Subgroups — #127×Dic₁₅ #14322256

Order 1920.

1848 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Dic₁₅	C₂	1	6	yes	maximal, complemented
3	#11751322	C₂	1	6	yes	maximal, complemented
4	#11751036	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₅×#127	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	—
8	C₂³×Dic₁₅	C₂²	1	4	yes	—
9	C₃×C₅×#127	C₄	1	8	yes	complemented
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×#6719	C₂²	1	12	yes	—
12	D₄×Dic₁₅	—	2	24	no	—
13	C₂×C₄⋊Dic₁₅	C₂²	1	6	yes	—
14	C₂×C₄×Dic₁₅	C₂²	1	3	yes	—
15	C₂×C₄×Dic₁₅	C₂²	1	3	yes	—
16	Dic₃×#127	—	5	1	no	maximal, Hall
17	C₂×C₅×#127	D₃	1	1	yes	characteristic
18	C₂×D₄×Dic₅	—	3	6	no	—
19	#65510	—	3	6	no	—
20	#65224	—	3	1	no	—
21	C₄²×Dic₅	—	3	1	no	—
22	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
23	C₂×C₃×C₅×D₄	—	2	12	no	—
24	C₂×C₃×C₅×D₄	—	2	12	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	24	yes	complemented
26	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
27	C₂²×Dic₁₅	—	2	24	no	—
28	C₂²×Dic₁₅	C₂³	1	4	yes	—
29	C₂²×Dic₁₅	C₂³	1	3	yes	—
30	C₂²×Dic₁₅	C₂³	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
32	#6719	—	2	24	no	—
33	C₄⋊Dic₁₅	—	2	12	no	—
34	C₄×Dic₁₅	—	2	6	no	—
35	C₄×Dic₁₅	D₄	1	12	yes	complemented
36	C₂×C₃×#127	D₅	1	1	yes	characteristic
37	C₂×D₄×Dic₃	—	5	6	no	—
38	#5221	—	5	6	no	—
39	#4935	—	5	1	no	—
40	C₄²×Dic₃	—	5	1	no	—
41	C₂²×C₅×D₄	C₂×D₃	1	6	yes	—
42	C₂³×Dic₅	—	3	4	no	—
43	C₅×#127	Dic₃	1	8	yes	complemented
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂×C₂²⋊Dic₅	—	3	12	no	—
46	D₄×Dic₅	—	6	24	no	—
47	C₂×C₄⋊Dic₅	—	3	6	no	—
48	C₂×C₄×Dic₅	—	3	3	no	—
49	C₂×C₄×Dic₅	—	3	3	no	—
50	C₄×#127	—	15	1	no	Sylow
51	C₂³×C₃×C₅	—	2	12	no	—
52	C₂³×C₃×C₅	—	2	12	no	—
53	C₂³×C₃×C₅	C₂²×C₄	1	8	yes	—
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×D₄	—	2	48	no	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂×C₃×C₄×C₅	#41	1	6	yes	—
59	C₂×Dic₁₅	—	2	6	no	—
60	C₂×Dic₁₅	—	2	12	no	—
61	C₂×Dic₁₅	—	4	16	no	—
62	C₂×Dic₁₅	C₂×D₄	1	6	yes	—
63	C₂×Dic₁₅	C₂×D₄	1	6	yes	—
64	C₂²×C₃×D₄	C₂×D₅	1	6	yes	—
65	C₂³×Dic₃	—	5	4	no	—
66	C₃×#127	Dic₅	1	8	yes	complemented
67	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
68	C₂×#217	—	5	12	no	—
69	D₄×Dic₃	—	10	24	no	—
70	C₂×C₄⋊Dic₃	—	5	6	no	—
71	C₂×C₄×Dic₃	—	5	3	no	—
72	C₂×C₄×Dic₃	—	5	3	no	—
73	C₂⁴×C₅	C₂²×D₃	1	4	yes	—
74	C₂×C₅×D₄	—	2	12	no	—
75	C₂×C₅×D₄	—	2	12	no	—
76	C₂×C₅×D₄	C₂×Dic₃	1	24	yes	complemented
77	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
78	C₂²×Dic₅	—	3	1	no	—
79	C₂²×Dic₅	—	3	3	no	—
80	C₂²×Dic₅	—	3	4	no	—
81	C₂²×Dic₅	—	6	24	no	—
82	C₄²×C₅	C₂×Dic₃	1	4	yes	complemented
83	C₂²⋊Dic₅	—	6	24	no	—
84	C₄⋊Dic₅	—	6	12	no	—
85	C₄×Dic₅	—	3	12	no	—
86	C₄×Dic₅	—	6	6	no	—
