Subgroups — C₄×Q₈×Dic₁₅ #14319618

Order 1920.

1368 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₅	C₂	1	2	yes	maximal
3	C₄⋊C₄×Dic₁₅	C₂	1	6	yes	maximal
4	C₄×C₄⋊Dic₁₅	C₂	1	3	yes	maximal
5	C₄²×Dic₁₅	C₂	1	3	yes	maximal
6	C₄×Q₈×Dic₅	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	C₄	1	8	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
11	Q₈×Dic₁₅	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊Dic₁₅	C₂²	1	6	yes	—
13	C₂×C₄×Dic₁₅	C₂²	1	3	yes	—
14	C₂×C₄×Dic₁₅	C₂²	1	6	yes	—
15	C₂×C₄×Dic₁₅	C₂²	1	6	yes	—
16	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
17	#91115	C₂²	1	6	yes	—
18	C₄×Q₈×Dic₃	—	5	1	no	maximal, Hall
19	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic
20	C₂×Q₈×Dic₅	—	3	2	no	—
21	C₄⋊C₄×Dic₅	—	3	6	no	—
22	C₄×C₄⋊Dic₅	—	3	3	no	—
23	C₄²×Dic₅	—	3	3	no	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	8	yes	—
25	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₅	C₂³	1	6	yes	—
30	C₂²×Dic₁₅	C₂³	1	2	yes	—
31	C₃×C₅×C₄⋊C₄	C₂×C₄	1	12	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	12	yes	—
33	C₄⋊Dic₁₅	C₂×C₄	1	24	yes	—
34	C₄×Dic₁₅	—	2	6	no	—
35	C₄×Dic₁₅	—	2	12	no	—
36	C₄×Dic₁₅	C₂×C₄	1	24	yes	—
37	C₄×Dic₁₅	Q₈	1	4	yes	direct product factor
38	C₂×C₃×C₄×Q₈	D₅	1	1	yes	characteristic
39	C₂×Q₈×Dic₃	—	5	2	no	—
40	Dic₃×C₄⋊C₄	—	5	6	no	—
41	C₄×C₄⋊Dic₃	—	5	3	no	—
42	C₄²×Dic₃	—	5	3	no	—
43	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
44	C₄×C₅×Q₈	Dic₃	1	8	yes	complemented
45	C₂×C₅×C₄⋊C₄	C₂×D₃	1	3	yes	—
46	C₂×C₄²×C₅	C₂×D₃	1	3	yes	—
47	Q₈×Dic₅	—	3	16	no	—
48	C₂×C₄⋊Dic₅	—	3	6	no	—
49	C₂×C₄×Dic₅	—	3	6	no	—
50	C₂×C₄×Dic₅	—	3	1	no	—
51	C₂×C₄×Dic₅	—	3	3	no	—
52	C₂×C₄×Dic₅	—	3	6	no	—
53	#3963	—	3	6	no	—
54	C₄²×Q₈	—	15	1	no	Sylow
55	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
56	C₃×C₅×Q₈	C₄²	1	16	yes	complemented
57	C₂×C₃×C₄×C₅	—	2	6	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	12	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
63	C₂×C₃×C₄×C₅	#41	1	2	yes	—
64	C₂×Dic₁₅	—	2	6	no	—
65	C₂×Dic₁₅	—	2	6	no	—
66	C₂×Dic₁₅	C₂²×C₄	1	12	yes	—
67	C₂×Dic₁₅	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₅	C₂×Q₈	1	4	yes	—
69	C₂×Dic₁₅	#41	1	4	yes	—
70	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
71	C₃×C₄×Q₈	Dic₅	1	8	yes	complemented
72	C₂×C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
73	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
74	Q₈×Dic₃	—	5	16	no	—
75	C₂×C₄⋊Dic₃	—	5	6	no	—
76	C₂×C₄×Dic₃	—	5	1	no	—
77	C₂×C₄×Dic₃	—	5	6	no	—
78	C₂×C₄×Dic₃	—	5	3	no	—
79	C₂×C₄×Dic₃	—	5	6	no	—
80	C₃⋊#95	—	5	6	no	—
81	C₂×C₅×Q₈	C₄×D₃	1	8	yes	—
82	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
83	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
84	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
85	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₅	—	3	2	no	—
87	C₂²×Dic₅	—	3	6	no	—
88	C₅×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
89	C₄²×C₅	C₂×Dic₃	1	12	yes	—
90	C₄⋊Dic₅	—	3	24	no	—
91	C₄×Dic₅	—	3	24	no	—
92	C₄×Dic₅	—	3	4	no	—
93	C₄×Dic₅	—	6	12	no	—
