Subgroups — C₂²×D₁₃×M₃₂ #16769378

Order 3328.

2334 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88052	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88038	C₂	1	2	yes	maximal, complemented
4	C₂×D₁₃×M₃₂	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₁₆×D₁₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₁₃	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×#1055	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₄	1	12	yes	—
11	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
12	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
13	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×D₁₃	C₂²	1	12	yes	—
15	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	12	yes	complemented
17	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	complemented
18	C₂×#88052	C₂²	1	12	yes	complemented
19	C₂²×#6090	C₂²	1	2	yes	complemented
20	D₁₃×M₃₂	C₂²	1	64	yes	direct product factor
21	C₂×#88038	C₂²	1	24	yes	complemented
22	C₂×C₁₆×D₁₃	C₂²	1	24	yes	complemented
23	C₂⁴×D₁₃	C₈	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×Dic₁₃	C₈	1	1	yes	characteristic
26	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
27	C₂²×C₄×D₁₃	C₈	1	12	yes	—
28	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
29	C₂²×C₄×D₁₃	C₂×C₄	1	12	yes	—
30	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂×C₄	1	12	yes	—
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	C₂×C₄	1	6	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	48	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	12	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	12	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	6	yes	—
44	C₂×C₈×D₁₃	C₂³	1	6	yes	—
45	C₂×C₈×D₁₃	C₂³	1	6	yes	—
46	C₂×C₈×D₁₃	C₂³	1	16	yes	—
47	C₁₃×M₃₂	C₂³	1	16	yes	complemented
48	C₂×C₁₃×C₁₆	C₂³	1	12	yes	complemented
49	#88052	C₂³	1	16	yes	complemented
50	C₂×#6090	C₂³	1	12	yes	complemented
51	#88038	C₂³	1	32	yes	complemented
52	C₁₆×D₁₃	C₂³	1	32	yes	complemented
53	C₂³×M₃₂	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂³×D₁₃	—	2	8	no	—
56	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂³×D₁₃	C₂×C₈	1	12	yes	—
58	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
60	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×C₈	1	12	yes	—
64	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₁₃	C₂×C₈	1	12	yes	—
67	C₂×C₄×D₁₃	C₂×C₈	1	6	yes	—
68	C₂×C₄×D₁₃	C₂×C₈	1	48	yes	—
69	C₂×C₄×D₁₃	C₂×C₈	1	6	yes	—
70	C₂×C₄×D₁₃	C₂×C₈	1	12	yes	—
71	C₂×C₄×D₁₃	C₂²×C₄	1	6	yes	—
72	C₂×C₄×D₁₃	C₂²×C₄	1	16	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	6	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
79	C₂×#1055	C₂²×C₄	1	6	yes	—
80	C₂×#1055	C₂²×C₄	1	6	yes	—
81	C₂×#1055	C₂²×C₄	1	16	yes	—
82	C₈×D₁₃	C₂²×C₄	1	12	yes	—
83	C₈×D₁₃	C₂²×C₄	1	12	yes	—
84	C₈×D₁₃	C₂²×C₄	1	16	yes	—
85	C₈×D₁₃	C₂²×C₄	1	16	yes	—
86	C₈×D₁₃	C₂⁴	1	4	yes	—
87	C₈×D₁₃	C₂⁴	1	4	yes	—
88	C₁₃×C₁₆	C₂⁴	1	8	yes	complemented
89	#6090	C₂⁴	1	8	yes	complemented
90	C₂⁴×C₈	—	13	1	no	—
91	C₂²×M₃₂	—	13	24	no	—
92	C₂²×M₃₂	—	13	2	no	—
93	C₂²×M₃₂	—	13	1	no	—
94	C₂²×M₃₂	D₁₃	1	1	yes	characteristic, direct product factor
95	C₂³×C₁₆	—	13	2	no	—
96	C₂³×C₁₃	—	2	4	no	—
97	C₂³×C₁₃	C₂²×C₈	1	6	yes	—
98	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	C₂²×D₁₃	—	2	4	no	—
