Subgroups — M₃₂×Dic₂₆ #16016432

Order 3328.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	#13116744	C₂	1	2	yes	maximal, complemented
3	#13116742	C₂	1	4	yes	maximal, complemented
4	C₁₆×Dic₂₆	C₂	1	2	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic
6	#13116591	C₂	1	2	yes	maximal
7	M₃₂×Dic₁₃	C₂	1	2	yes	maximal
8	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88054	C₄	1	2	yes	—
14	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
15	C₂×#88052	C₂²	1	2	yes	—
16	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
17	C₈×Dic₂₆	C₄	1	2	yes	—
18	C₈×Dic₂₆	C₄	1	1	yes	characteristic
19	C₈×Dic₂₆	C₄	1	1	yes	characteristic
20	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
21	C₈×Dic₂₆	C₂²	1	2	yes	—
22	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
23	C₁₃×#346	C₂²	1	2	yes	—
24	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
25	#11718094	C₂²	1	4	yes	—
26	#11718093	C₂²	1	4	yes	—
27	#11718093	C₂²	1	4	yes	—
28	C₁₆×Dic₁₃	C₂²	1	4	yes	—
29	C₄⋊#6090	C₂²	1	2	yes	—
30	C₄⋊#6090	C₂²	1	2	yes	—
31	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
36	C₂×#6101	C₈	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂²×#1055	C₂×C₄	1	2	yes	—
39	C₄×Dic₂₆	C₈	1	1	yes	characteristic
40	C₄×Dic₂₆	C₈	1	2	yes	—
41	C₄×Dic₂₆	C₈	1	1	yes	characteristic
42	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
45	C₁₃×M₃₂	—	2	2	no	—
46	C₁₃×M₃₂	Q₈	1	4	yes	complemented
47	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
48	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	#88054	C₂×C₄	1	2	yes	—
54	#88054	C₂×C₄	1	2	yes	—
55	#88054	C₂×C₄	1	2	yes	—
56	#88054	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂³	1	2	yes	—
60	C₈×Dic₁₃	C₂³	1	2	yes	—
61	#88052	—	2	4	no	—
62	C₂×#6090	C₂³	1	4	yes	—
63	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
64	C₄⋊#1055	C₂×C₄	1	2	yes	—
65	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
66	Q₈×M₃₂	—	13	1	no	maximal, Sylow
67	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
71	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
72	C₂×Dic₂₆	—	2	4	no	—
73	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
74	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
75	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₁₃	#41	1	2	yes	—
82	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
90	#6101	C₂×C₈	1	2	yes	—
91	#6101	C₂×C₈	1	2	yes	—
92	#6101	C₂×C₈	1	2	yes	—
93	#6101	C₂×C₈	1	2	yes	—
94	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
96	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	C₂²×C₄	1	2	yes	—
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₁₃×C₁₆	—	2	2	no	—
102	C₁₃×C₁₆	C₂×Q₈	1	4	yes	complemented
103	#6090	—	2	4	no	—
104	C₂×C₈×Q₈	—	13	1	no	—
105	#2183	—	13	4	no	—
106	#2183	—	13	2	no	—
107	Q₈×C₁₆	—	13	2	no	—
108	#2150	—	13	2	no	—
109	#2150	—	13	1	no	—
110	C₄×M₃₂	—	13	2	no	—
111	C₄×M₃₂	D₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#131	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	4	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
126	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
129	Dic₂₆	—	2	2	no	—
130	Dic₂₆	—	2	4	no	—
131	Dic₂₆	M₃₂	1	4	yes	direct product factor
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	C₄×Q₈	1	2	yes	—
135	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
137	#1055	—	2	2	no	—
138	#1055	—	2	2	no	—
139	C₂×C₄×Q₈	—	13	1	no	—
140	C₂×M₃₂	—	13	2	no	—
141	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
142	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
143	C₈×Q₈	—	13	2	no	—
144	C₈×Q₈	—	13	1	no	—
145	C₈×Q₈	—	13	1	no	—
146	C₈×Q₈	—	13	1	no	—
147	C₈×Q₈	—	13	2	no	—
148	C₈×Q₈	—	13	1	no	—
149	C₂×C₄⋊C₈	—	13	2	no	—
150	C₂×C₄⋊C₈	—	13	1	no	—
151	C₂×C₄×C₈	—	13	2	no	—
152	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₄⋊C₁₆	—	13	4	no	—
