Subgroups — C₄×D₁₃×Q₃₂ #16016190

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₁₃	C₂	1	2	yes	maximal
3	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116694	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#88069	C₂	1	2	yes	maximal, complemented
7	#13116692	C₂	1	2	yes	maximal, complemented
8	D₁₃×#366	C₂	1	1	yes	maximal, characteristic
9	D₁₃×#358	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
11	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₂²	1	2	yes	—
13	C₄×Q₈×D₁₃	—	2	2	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
17	C₂×#88069	C₂²	1	2	yes	—
18	C₄×#6107	—	2	2	no	—
19	D₁₃×Q₃₂	C₄	1	8	yes	direct product factor
20	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	D₁₃×#107	C₂²	1	2	yes	—
24	#11718534	C₂²	1	2	yes	—
25	D₁₃×#103	—	2	2	no	—
26	#11718451	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	2	yes	—
28	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
32	#11718156	C₂²	1	2	yes	—
33	#11718096	C₂²	1	1	yes	characteristic
34	#11718095	C₂²	1	2	yes	—
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	#11718084	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	C₄×#6090	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₁₃	—	2	2	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	Q₈×Dic₁₃	—	2	2	no	—
44	C₂×#6107	—	2	2	no	—
45	Q₁₆×D₁₃	—	2	4	no	—
46	Q₁₆×D₁₃	C₂×C₄	1	8	yes	—
47	C₂×Dic₅₂	C₂³	1	2	yes	—
48	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×D₁₃	—	2	2	no	—
51	C₄⋊C₄×D₁₃	—	4	2	no	—
52	#88141	—	2	2	no	—
53	#88141	—	4	2	no	—
54	C₄²×D₁₃	—	4	2	no	—
55	C₄²×D₁₃	D₄	1	1	yes	characteristic
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	—	4	2	no	—
58	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#103	—	2	2	no	—
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88075	—	2	2	no	—
65	#88069	C₂×C₄	1	8	yes	complemented
66	#88058	C₂³	1	2	yes	—
67	#88056	—	2	2	no	—
68	C₈×Dic₁₃	D₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88050	—	2	2	no	—
73	#88047	—	2	2	no	—
74	C₄×#1055	D₄	1	1	yes	characteristic
75	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
76	C₁₆×D₁₃	—	2	2	no	—
77	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
78	C₂×C₄×Q₃₂	—	13	1	no	maximal, Sylow
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	Q₈×D₁₃	—	2	8	no	—
81	Q₈×D₁₃	—	4	4	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₂×C₄×D₁₃	—	4	2	no	—
84	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₂₆	—	2	2	no	—
88	C₂×Dic₂₆	—	4	2	no	—
89	C₁₃×Q₁₆	—	2	2	no	—
90	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₄²×C₁₃	—	4	2	no	—
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	#6107	—	2	8	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	4	2	no	—
100	#6101	—	4	2	no	—
101	#6101	—	4	2	no	—
102	#6101	—	4	2	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	C₄×Dic₁₃	—	4	2	no	—
105	C₄×Dic₁₃	—	4	2	no	—
106	C₄×Dic₁₃	D₈	1	1	yes	characteristic
107	C₄×Dic₁₃	D₈	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×#1055	C₂×D₄	1	1	yes	characteristic
110	C₂×#1055	C₂×D₄	1	1	yes	characteristic
111	Dic₅₂	—	2	2	no	—