100	C₂²×D₁₃	—	2	48	no	—
101	C₂²×D₁₃	M₃₂	1	8	yes	direct product factor
102	C₂²×D₁₃	C₂²×C₈	1	6	yes	—
103	C₂²×D₁₃	C₂²×C₈	1	16	yes	—
104	C₂²×D₁₃	C₂²×C₈	1	6	yes	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
107	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
110	C₂×Dic₁₃	C₂²×C₈	1	16	yes	—
111	C₂×Dic₁₃	C₂²×C₈	1	6	yes	—
112	C₂×Dic₁₃	C₂²×C₈	1	6	yes	—
113	C₄×D₁₃	C₂²×C₈	1	12	yes	—
114	C₄×D₁₃	C₂²×C₈	1	16	yes	—
115	C₄×D₁₃	C₂²×C₈	1	12	yes	—
116	C₄×D₁₃	C₂²×C₈	1	16	yes	—
117	C₄×D₁₃	C₂³×C₄	1	4	yes	—
118	C₄×D₁₃	C₂³×C₄	1	4	yes	—
119	C₈×C₁₃	C₂³×C₄	1	3	yes	—
120	C₈×C₁₃	C₂³×C₄	1	4	yes	—
121	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
122	#1055	C₂³×C₄	1	4	yes	—
123	#1055	C₂³×C₄	1	4	yes	—
124	C₂⁴×C₄	—	13	1	no	—
125	C₂³×C₈	—	13	1	no	—
126	C₂³×C₈	—	13	12	no	—
127	C₂³×C₈	—	13	1	no	—
128	C₂³×C₈	—	13	1	no	—
129	C₂³×C₈	—	13	12	no	—
130	C₂³×C₈	—	13	1	no	—
131	C₂³×C₈	—	13	1	no	—
132	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂×M₃₂	—	13	12	no	—
134	C₂×M₃₂	—	13	64	no	—
135	C₂×M₃₂	—	13	24	no	—
136	C₂×M₃₂	C₂×D₁₃	1	12	yes	direct product factor
137	C₂²×C₁₆	—	13	2	no	—
138	C₂²×C₁₆	—	13	24	no	—
139	C₂²×C₁₆	C₂×D₁₃	1	2	yes	complemented
140	C₂²×C₁₃	—	2	12	no	—
141	C₂²×C₁₃	C₂×M₃₂	1	4	yes	complemented
142	C₂²×C₁₃	C₂³×C₈	1	4	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	3	yes	—
144	C₂×D₁₃	—	2	12	no	—
145	C₂×D₁₃	—	2	32	no	—
146	C₂×D₁₃	C₂×M₃₂	1	24	yes	direct product factor
147	C₂×D₁₃	C₂³×C₈	1	4	yes	—
148	C₂×D₁₃	C₂³×C₈	1	4	yes	—
149	C₄×C₁₃	C₂³×C₈	1	3	yes	—
150	C₄×C₁₃	C₂³×C₈	1	4	yes	—
151	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
152	Dic₁₃	C₂³×C₈	1	4	yes	—
153	Dic₁₃	C₂³×C₈	1	4	yes	—
154	C₂⁵	—	13	1	no	—
155	C₂³×C₄	—	13	1	no	—
156	C₂³×C₄	—	13	1	no	—
157	C₂³×C₄	—	13	12	no	—
158	C₂³×C₄	—	13	1	no	—
159	C₂³×C₄	—	13	12	no	—
160	C₂³×C₄	—	13	1	no	—
161	C₂³×C₄	—	13	1	no	—
162	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
163	C₂²×C₈	—	13	12	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	13	12	no	—
167	C₂²×C₈	—	13	48	no	—
168	C₂²×C₈	—	13	12	no	—
169	C₂²×C₈	—	13	6	no	—
170	C₂²×C₈	—	13	6	no	—
171	C₂²×C₈	—	13	6	no	—
172	C₂²×C₈	—	13	16	no	—
173	C₂²×C₈	—	13	6	no	—
174	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₄×D₁₃	1	6	yes	—
176	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	M₃₂	—	13	16	no	—
179	M₃₂	—	13	32	no	—
180	M₃₂	C₂²×D₁₃	1	16	yes	direct product factor
181	C₂×C₁₆	—	13	12	no	—
182	C₂×C₁₆	—	13	32	no	—
183	C₂×C₁₆	C₂²×D₁₃	1	12	yes	complemented
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	C₂²×M₃₂	1	6	yes	complemented
186	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
187	D₁₃	—	2	4	no	—
188	D₁₃	C₂²×M₃₂	1	8	yes	direct product factor
189	C₂⁴	—	13	1	no	—
190	C₂⁴	—	13	12	no	—
191	C₂⁴	—	13	1	no	—
192	C₂⁴	—	26	8	no	—
193	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	—	13	12	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	12	no	—
198	C₂²×C₄	—	13	6	no	—
199	C₂²×C₄	—	13	48	no	—
200	C₂²×C₄	—	13	12	no	—
201	C₂²×C₄	—	13	6	no	—
202	C₂²×C₄	—	13	6	no	—
203	C₂²×C₄	—	13	16	no	—
204	C₂²×C₄	—	13	6	no	—
205	C₂²×C₄	C₈×D₁₃	1	6	yes	—
206	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
208	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
209	C₂×C₈	—	13	6	no	—
210	C₂×C₈	—	13	16	no	—
