Subgroups — D₄×C₁₃×Q₃₂ #15720191

Order 3328.

664 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×Q₃₂	C₂	1	2	yes	maximal, complemented
2	D₄×C₁₃×Q₁₆	C₂	1	2	yes	maximal
3	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2216	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2188	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
10	D₄×Q₈×C₁₃	—	2	2	no	—
11	C₂×C₁₃×Q₃₂	—	2	8	no	—
12	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
17	C₁₃×#494	C₂²	1	2	yes	—
18	C₁₃×#467	C₂²	1	4	yes	—
19	C₁₃×#462	—	2	2	no	—
20	C₁₃×#451	—	2	4	no	—
21	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
22	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#366	C₂²	1	1	yes	characteristic
24	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₃×#358	C₂²	1	2	yes	—
26	C₁₃×#358	C₂²	1	4	yes	—
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₁₃	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	2	8	no	—
31	C₂×C₁₃×Q₁₆	—	2	4	no	—
32	C₂×C₁₃×Q₁₆	—	2	2	no	—
33	C₂×C₁₃×Q₁₆	—	4	4	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
35	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₁₃×C₄⋊Q₈	—	2	2	no	—
38	C₁₃×C₄⋊Q₈	—	4	2	no	—
39	C₁₃×C₂²⋊Q₈	—	2	4	no	—
40	C₁₃×C₂²⋊Q₈	—	4	4	no	—
41	C₄×Q₈×C₁₃	—	2	2	no	—
42	C₄×D₄×C₁₃	—	4	2	no	—
43	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×Q₃₂	—	2	8	no	—
45	C₁₃×Q₃₂	—	2	4	no	—
46	C₁₃×Q₃₂	—	4	2	no	—
47	C₁₃×Q₃₂	D₄	1	4	yes	direct product factor
48	C₂×C₁₃×C₁₆	—	2	4	no	—
49	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×#107	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₁₃×#103	—	2	2	no	—
55	C₁₃×#103	—	2	4	no	—
56	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	D₄×Q₃₂	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	—	4	8	no	—
62	C₂×Q₈×C₁₃	—	4	4	no	—
63	C₂×Q₈×C₁₃	—	8	4	no	—
64	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₁₃×Q₁₆	—	2	4	no	—
68	C₁₃×Q₁₆	—	2	2	no	—
69	C₁₃×Q₁₆	—	2	4	no	—
70	C₁₃×Q₁₆	—	4	8	no	—
71	C₁₃×Q₁₆	—	8	2	no	—
72	C₁₃×Q₁₆	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₁₃×C₁₆	—	2	2	no	—
89	C₁₃×C₁₆	—	2	1	no	—
90	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
91	C₂²×Q₃₂	C₂×C₁₃	1	2	yes	complemented
92	D₄×Q₁₆	C₂×C₁₃	1	2	yes	—
93	#2247	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#2216	C₂×C₁₃	1	2	yes	complemented
95	#2209	C₂×C₁₃	1	2	yes	complemented
96	#2188	C₂×C₁₃	1	4	yes	complemented
97	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
98	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×D₈	1	2	yes	—
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	4	2	no	—
102	Q₈×C₁₃	—	4	4	no	—
103	Q₈×C₁₃	—	4	4	no	—
104	Q₈×C₁₃	—	8	8	no	—
105	D₄×C₁₃	Q₃₂	1	4	yes	direct product factor
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	8	8	no	—
112	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
121	D₄×Q₈	—	2	2	no	—
122	C₂×Q₃₂	—	2	8	no	—
123	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
124	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
125	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
126	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
127	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
128	#494	C₂²×C₁₃	1	2	yes	—
129	#467	C₂²×C₁₃	1	4	yes	—
130	#462	—	2	2	no	—
131	#451	—	2	4	no	—
132	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
133	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
134	#366	C₂²×C₁₃	1	1	yes	characteristic
135	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
136	#358	C₂²×C₁₃	1	2	yes	—
137	#358	C₂²×C₁₃	1	4	yes	—
138	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
139	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	C₂×Q₃₂	1	4	yes	complemented
143	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	4	4	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	—	8	4	no	—
149	C₄×C₁₃	C₂×Q₃₂	1	2	yes	complemented