87	C₂×#127	D₁₅	1	1	yes	characteristic
88	C₂×C₄×D₄	—	15	6	no	—
89	#390	—	15	6	no	—
90	#384	—	15	1	no	—
91	C₄³	—	15	1	no	—
92	C₂²×C₃×C₅	—	2	24	no	—
93	C₂²×C₃×C₅	—	4	4	no	—
94	C₂²×C₃×C₅	—	4	12	no	—
95	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
97	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
98	C₃×C₄×C₅	C₄×D₄	1	12	yes	complemented
99	Dic₁₅	—	2	6	no	—
100	Dic₁₅	—	4	4	no	—
101	Dic₁₅	#127	1	4	yes	direct product factor
102	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
103	C₂×C₃×D₄	—	2	12	no	—
104	C₂×C₃×D₄	—	2	12	no	—
105	C₂×C₃×D₄	C₂×Dic₅	1	24	yes	complemented
106	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
107	C₂²×Dic₃	—	5	1	no	—
108	C₂²×Dic₃	—	5	3	no	—
109	C₂²×Dic₃	—	5	4	no	—
110	C₂²×Dic₃	—	10	24	no	—
111	C₃×C₄²	C₂×Dic₅	1	4	yes	complemented
112	#217	—	10	24	no	—
113	C₄⋊Dic₃	—	10	12	no	—
114	C₄×Dic₃	—	5	12	no	—
115	C₄×Dic₃	—	10	6	no	—
116	C₂³×C₅	—	2	12	no	—
117	C₂³×C₅	—	2	12	no	—
118	C₂³×C₅	C₂²×Dic₃	1	8	yes	—
119	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
120	C₅×D₄	—	2	48	no	—
121	C₂×C₄×C₅	D₃×D₄	1	6	yes	—
122	C₂×C₄×C₅	#237	1	6	yes	—
123	C₂×C₄×C₅	C₂²×Dic₃	1	6	yes	—
124	C₂×Dic₅	—	3	6	no	—
125	C₂×Dic₅	—	3	6	no	—
126	C₂×Dic₅	—	6	6	no	—
127	C₂×Dic₅	—	6	12	no	—
128	C₂×Dic₅	—	12	16	no	—
129	C₂²×D₄	C₂×D₁₅	1	6	yes	—
130	C₂³×C₄	—	15	4	no	—
131	#127	Dic₁₅	1	8	yes	direct product factor
132	C₄×D₄	—	30	24	no	—
133	C₂×C₄⋊C₄	—	15	6	no	—
134	C₂×C₂²⋊C₄	—	15	12	no	—
135	C₂×C₄²	—	15	3	no	—
136	C₂×C₄²	—	15	3	no	—
137	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
138	C₂×C₃×C₅	—	4	8	no	—
139	C₂×C₃×C₅	C₂×C₄×D₄	1	3	yes	—
140	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
141	C₂×C₃×C₅	#532	1	3	yes	—
142	C₂³×C₃	—	2	12	no	—
143	C₂³×C₃	—	2	12	no	—
144	C₂³×C₃	C₂²×Dic₅	1	8	yes	—
145	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
146	C₃×D₄	—	2	48	no	—
147	C₂×C₃×C₄	D₄×D₅	1	6	yes	—
148	C₂×C₃×C₄	#711	1	6	yes	—
149	C₂×C₃×C₄	C₂²×Dic₅	1	6	yes	—
150	C₂×Dic₃	—	5	6	no	—
151	C₂×Dic₃	—	5	6	no	—
152	C₂×Dic₃	—	10	6	no	—
153	C₂×Dic₃	—	10	12	no	—
154	C₂×Dic₃	—	20	16	no	—
155	C₂²×C₅	—	2	24	no	—
156	C₂²×C₅	—	4	4	no	—
157	C₂²×C₅	—	4	12	no	—
158	C₂²×C₅	C₂×D₃×D₄	1	3	yes	—
159	C₂²×C₅	C₂×#237	1	3	yes	—
160	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
161	C₄×C₅	D₄×Dic₃	1	12	yes	complemented
162	Dic₅	—	3	4	no	—
163	Dic₅	—	6	6	no	—
164	Dic₅	—	12	4	no	—
165	C₂⁴	C₂²×D₁₅	1	4	yes	—
166	C₂×D₄	—	2	12	no	—
167	C₂×D₄	—	2	12	no	—
168	C₂×D₄	C₂×Dic₁₅	1	24	yes	complemented
169	C₂²×C₄	—	15	1	no	—
170	C₂²×C₄	—	15	3	no	—
171	C₂²×C₄	—	15	4	no	—
172	C₂²×C₄	—	30	24	no	—
173	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
174	C₄⋊C₄	—	30	12	no	—
175	C₂²⋊C₄	—	30	24	no	—
176	C₄²	—	15	12	no	—
177	C₄²	—	30	6	no	—
178	C₄²	C₂×Dic₁₅	1	4	yes	complemented