112	Dic₅₂	C₂²×C₄	1	4	yes	—
113	C₈×D₁₃	—	2	2	no	—
114	C₈×D₁₃	C₂²×C₄	1	2	yes	—
115	C₈×D₁₃	#41	1	2	yes	—
116	C₁₃×C₁₆	—	2	1	no	—
117	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
118	#6090	—	2	1	no	—
119	#6090	C₂²×C₄	1	2	yes	—
120	C₂²×Q₃₂	—	13	1	no	—
121	C₂×C₄×Q₁₆	—	13	2	no	—
122	C₄×Q₃₂	—	13	1	no	—
123	C₄×Q₃₂	—	13	2	no	—
124	C₄×Q₃₂	—	13	1	no	—
125	C₄×Q₃₂	—	13	2	no	—
126	C₄×Q₃₂	—	13	1	no	—
127	C₄×Q₃₂	D₁₃	1	1	yes	characteristic, direct product factor
128	C₂×#366	—	13	1	no	—
129	C₂×#358	—	13	2	no	—
130	C₂×C₄×C₁₆	—	13	1	no	—
131	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
132	Q₈×C₁₃	—	2	4	no	—
133	Q₈×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
144	C₄×D₁₃	—	2	2	no	—
145	C₄×D₁₃	—	4	8	no	—
146	C₄×D₁₃	—	8	4	no	—
147	C₄×D₁₃	Q₃₂	1	4	yes	direct product factor
148	C₄×D₁₃	C₄×D₄	1	2	yes	—
149	C₄×D₁₃	#135	1	2	yes	—
150	Dic₂₆	—	2	4	no	—
151	Dic₂₆	—	4	2	no	—
152	Dic₂₆	—	4	8	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	#1055	—	2	1	no	—
157	#1055	C₄×D₄	1	2	yes	—
158	C₂²×Q₁₆	—	13	2	no	—
159	C₂×C₄×Q₈	—	26	2	no	—
160	C₂×Q₃₂	—	13	2	no	—
161	C₂×Q₃₂	—	13	8	no	—
162	C₂×Q₃₂	—	13	1	no	—
163	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
164	C₂²×C₁₆	—	13	1	no	—
165	C₂²×C₁₆	—	13	1	no	—
166	C₄×Q₁₆	—	13	2	no	—
167	C₄×Q₁₆	—	13	2	no	—
168	C₄×Q₁₆	—	13	2	no	—
169	C₄×Q₁₆	—	26	2	no	—
170	C₄×Q₁₆	—	26	2	no	—
171	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
172	C₂×#107	—	13	2	no	—
173	C₂×#103	—	26	2	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	#366	—	13	1	no	—
176	#366	—	13	2	no	—
177	#366	C₂×D₁₃	1	1	yes	characteristic
178	#358	—	13	2	no	—
179	#358	—	13	2	no	—
180	#358	—	13	2	no	—
181	#358	C₂×D₁₃	1	2	yes	—
182	C₄×C₁₆	—	13	1	no	—
183	C₄×C₁₆	—	13	1	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
186	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
187	C₂×D₁₃	C₄×D₈	1	2	yes	—
188	C₂×D₁₃	C₂×Q₃₂	1	2	yes	—
189	C₂×D₁₃	#508	1	2	yes	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	4	4	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	C₂×Q₃₂	1	2	yes	complemented
194	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
196	Dic₁₃	—	2	1	no	—
197	Dic₁₃	—	4	4	no	—
198	Dic₁₃	—	8	2	no	—
199	Dic₁₃	C₄×D₈	1	2	yes	—
200	Dic₁₃	C₂×Q₃₂	1	2	yes	complemented
201	C₂²×Q₈	—	26	2	no	—
202	C₂×Q₁₆	—	13	8	no	—
203	C₂×Q₁₆	—	13	2	no	—
204	C₂×Q₁₆	—	26	4	no	—
205	C₂×Q₁₆	—	26	2	no	—
206	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	26	2	no	—
211	C₄×Q₈	—	26	2	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	—	52	2	no	—
214	C₄×Q₈	—	52	2	no	—
215	C₂×C₄⋊C₄	—	26	2	no	—
216	C₂×C₄⋊C₄	—	52	2	no	—
217	C₂×C₄²	—	13	1	no	—
218	C₂×C₄²	—	52	2	no	—
219	Q₃₂	—	13	8	no	—
220	Q₃₂	—	13	4	no	—
221	Q₃₂	C₄×D₁₃	1	4	yes	direct product factor
222	C₂×C₁₆	—	13	1	no	—
223	C₂×C₁₆	—	13	4	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	26	2	no	—
226	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
228	#107	—	13	2	no	—
229	#107	—	26	2	no	—
230	#107	C₂²×D₁₃	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	26	2	no	—
233	#103	—	26	2	no	—
234	#103	—	26	2	no	—
