Subgroups — #14312583

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91082	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×Dic₁₅	C₂	1	2	yes	maximal, complemented
5	#11746282	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599864	—	3	1	no	maximal, Hall
7	C₂×C₈×D₁₅	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
9	C₄²×D₁₅	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₅	C₄	1	1	yes	characteristic
13	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊#672	C₂²	1	2	yes	—
15	C₄×C₃⋊#168	C₄	1	1	yes	characteristic
16	#91082	C₄	1	8	yes	complemented
17	#69071	—	5	1	no	maximal, Hall
18	C₂×#3930	—	3	2	no	—
19	C₄×C₈×D₅	—	3	1	no	—
20	C₅×#346	D₃	1	1	yes	characteristic, complemented
21	C₁₆×Dic₅	—	3	2	no	—
22	#64681	—	3	1	no	—
23	C₂×C₄×D₁₅	C₈	1	2	yes	—
24	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₅	C₈	1	2	yes	—
29	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
32	C₈×D₁₅	C₈	1	4	yes	—
33	C₈×D₁₅	C₂×C₄	1	4	yes	—
34	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
35	C₃⋊#672	C₂×C₄	1	4	yes	complemented
36	C₂×#798	—	5	2	no	—
37	C₄×D₃×C₈	—	5	1	no	—
38	C₃×#346	D₅	1	1	yes	characteristic, complemented
39	Dic₃×C₁₆	—	5	2	no	—
40	#4461	—	5	1	no	—
41	C₂×C₈×D₅	—	3	1	no	—
42	C₂×C₈×D₅	—	3	1	no	—
43	C₄²×D₅	—	3	1	no	—
44	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
45	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₈×Dic₅	—	3	1	no	—
47	C₈×Dic₅	—	3	1	no	—
48	C₂×#672	—	3	2	no	—
49	C₄×#168	—	3	1	no	—
50	#3930	—	3	8	no	—
51	C₄×M₃₂	—	15	1	no	Sylow
52	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂×C₈	1	2	yes	—
57	C₄×D₁₅	—	2	4	no	—
58	C₄×D₁₅	C₄²	1	2	yes	—
59	C₄×D₁₅	C₂×C₈	1	2	yes	—
60	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
61	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
62	C₃⋊#168	C₄²	1	2	yes	—
63	C₃⋊#168	C₂×C₈	1	2	yes	—
64	C₂×D₃×C₈	—	5	1	no	—
65	C₂×D₃×C₈	—	5	1	no	—
66	C₄²×D₃	—	5	1	no	—
67	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
68	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
69	C₈×Dic₃	—	5	1	no	—
70	C₈×Dic₃	—	5	1	no	—
71	C₂×C₃⋊C₁₆	—	5	2	no	—
72	C₄×C₃⋊C₈	—	5	1	no	—
73	#798	—	5	8	no	—
74	C₂×C₄×D₅	—	3	1	no	—
75	C₂×C₄×D₅	—	3	2	no	—
76	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
78	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
79	C₄×Dic₅	—	3	2	no	—
80	C₄×Dic₅	—	3	1	no	—
81	C₂×#168	—	3	1	no	—
82	C₂×#168	—	3	1	no	—
83	C₈×D₅	—	3	4	no	—
84	C₈×D₅	—	3	4	no	—
85	C₅×C₁₆	C₄×D₃	1	4	yes	complemented
86	#672	—	3	4	no	—
87	C₂×M₃₂	—	15	2	no	—
88	C₂×C₄×C₈	—	15	1	no	—
89	#346	—	15	1	no	—
90	#346	D₁₅	1	1	yes	characteristic, complemented
91	C₄×C₁₆	—	15	2	no	—
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₂×D₁₅	—	2	2	no	—
94	C₂×D₁₅	C₄×C₈	1	2	yes	—
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	Dic₁₅	C₄×C₈	1	2	yes	—
99	Dic₁₅	M₃₂	1	4	yes	complemented
100	C₂×C₄×D₃	—	5	1	no	—
101	C₂×C₄×D₃	—	5	2	no	—
102	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
104	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
105	C₄×Dic₃	—	5	2	no	—
106	C₄×Dic₃	—	5	1	no	—
107	C₂×C₃⋊C₈	—	5	1	no	—
108	C₂×C₃⋊C₈	—	5	1	no	—
109	D₃×C₈	—	5	4	no	—
110	D₃×C₈	—	5	4	no	—
111	C₃×C₁₆	C₄×D₅	1	4	yes	complemented
112	C₃⋊C₁₆	—	5	4	no	—
113	C₂²×D₅	—	3	1	no	—
114	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
115	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂×Dic₅	—	3	2	no	—
117	C₂×Dic₅	—	3	1	no	—