179	C₃×C₅	C₄×#127	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	—	2	24	no	—
181	C₂²×C₃	—	4	4	no	—
182	C₂²×C₃	—	4	12	no	—
183	C₂²×C₃	C₂×D₄×D₅	1	3	yes	—
184	C₂²×C₃	C₂×#711	1	3	yes	—
185	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
186	C₃×C₄	D₄×Dic₅	1	12	yes	complemented
187	Dic₃	—	5	4	no	—
188	Dic₃	—	10	6	no	—
189	Dic₃	—	20	4	no	—
190	C₂×C₅	—	4	8	no	—
191	C₂×C₅	D₃×#127	1	1	yes	characteristic
192	C₂×C₅	#5716	1	3	yes	—
193	C₂×C₅	C₂×D₄×Dic₃	1	3	yes	—
194	C₂³	—	2	12	no	—
195	C₂³	—	2	12	no	—
196	C₂³	C₂²×Dic₁₅	1	8	yes	—
197	C₂³	C₂³×D₁₅	1	1	yes	characteristic
198	D₄	—	2	48	no	—
199	C₂×C₄	—	15	6	no	—
200	C₂×C₄	—	15	6	no	—
201	C₂×C₄	—	30	6	no	—
202	C₂×C₄	—	30	12	no	—
203	C₂×C₄	—	60	16	no	—
204	C₂×C₄	D₄×D₁₅	1	6	yes	—
205	C₂×C₄	#6819	1	6	yes	—
206	C₂×C₄	C₂²×Dic₁₅	1	6	yes	—
207	C₂×C₃	—	4	8	no	—
208	C₂×C₃	D₅×#127	1	1	yes	characteristic
209	C₂×C₃	#66050	1	3	yes	—
210	C₂×C₃	C₂×D₄×Dic₅	1	3	yes	—
211	C₅	Dic₃×#127	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	24	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	12	no	—
215	C₂²	C₂×D₄×D₁₅	1	3	yes	—
216	C₂²	C₂×#6819	1	3	yes	—
217	C₂²	C₂³×Dic₁₅	1	1	yes	characteristic
218	C₄	—	15	4	no	—
219	C₄	—	30	6	no	—
220	C₄	—	60	4	no	—
221	C₄	D₄×Dic₁₅	1	12	yes	complemented
222	C₃	Dic₅×#127	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	4	8	no	—
224	C₂	D₁₅×#127	1	1	yes	characteristic
225	C₂	#11756475	1	3	yes	—
226	C₂	C₂×D₄×Dic₁₅	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅
commutator	C₂²×C₃×C₅	C₂³×C₄
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle, cosocle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×#127	D₁₅
hypocenter	C₃×C₅	C₄×#127
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
2. C₂×D₄×Dic₁₅	C₂	semidirect
3. #11751322	C₂	semidirect
4. #11751036	C₂	semidirect
5. C₄²×Dic₁₅	C₂	semidirect
9. C₃×C₅×#127	C₄	semidirect
25. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
31. C₃×C₄²×C₅	C₂×C₄	semidirect
35. C₄×Dic₁₅	D₄	semidirect
43. C₅×#127	Dic₃	semidirect
66. C₃×#127	Dic₅	semidirect
76. C₂×C₅×D₄	C₂×Dic₃	semidirect
82. C₄²×C₅	C₂×Dic₃	semidirect
98. C₃×C₄×C₅	C₄×D₄	semidirect
101. Dic₁₅	#127	direct (N × H)
105. C₂×C₃×D₄	C₂×Dic₅	semidirect
111. C₃×C₄²	C₂×Dic₅	semidirect
131. #127	Dic₁₅	direct (N × H)
161. C₄×C₅	D₄×Dic₃	semidirect
168. C₂×D₄	C₂×Dic₁₅	semidirect
178. C₄²	C₂×Dic₁₅	semidirect
179. C₃×C₅	C₄×#127	semidirect
186. C₃×C₄	D₄×Dic₅	semidirect
211. C₅	Dic₃×#127	semidirect
221. C₄	D₄×Dic₁₅	semidirect
222. C₃	Dic₅×#127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#127	2
C₂×D₄×Dic₁₅	2
#11751322	2
#11751036	2
C₄²×Dic₁₅	2
Dic₅×#127	3
Dic₃×#127	5

Subgroups — #127×Dic15 #14322256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #127×Dic₁₅ #14322256