—	6	2	no	—
174	C₂×Q₈	—	6	4	no	—
175	C₂×Q₈	—	6	4	no	—
176	C₂²×C₄	—	6	2	no	—
177	C₂²×C₄	—	30	2	no	—
178	C₂²×C₄	#1203	1	2	yes	—
179	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
180	C₂×C₈	—	5	4	no	—
181	C₂×C₈	—	10	2	no	—
182	C₂×C₈	—	10	2	no	—
183	C₄⋊C₄	—	6	8	no	—
184	C₄⋊C₄	—	30	4	no	—
185	C₄²	D₃×Dic₅	1	2	yes	—
186	C₄²	#1201	1	2	yes	—
187	C₃×C₅	#1960	1	1	yes	characteristic, complemented, Hall
188	C₂²×C₃	#4082	1	1	yes	characteristic
189	C₂²×C₃	#4086	1	1	yes	characteristic
190	C₂²×C₃	#4088	1	4	yes	—
191	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	D₄×Dic₅	1	2	yes	—
194	C₃×C₄	#4084	1	2	yes	—
195	Dic₃	—	4	4	no	—
196	Dic₃	—	40	2	no	—
197	C₂×C₅	#4703	1	2	yes	—
198	C₂×C₅	#4715	1	2	yes	—
199	C₂×C₅	#4939	1	1	yes	characteristic
200	C₂×C₅	#5120	1	2	yes	—
201	C₂³	C₂×#1203	1	1	yes	characteristic
202	Q₈	—	6	8	no	—
203	Q₈	—	12	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	6	4	no	—
206	C₂×C₄	—	12	2	no	—
207	C₂×C₄	—	12	2	no	—
208	C₂×C₄	—	60	2	no	—
209	C₂×C₄	—	60	4	no	—
210	C₂×C₄	(C₂×D₃)⋊Dic₅	1	4	yes	—
211	C₂×C₄	#6772	1	2	yes	—
212	C₂×C₄	D₅×D₁₂	1	2	yes	—
213	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
214	C₂×C₄	C₂×#1201	1	1	yes	characteristic
215	C₈	—	10	4	no	—
216	C₂×C₃	#65371	1	2	yes	—
217	C₂×C₃	#65375	1	2	yes	—
218	C₂×C₃	#65510	1	1	yes	characteristic
219	C₂×C₃	#65514	1	2	yes	—
220	C₅	#69822	1	1	yes	characteristic, complemented, Sylow
221	C₂²	#91321	1	4	yes	—
222	C₂²	#91373	1	1	yes	characteristic
223	C₂²	#91451	1	1	yes	characteristic
224	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	12	4	no	—
227	C₄	—	120	2	no	—
228	C₄	Dic₅×D₁₂	1	2	yes	—
229	C₄	#91449	1	2	yes	—
230	C₃	#10602579	1	1	yes	characteristic, complemented, Sylow
231	C₂	#11746931	1	2	yes	—
232	C₂	#11747097	1	2	yes	—
233	C₂	#11748409	1	2	yes	—
234	C₂	#11748764	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1203
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1960
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. C₅×#869	C₄	semidirect
72. #869	Dic₅	semidirect
187. C₃×C₅	#1960	semidirect
220. C₅	#69822	semidirect
230. C₃	#10602579	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#869	2
#11751037	2
C₂×C₃×#3935	2
C₂×#90973	2
#10602579	3
#69822	5