211	C₂×C₈	—	13	6	no	—
212	C₂×C₈	—	13	12	no	—
213	C₂×C₈	—	13	16	no	—
214	C₂×C₈	—	13	16	no	—
215	C₂×C₈	—	13	12	no	—
216	C₂×C₈	—	13	4	no	—
217	C₂×C₈	—	13	4	no	—
218	C₂×C₈	C₂×C₄×D₁₃	1	6	yes	—
219	C₂×C₈	C₂×C₄×D₁₃	1	12	yes	—
220	C₂×C₈	C₂×C₄×D₁₃	1	3	yes	—
221	C₂×C₈	C₂³×D₁₃	1	3	yes	—
222	C₂×C₈	C₂³×D₁₃	1	4	yes	—
223	C₁₆	—	13	8	no	—
224	C₁₆	C₂³×D₁₃	1	8	yes	complemented
225	C₁₃	C₂³×M₃₂	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	2	4	no	—
227	C₂³	—	13	6	no	—
228	C₂³	—	13	8	no	—
229	C₂³	—	13	16	no	—
230	C₂³	—	13	6	no	—
231	C₂³	—	26	48	no	—
232	C₂³	—	26	4	no	—
233	C₂³	C₂×C₈×D₁₃	1	6	yes	—
234	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
235	C₂×C₄	—	13	16	no	—
236	C₂×C₄	—	13	6	no	—
237	C₂×C₄	—	13	6	no	—
238	C₂×C₄	—	13	16	no	—
239	C₂×C₄	—	13	12	no	—
240	C₂×C₄	—	13	12	no	—
241	C₂×C₄	—	13	16	no	—
242	C₂×C₄	—	13	4	no	—
243	C₂×C₄	—	13	4	no	—
244	C₂×C₄	C₂×C₈×D₁₃	1	12	yes	—
245	C₂×C₄	C₂×C₈×D₁₃	1	6	yes	—
246	C₂×C₄	C₂×C₈×D₁₃	1	3	yes	—
247	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
248	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
249	C₈	—	13	4	no	—
250	C₈	—	13	4	no	—
251	C₈	C₂²×C₄×D₁₃	1	4	yes	—
252	C₈	C₂²×C₄×D₁₃	1	3	yes	—
253	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
254	C₂²	—	2	12	no	—
255	C₂²	—	13	4	no	—
256	C₂²	—	13	24	no	—
257	C₂²	—	13	4	no	—
258	C₂²	—	26	32	no	—
259	C₂²	—	26	12	no	—
260	C₂²	D₁₃×M₃₂	1	4	yes	direct product factor
261	C₂²	C₂²×C₈×D₁₃	1	4	yes	—
262	C₂²	C₂²×C₈×D₁₃	1	3	yes	—
263	C₄	—	13	4	no	—
264	C₄	—	13	4	no	—
265	C₄	C₂²×C₈×D₁₃	1	3	yes	—
266	C₄	C₂²×C₈×D₁₃	1	4	yes	—
267	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
268	C₂	—	2	4	no	—
269	C₂	—	13	8	no	—
270	C₂	—	26	4	no	—
271	C₂	C₂×D₁₃×M₃₂	1	6	yes	direct product factor
272	C₂	C₂³×C₈×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×M₃₂	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×D₁₃	C₈
cosocle	C₈×C₁₃	C₂⁵
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	C₂³×M₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂²×#88052	C₂	semidirect
3. C₂²×#88038	C₂	semidirect
4. C₂×D₁₃×M₃₂	C₂	direct (N × H)
5. C₂²×C₁₆×D₁₃	C₂	semidirect
6. C₂²×C₁₃×M₃₂	C₂	semidirect
16. C₂×C₁₃×M₃₂	C₂²	semidirect
17. C₂²×C₁₃×C₁₆	C₂²	semidirect
18. C₂×#88052	C₂²	semidirect
19. C₂²×#6090	C₂²	semidirect
20. D₁₃×M₃₂	C₂²	direct (N × H)
21. C₂×#88038	C₂²	semidirect
22. C₂×C₁₆×D₁₃	C₂²	semidirect
47. C₁₃×M₃₂	C₂³	semidirect
48. C₂×C₁₃×C₁₆	C₂³	semidirect
49. #88052	C₂³	semidirect
50. C₂×#6090	C₂³	semidirect
51. #88038	C₂³	semidirect
52. C₁₆×D₁₃	C₂³	semidirect
88. C₁₃×C₁₆	C₂⁴	semidirect
89. #6090	C₂⁴	semidirect
94. C₂²×M₃₂	D₁₃	direct (N × H)
101. C₂²×D₁₃	M₃₂	direct (N × H)
136. C₂×M₃₂	C₂×D₁₃	direct (N × H)
139. C₂²×C₁₆	C₂×D₁₃	semidirect
141. C₂²×C₁₃	C₂×M₃₂	semidirect
146. C₂×D₁₃	C₂×M₃₂	direct (N × H)
180. M₃₂	C₂²×D₁₃	direct (N × H)
183. C₂×C₁₆	C₂²×D₁₃	semidirect
185. C₂×C₁₃	C₂²×M₃₂	semidirect
188. D₁₃	C₂²×M₃₂	direct (N × H)
224. C₁₆	C₂³×D₁₃	semidirect
225. C₁₃	C₂³×M₃₂	semidirect
260. C₂²	D₁₃×M₃₂	direct (N × H)
271. C₂	C₂×D₁₃×M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₁₃	2
C₂²×#88052	2
C₂²×#88038	2
C₂×D₁₃×M₃₂	2
C₂²×C₁₆×D₁₃	2
C₂²×C₁₃×M₃₂	2
C₂³×M₃₂	13

Subgroups — C22×D13×M32 #16769378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₃×M₃₂ #16769378