150	C₄×C₁₃	D₄²	1	1	yes	characteristic
151	C₄×C₁₃	#576	1	1	yes	characteristic
152	C₂²×Q₈	—	2	4	no	—
153	C₂×Q₁₆	—	2	8	no	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	—	2	2	no	—
156	C₂×Q₁₆	—	4	4	no	—
157	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
158	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
159	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	4	2	no	—
162	C₂²⋊Q₈	—	2	4	no	—
163	C₂²⋊Q₈	—	4	4	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×D₄	—	4	2	no	—
166	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
167	Q₃₂	—	2	8	no	—
168	Q₃₂	—	2	4	no	—
169	Q₃₂	—	4	2	no	—
170	Q₃₂	D₄×C₁₃	1	4	yes	direct product factor
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
173	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
174	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
175	#107	C₂³×C₁₃	1	2	yes	—
176	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
177	#103	—	2	2	no	—
178	#103	—	2	4	no	—
179	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
180	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	C₂×C₁₃	—	2	4	no	—
182	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
183	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
184	C₂×C₁₃	#3417	1	1	yes	characteristic
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	4	8	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂×Q₈	—	8	4	no	—
190	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	2	2	no	—
195	Q₁₆	—	2	4	no	—
196	Q₁₆	—	4	8	no	—
197	Q₁₆	—	8	2	no	—
198	Q₁₆	C₂×D₄×C₁₃	1	4	yes	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
203	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
210	C₂²⋊C₄	—	4	4	no	—
211	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
212	C₄²	—	4	2	no	—
213	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₁₆	—	2	2	no	—
215	C₁₆	—	2	1	no	—
216	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
217	C₁₃	D₄×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
218	C₂³	C₂×C₁₃×D₈	1	2	yes	—
219	Q₈	—	2	4	no	—
220	Q₈	—	4	2	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	8	8	no	—
224	D₄	C₁₃×Q₃₂	1	4	yes	direct product factor
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
232	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
233	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	1	no	—
237	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
238	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×C₁₃×Q₃₂	1	4	yes	complemented
242	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	4	no	—
246	C₄	—	8	4	no	—
247	C₄	—	8	2	no	—
248	C₄	C₂×C₁₃×Q₃₂	1	2	yes	complemented
249	C₄	D₄²×C₁₃	1	1	yes	characteristic
250	C₄	C₁₃×#576	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	D₄×C₁₃×D₈	1	1	yes	characteristic
253	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic
254	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×D₄×C₁₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×Q₃₂	C₂	semidirect
3. C₁₃×#2247	C₂	semidirect
4. C₁₃×#2216	C₂	semidirect
5. C₁₃×#2209	C₂	semidirect
6. C₁₃×#2188	C₂	semidirect
7. C₄×C₁₃×Q₃₂	C₂	semidirect
47. C₁₃×Q₃₂	D₄	direct (N × H)
58. D₄×Q₃₂	C₁₃	direct (N × H)
91. C₂²×Q₃₂	C₂×C₁₃	semidirect
93. #2247	C₂×C₁₃	semidirect
94. #2216	C₂×C₁₃	semidirect
95. #2209	C₂×C₁₃	semidirect
96. #2188	C₂×C₁₃	semidirect
97. C₄×Q₃₂	C₂×C₁₃	semidirect
105. D₄×C₁₃	Q₃₂	direct (N × H)
142. C₂²×C₁₃	C₂×Q₃₂	semidirect
149. C₄×C₁₃	C₂×Q₃₂	semidirect
170. Q₃₂	D₄×C₁₃	direct (N × H)
217. C₁₃	D₄×Q₃₂	direct (N × H)
224. D₄	C₁₃×Q₃₂	direct (N × H)
241. C₂²	C₂×C₁₃×Q₃₂	semidirect
248. C₄	C₂×C₁₃×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×Q₃₂	2
D₄×C₁₃×Q₁₆	2
C₁₃×#2247	2
C₁₃×#2216	2
C₁₃×#2209	2
C₁₃×#2188	2
C₄×C₁₃×Q₃₂	2
D₄×C₁₃×C₁₆	2
D₄×Q₃₂	13

Subgroups — D4×C13×Q32 #15720191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₁₃×Q₃₂ #15720191