118	C₄×D₅	—	3	2	no	—
119	C₄×D₅	—	3	2	no	—
120	C₄×D₅	—	6	4	no	—
121	C₅×C₈	D₃×C₈	1	2	yes	—
122	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
123	#168	—	3	2	no	—
124	#168	—	3	2	no	—
125	C₂²×C₈	—	15	1	no	—
126	C₂²×C₈	—	15	1	no	—
127	C₂×C₄²	—	15	1	no	—
128	M₃₂	—	15	8	no	—
129	C₂×C₁₆	—	15	2	no	—
130	C₂×C₁₆	C₂×D₁₅	1	2	yes	—
131	C₄×C₈	—	15	1	no	—
132	C₄×C₈	—	15	1	no	—
133	C₄×C₈	—	15	1	no	—
134	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
137	D₁₅	—	2	2	no	—
138	C₂²×D₃	—	5	1	no	—
139	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
141	C₂×Dic₃	—	5	2	no	—
142	C₂×Dic₃	—	5	1	no	—
143	C₄×D₃	—	5	2	no	—
144	C₄×D₃	—	5	2	no	—
145	C₄×D₃	—	10	4	no	—
146	C₃×C₈	C₈×D₅	1	2	yes	—
147	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
148	C₃⋊C₈	—	5	2	no	—
149	C₃⋊C₈	—	5	2	no	—
150	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
151	C₂×D₅	—	3	2	no	—
152	C₂×D₅	—	6	2	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
155	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
156	Dic₅	—	3	4	no	—
157	Dic₅	—	3	2	no	—
158	C₂²×C₄	—	15	1	no	—
159	C₂²×C₄	—	15	2	no	—
160	C₂×C₈	—	15	1	no	—
161	C₂×C₈	—	15	1	no	—
162	C₂×C₈	—	15	4	no	—
163	C₂×C₈	—	15	4	no	—
164	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
166	C₄²	—	15	2	no	—
167	C₄²	—	15	1	no	—
168	C₄²	C₄×D₁₅	1	1	yes	characteristic
169	C₁₆	—	15	4	no	—
170	C₁₆	C₄×D₁₅	1	4	yes	complemented
171	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
173	C₂×D₃	—	5	2	no	—
174	C₂×D₃	—	10	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
177	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
178	Dic₃	—	5	4	no	—
179	Dic₃	—	5	2	no	—
180	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
181	C₂×C₅	D₃×M₃₂	1	2	yes	—
182	D₅	—	6	2	no	—
183	C₂³	—	15	1	no	—
184	C₂×C₄	—	15	2	no	—
185	C₂×C₄	—	15	1	no	—
186	C₂×C₄	—	15	2	no	—
187	C₂×C₄	—	15	2	no	—
188	C₂×C₄	—	30	4	no	—
189	C₂×C₄	C₈×D₁₅	1	2	yes	—
190	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
191	C₈	—	15	2	no	—
192	C₈	—	15	2	no	—
193	C₈	C₈×D₁₅	1	2	yes	—
194	C₈	C₂×C₄×D₁₅	1	2	yes	—
195	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
196	C₂×C₃	D₅×M₃₂	1	2	yes	—
197	D₃	—	10	2	no	—
198	C₅	#69071	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	15	2	no	—
200	C₂²	—	30	2	no	—
201	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	15	4	no	—
204	C₄	—	15	2	no	—
205	C₄	C₄²×D₁₅	1	1	yes	characteristic
206	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
207	C₃	#10599864	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	30	2	no	—
209	C₂	C₄×C₈×D₁₅	1	1	yes	characteristic
210	C₂	D₁₅×M₃₂	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₅×#346	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×D₁₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	#346	D₁₅
hypocenter	C₃×C₅	C₄×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#346	C₂	semidirect
4. C₁₆×Dic₁₅	C₂	semidirect
5. #11746282	C₂	semidirect
16. #91082	C₄	semidirect
20. C₅×#346	D₃	semidirect
34. C₃×C₅×C₁₆	C₂×C₄	semidirect
35. C₃⋊#672	C₂×C₄	semidirect
38. C₃×#346	D₅	semidirect
85. C₅×C₁₆	C₄×D₃	semidirect
90. #346	D₁₅	semidirect
99. Dic₁₅	M₃₂	semidirect
111. C₃×C₁₆	C₄×D₅	semidirect
170. C₁₆	C₄×D₁₅	semidirect
171. C₃×C₅	C₄×M₃₂	semidirect
198. C₅	#69071	semidirect
207. C₃	#10599864	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91082	2
C₄×C₈×D₁₅	2
C₃×C₅×#346	2
C₁₆×Dic₁₅	2
#11746282	2
#10599864	3
#69071	5