94	C₄×Dic₅	—	6	6	no	—
95	C₂×C₄×Q₈	—	15	2	no	—
96	C₂×C₄×Q₈	D₁₅	1	1	yes	characteristic
97	C₄×C₄⋊C₄	—	15	6	no	—
98	C₄×C₄⋊C₄	—	15	3	no	—
99	C₄³	—	15	3	no	—
100	C₂²×C₃×C₅	C₂³×C₄	1	2	yes	—
101	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
105	C₃×C₄×C₅	—	2	6	no	—
106	C₃×C₄×C₅	C₂×C₄²	1	12	yes	—
107	C₃×C₄×C₅	C₄×Q₈	1	4	yes	complemented
108	Dic₁₅	—	2	12	no	—
109	Dic₁₅	C₄×Q₈	1	8	yes	direct product factor
110	C₂×C₃×Q₈	C₄×D₅	1	8	yes	—
111	C₂×C₃×Q₈	C₂×Dic₅	1	4	yes	—
112	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
113	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
114	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂²×Dic₃	—	5	2	no	—
116	C₂²×Dic₃	—	5	6	no	—
117	C₃×C₄⋊C₄	C₂×Dic₅	1	12	yes	—
118	C₃×C₄²	C₂×Dic₅	1	12	yes	—
119	C₄⋊Dic₃	—	5	24	no	—
120	C₄×Dic₃	—	5	24	no	—
121	C₄×Dic₃	—	5	4	no	—
122	C₄×Dic₃	—	10	12	no	—
123	C₄×Dic₃	—	10	6	no	—
124	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
125	C₅×Q₈	C₄×Dic₃	1	16	yes	complemented
126	C₂×C₄×C₅	—	2	6	no	—
127	C₂×C₄×C₅	C₂×C₄×D₃	1	12	yes	—
128	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
129	C₂×C₄×C₅	#239	1	2	yes	—
130	C₂×C₄×C₅	C₂²×Dic₃	1	6	yes	—
131	C₂×C₄×C₅	C₂²×Dic₃	1	6	yes	—
132	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
133	C₂×Dic₅	—	3	4	no	—
134	C₂×Dic₅	—	3	4	no	—
135	C₂×Dic₅	—	3	4	no	—
136	C₂×Dic₅	—	3	12	no	—
137	C₂×Dic₅	—	6	6	no	—
138	C₂×Dic₅	—	6	6	no	—
139	C₂²×Q₈	C₂×D₁₅	1	1	yes	characteristic
140	C₄×Q₈	—	15	16	no	—
141	C₄×Q₈	Dic₁₅	1	8	yes	direct product factor
142	C₂×C₄⋊C₄	—	15	6	no	—
143	C₂×C₄⋊C₄	C₂×D₁₅	1	3	yes	—
144	C₂×C₄²	—	15	6	no	—
145	C₂×C₄²	—	15	1	no	—
146	C₂×C₄²	—	15	6	no	—
147	C₂×C₄²	—	15	3	no	—
148	C₂×C₄²	C₂×D₁₅	1	3	yes	—
149	#95	—	15	6	no	—
150	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
151	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₂×C₃×C₅	C₂×C₄×Q₈	1	2	yes	—
153	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
154	C₂×C₃×C₅	C₄×#41	1	2	yes	—
155	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
156	C₃×Q₈	C₄×Dic₅	1	16	yes	complemented
157	C₂×C₃×C₄	—	2	6	no	—
158	C₂×C₃×C₄	C₂×C₄×D₅	1	12	yes	—
159	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
160	C₂×C₃×C₄	#713	1	2	yes	—
161	C₂×C₃×C₄	C₂²×Dic₅	1	6	yes	—
162	C₂×C₃×C₄	C₂²×Dic₅	1	6	yes	—
163	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
164	C₂×Dic₃	—	5	4	no	—
165	C₂×Dic₃	—	5	4	no	—
166	C₂×Dic₃	—	5	4	no	—
167	C₂×Dic₃	—	5	12	no	—
168	C₂×Dic₃	—	10	6	no	—
169	C₂×Dic₃	—	10	6	no	—
170	C₂²×C₅	C₂²×C₄×D₃	1	2	yes	—
171	C₂²×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
172	C₂²×C₅	C₂×#239	1	1	yes	characteristic
173	C₂²×C₅	D₃×#41	1	2	yes	—
174	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
175	C₄×C₅	—	2	6	no	—
176	C₄×C₅	C₂×C₄×Dic₃	1	12	yes	—
177	C₄×C₅	Q₈×Dic₃	1	4	yes	complemented
178	Dic₅	—	3	8	no	—
179	Dic₅	—	6	12	no	—
180	C₂×Q₈	C₄×D₁₅	1	8	yes	—
181	C₂×Q₈	C₂×Dic₁₅	1	4	yes	—
182	C₂²×C₄	—	15	2	no	—
183	C₂²×C₄	—	15	6	no	—
184	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
185	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