235	C₄×C₈	—	13	1	no	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
240	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
242	D₁₃	C₄×Q₃₂	1	4	yes	direct product factor
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	26	8	no	—
245	C₂×Q₈	—	26	2	no	—
246	C₂×Q₈	—	52	2	no	—
247	C₂×Q₈	—	52	4	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	52	2	no	—
252	C₂²×C₄	—	52	2	no	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	13	4	no	—
255	Q₁₆	—	26	8	no	—
256	Q₁₆	—	26	2	no	—
257	Q₁₆	C₂×C₄×D₁₃	1	4	yes	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	13	1	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	52	2	no	—
269	C₄⋊C₄	—	52	2	no	—
270	C₄⋊C₄	—	52	2	no	—
271	C₄⋊C₄	—	52	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	13	1	no	—
274	C₄²	—	13	1	no	—
275	C₄²	—	13	1	no	—
276	C₄²	—	52	2	no	—
277	C₄²	—	52	2	no	—
278	C₄²	—	52	2	no	—
279	C₄²	D₄×D₁₃	1	1	yes	characteristic
280	C₁₆	—	2	1	no	—
281	C₁₆	—	13	2	no	—
282	C₁₆	—	26	1	no	—
283	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
284	C₁₃	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	13	1	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	26	4	no	—
289	Q₈	—	52	8	no	—
290	Q₈	—	52	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	4	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	52	8	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	104	4	no	—
304	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
305	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	13	2	no	—
309	C₈	—	26	1	no	—
310	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
311	C₈	#41×D₁₃	1	1	yes	characteristic
312	C₂²	—	13	2	no	—
313	C₂²	—	13	2	no	—
314	C₂²	—	13	2	no	—
315	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	13	2	no	—
320	C₄	—	13	2	no	—
321	C₄	—	26	1	no	—
322	C₄	—	52	4	no	—
323	C₄	—	104	2	no	—
324	C₄	D₁₃×Q₃₂	1	2	yes	direct product factor
325	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
326	C₄	D₁₃×#135	1	1	yes	characteristic
327	C₂	—	13	4	no	—
328	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic
329	C₂	C₂×D₁₃×Q₃₂	1	1	yes	characteristic
330	C₂	D₁₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×Q₃₂	D₁₃
hypocenter	C₁₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₁₀₄	C₂	semidirect
4. Q₃₂×Dic₁₃	C₂	semidirect
5. #13116694	C₂	semidirect
6. C₄×#88069	C₂	semidirect
7. #13116692	C₂	semidirect
11. C₄×C₁₃×Q₃₂	C₂	semidirect
19. D₁₃×Q₃₂	C₄	direct (N × H)
58. C₁₃×Q₃₂	C₂×C₄	semidirect
65. #88069	C₂×C₄	semidirect
75. Dic₁₀₄	C₂×C₄	semidirect
127. C₄×Q₃₂	D₁₃	direct (N × H)
147. C₄×D₁₃	Q₃₂	direct (N × H)
193. C₄×C₁₃	C₂×Q₃₂	semidirect
200. Dic₁₃	C₂×Q₃₂	semidirect
221. Q₃₂	C₄×D₁₃	direct (N × H)
242. D₁₃	C₄×Q₃₂	direct (N × H)
284. C₁₃	C₂×C₄×Q₃₂	semidirect
324. C₄	D₁₃×Q₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₁₃×Q₃₂	2
C₄×Q₁₆×D₁₃	2
C₄×Dic₁₀₄	2
Q₃₂×Dic₁₃	2
#13116694	2
C₄×#88069	2
#13116692	2
D₁₃×#366	2
D₁₃×#358	2
C₄×C₁₆×D₁₃	2
C₄×C₁₃×Q₃₂	2
C₂×C₄×Q₃₂	13

Subgroups — C4×D13×Q32 #16016190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₁₃×Q₃₂ #16016190