154	C₄⋊C₁₆	—	13	4	no	—
155	C₄⋊C₁₆	—	13	2	no	—
156	C₄⋊C₁₆	—	13	2	no	—
157	#346	—	13	4	no	—
158	#346	C₂×D₁₃	1	2	yes	—
159	C₄×C₁₆	—	13	4	no	—
160	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
164	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	C₈×Q₈	1	2	yes	—
169	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
170	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
172	Dic₁₃	—	2	2	no	—
173	Dic₁₃	—	2	2	no	—
174	Dic₁₃	—	2	2	no	—
175	Dic₁₃	C₂×M₃₂	1	4	yes	—
176	C₂²×Q₈	—	13	1	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₄×Q₈	—	13	1	no	—
181	C₄×Q₈	—	13	2	no	—
182	C₄×Q₈	—	13	1	no	—
183	C₄×Q₈	—	13	1	no	—
184	C₄×Q₈	—	13	2	no	—
185	C₄×Q₈	—	13	1	no	—
186	C₂×C₄⋊C₄	—	13	2	no	—
187	C₂×C₄⋊C₄	—	13	1	no	—
188	C₂×C₄²	—	13	2	no	—
189	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
190	M₃₂	—	2	2	no	—
191	M₃₂	—	26	4	no	—
192	M₃₂	Dic₂₆	1	4	yes	direct product factor
193	C₂×C₁₆	—	13	4	no	—
194	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
195	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
196	C₄⋊C₈	—	13	2	no	—
197	C₄⋊C₈	—	13	2	no	—
198	C₄⋊C₈	—	13	2	no	—
199	C₄⋊C₈	—	13	2	no	—
200	C₄⋊C₈	—	13	1	no	—
201	C₄⋊C₈	—	13	2	no	—
202	C₄⋊C₈	—	13	1	no	—
203	C₄×C₈	—	13	2	no	—
204	C₄×C₈	—	13	2	no	—
205	C₄×C₈	—	13	2	no	—
206	C₄×C₈	—	13	2	no	—
207	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
208	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
213	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
214	C₂×C₁₃	#3407	1	1	yes	characteristic
215	C₂×Q₈	—	13	1	no	—
216	C₂×Q₈	—	13	2	no	—
217	C₂×Q₈	—	13	1	no	—
218	C₂×Q₈	—	26	4	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
232	C₂×C₈	#6127	1	2	yes	—
233	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
234	C₄⋊C₄	—	13	2	no	—
235	C₄⋊C₄	—	13	2	no	—
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	13	2	no	—
239	C₄⋊C₄	—	13	1	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₄²	—	13	2	no	—
242	C₄²	—	13	2	no	—
243	C₄²	—	13	2	no	—
244	C₄²	—	13	2	no	—
245	C₄²	C₈×D₁₃	1	1	yes	characteristic
246	C₄²	C₈×D₁₃	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
249	C₁₆	—	2	2	no	—
250	C₁₆	—	26	4	no	—
251	C₁₆	C₂×Dic₂₆	1	4	yes	complemented
252	C₁₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
253	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
254	Q₈	—	13	4	no	—
255	Q₈	—	26	4	no	—
256	Q₈	—	26	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	26	4	no	—
264	C₂×C₄	—	52	2	no	—
265	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
271	C₂×C₄	#88155	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	26	2	no	—
276	C₈	—	26	2	no	—
277	C₈	C₄×Dic₂₆	1	2	yes	—
278	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
279	C₈	C₂×#6127	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	C₈×Dic₂₆	1	2	yes	—
283	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	13	4	no	—
288	C₄	—	26	2	no	—
289	C₄	—	26	2	no	—
290	C₄	—	26	2	no	—
291	C₄	C₈×Dic₂₆	1	2	yes	—
292	C₄	D₁₃×M₃₂	1	2	yes	—
293	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
294	C₄	C₂×#88155	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×C₈×Dic₂₆	1	1	yes	characteristic
297	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic
298	C₂	#13127938	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₄×C₁₃×M₃₂	C₂
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13116744	C₂	semidirect
3. #13116742	C₂	semidirect
4. C₁₆×Dic₂₆	C₂	semidirect
46. C₁₃×M₃₂	Q₈	semidirect
102. C₁₃×C₁₆	C₂×Q₈	semidirect
131. Dic₂₆	M₃₂	direct (N × H)
192. M₃₂	Dic₂₆	direct (N × H)
251. C₁₆	C₂×Dic₂₆	semidirect
252. C₁₃	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₆	2
#13116744	2
#13116742	2
C₁₆×Dic₂₆	2
#13116592	2
#13116591	2
M₃₂×Dic₁₃	2
C₄×C₁₃×M₃₂	2
Q₈×M₃₂	13

Subgroups — M32×Dic26 #16016432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×Dic₂₆ #16016432