186	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
187	C₄⋊C₄	—	15	24	no	—
188	C₄⋊C₄	C₂×Dic₁₅	1	12	yes	—
189	C₄²	—	15	24	no	—
190	C₄²	—	15	4	no	—
191	C₄²	—	30	12	no	—
192	C₄²	—	30	6	no	—
193	C₄²	C₂×Dic₁₅	1	12	yes	—
194	C₃×C₅	C₄²×Q₈	1	1	yes	characteristic, complemented, Hall
195	C₂²×C₃	C₂²×C₄×D₅	1	2	yes	—
196	C₂²×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
197	C₂²×C₃	C₂×#713	1	1	yes	characteristic
198	C₂²×C₃	D₅×#41	1	2	yes	—
199	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
200	C₃×C₄	—	2	6	no	—
201	C₃×C₄	C₂×C₄×Dic₅	1	12	yes	—
202	C₃×C₄	Q₈×Dic₅	1	4	yes	complemented
203	Dic₃	—	5	8	no	—
204	Dic₃	—	10	12	no	—
205	C₂×C₅	C₄×D₃×Q₈	1	2	yes	—
206	C₂×C₅	C₄×#239	1	2	yes	—
207	C₂×C₅	C₂²×C₄×Dic₃	1	1	yes	characteristic
208	C₂×C₅	C₂×Q₈×Dic₃	1	1	yes	characteristic
209	C₂×C₅	Dic₃×#41	1	1	yes	characteristic
210	C₂³	C₂³×D₁₅	1	1	yes	characteristic
211	Q₈	C₄×Dic₁₅	1	16	yes	direct product factor
212	C₂×C₄	—	2	6	no	—
213	C₂×C₄	—	15	4	no	—
214	C₂×C₄	—	15	4	no	—
215	C₂×C₄	—	15	4	no	—
216	C₂×C₄	—	15	12	no	—
217	C₂×C₄	—	30	6	no	—
218	C₂×C₄	—	30	6	no	—
219	C₂×C₄	C₂×C₄×D₁₅	1	12	yes	—
220	C₂×C₄	Q₈×D₁₅	1	2	yes	—
221	C₂×C₄	#6821	1	2	yes	—
222	C₂×C₄	C₂²×Dic₁₅	1	6	yes	—
223	C₂×C₄	C₂²×Dic₁₅	1	6	yes	—
224	C₂×C₄	C₂²×Dic₁₅	1	2	yes	—
225	C₂×C₃	C₄×Q₈×D₅	1	2	yes	—
226	C₂×C₃	C₄×#713	1	2	yes	—
227	C₂×C₃	C₂²×C₄×Dic₅	1	1	yes	characteristic
228	C₂×C₃	C₂×Q₈×Dic₅	1	1	yes	characteristic
229	C₂×C₃	#41×Dic₅	1	1	yes	characteristic
230	C₅	C₄×Q₈×Dic₃	1	1	yes	characteristic, complemented, Sylow
231	C₂²	C₂²×C₄×D₁₅	1	2	yes	—
232	C₂²	C₂×Q₈×D₁₅	1	1	yes	characteristic
233	C₂²	C₂×#6821	1	1	yes	characteristic
234	C₂²	#41×D₁₅	1	2	yes	—
235	C₂²	C₂³×Dic₁₅	1	1	yes	characteristic
236	C₄	—	2	6	no	—
237	C₄	—	15	8	no	—
238	C₄	—	30	12	no	—
239	C₄	C₂×C₄×Dic₁₅	1	12	yes	—
240	C₄	Q₈×Dic₁₅	1	4	yes	direct product factor
241	C₃	C₄×Q₈×Dic₅	1	1	yes	characteristic, complemented, Sylow
242	C₂	C₄×Q₈×D₁₅	1	2	yes	—
243	C₂	C₄×#6821	1	2	yes	—
244	C₂	C₂²×C₄×Dic₁₅	1	1	yes	characteristic
245	C₂	C₂×Q₈×Dic₁₅	1	1	yes	characteristic
246	C₂	#41×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₅
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₅
hypocenter	C₃×C₅	C₄²×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
8. C₃×C₄×C₅×Q₈	C₄	semidirect
11. Q₈×Dic₁₅	C₄	direct (N × H)
37. C₄×Dic₁₅	Q₈	direct (N × H)
44. C₄×C₅×Q₈	Dic₃	semidirect
56. C₃×C₅×Q₈	C₄²	semidirect
71. C₃×C₄×Q₈	Dic₅	semidirect
107. C₃×C₄×C₅	C₄×Q₈	semidirect
109. Dic₁₅	C₄×Q₈	direct (N × H)
125. C₅×Q₈	C₄×Dic₃	semidirect
141. C₄×Q₈	Dic₁₅	direct (N × H)
156. C₃×Q₈	C₄×Dic₅	semidirect
177. C₄×C₅	Q₈×Dic₃	semidirect
194. C₃×C₅	C₄²×Q₈	semidirect
202. C₃×C₄	Q₈×Dic₅	semidirect
211. Q₈	C₄×Dic₁₅	direct (N × H)
230. C₅	C₄×Q₈×Dic₃	semidirect
240. C₄	Q₈×Dic₁₅	direct (N × H)
241. C₃	C₄×Q₈×Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×Q₈	2
C₂×Q₈×Dic₁₅	2
C₄⋊C₄×Dic₁₅	2
C₄×C₄⋊Dic₁₅	2
C₄²×Dic₁₅	2
C₄×Q₈×Dic₅	3
C₄×Q₈×Dic₃	5

Subgroups — C4×Q8×Dic15 #14319618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈×Dic₁₅ #14319618