Subgroups — M₃₂×Dic₁₇₈ #36707941

Order 22784.

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×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26022160	C₂	1	2	yes	maximal, complemented
3	#26022158	C₂	1	4	yes	maximal, complemented
4	C₁₆×Dic₁₇₈	C₂	1	2	yes	maximal, complemented
5	#26022008	C₂	1	1	yes	maximal, characteristic
6	#26022007	C₂	1	2	yes	maximal
7	M₃₂×Dic₈₉	C₂	1	2	yes	maximal
8	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
10	C₂×#15667952	C₂²	1	2	yes	—
11	C₈×Dic₁₇₈	C₄	1	2	yes	—
12	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
13	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
14	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₇₈	C₂²	1	2	yes	—
16	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
18	C₂×#15667941	C₄	1	2	yes	—
19	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
20	C₄⋊#13083549	C₂²	1	2	yes	—
21	C₄⋊#13083549	C₂²	1	2	yes	—
22	#19151416	C₂²	1	4	yes	—
23	#19151416	C₂²	1	4	yes	—
24	#19151373	C₂²	1	4	yes	—
25	C₁₆×Dic₈₉	C₂²	1	4	yes	—
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#346×C₈₉	C₂²	1	2	yes	—
30	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
31	C₂²×Dic₁₇₈	C₈	1	1	yes	characteristic
32	C₂²×#10623494	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₇₈	C₈	1	2	yes	—
34	C₄×Dic₁₇₈	C₈	1	1	yes	characteristic
35	C₄×Dic₁₇₈	C₈	1	1	yes	characteristic
36	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
40	C₂×#13083553	C₈	1	2	yes	—
41	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
42	#15667952	—	2	4	no	—
43	C₂×#13083549	C₂³	1	4	yes	—
44	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
45	C₄⋊#10623494	C₂×C₄	1	2	yes	—
46	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
47	#15667941	C₂×C₄	1	2	yes	—
48	#15667941	C₂×C₄	1	2	yes	—
49	#15667941	C₂×C₄	1	2	yes	—
50	#15667941	C₂×C₄	1	2	yes	—
51	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
52	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
53	C₈×Dic₈₉	C₂³	1	2	yes	—
54	C₈×Dic₈₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₈₉	—	2	2	no	—
59	M₃₂×C₈₉	Q₈	1	4	yes	complemented
60	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
62	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₇₈	—	2	4	no	—
67	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
68	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
69	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
70	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
71	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
72	C₂×#10623494	—	2	4	no	—
73	C₂×#10623494	C₂²×C₄	1	2	yes	—
74	C₂×#10623494	C₂²×C₄	1	2	yes	—
75	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
76	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
78	#13083553	C₂×C₈	1	2	yes	—
79	#13083553	C₂×C₈	1	2	yes	—
80	#13083553	C₂×C₈	1	2	yes	—
81	#13083553	C₂×C₈	1	2	yes	—
82	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
83	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
84	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
85	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
86	#13083549	—	2	4	no	—
87	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₈₉	#41	1	2	yes	—
96	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
98	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₁₆×C₈₉	—	2	2	no	—
102	C₁₆×C₈₉	C₂×Q₈	1	4	yes	complemented
103	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#131	1	2	yes	—
113	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₈₉	—	2	4	no	—
115	C₂×Dic₈₉	—	4	2	no	—
116	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
117	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
118	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
119	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
120	Dic₁₇₈	—	2	2	no	—
121	Dic₁₇₈	—	2	4	no	—
122	Dic₁₇₈	M₃₂	1	4	yes	direct product factor
123	C₈×C₈₉	—	2	1	no	—
124	C₈×C₈₉	—	2	1	no	—
125	C₈×C₈₉	C₄×Q₈	1	2	yes	—
126	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
128	#10623494	—	2	2	no	—
129	#10623494	—	2	2	no	—
130	C₂²×C₈₉	—	2	1	no	—
131	C₂²×C₈₉	—	2	1	no	—
132	C₂²×C₈₉	C₈×Q₈	1	2	yes	—
133	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	C₈×Q₈	1	2	yes	—
138	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
139	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
141	Dic₈₉	—	2	2	no	—
142	Dic₈₉	—	2	2	no	—
143	Dic₈₉	—	2	2	no	—
144	Dic₈₉	C₂×M₃₂	1	4	yes	—
145	Q₈×M₃₂	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
148	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
149	C₂×C₈₉	#3407	1	1	yes	characteristic
150	C₂×C₈×Q₈	—	89	1	no	—
151	#2183	—	89	4	no	—
152	#2183	—	89	2	no	—
153	Q₈×C₁₆	—	89	2	no	—
154	#2150	—	89	2	no	—
155	#2150	—	89	1	no	—
156	C₄×M₃₂	—	89	2	no	—
157	C₄×M₃₂	D₈₉	1	1	yes	characteristic
158	C₈₉	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
159	C₂×C₄×Q₈	—	89	1	no	—
160	C₂×M₃₂	—	89	2	no	—
161	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
162	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
163	C₈×Q₈	—	89	2	no	—
164	C₈×Q₈	—	89	1	no	—
165	C₈×Q₈	—	89	1	no	—
166	C₈×Q₈	—	89	1	no	—
167	C₈×Q₈	—	89	2	no	—
168	C₈×Q₈	—	89	1	no	—
169	C₂×C₄⋊C₈	—	89	2	no	—
170	C₂×C₄⋊C₈	—	89	1	no	—
171	C₂×C₄×C₈	—	89	2	no	—
172	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
173	C₄⋊C₁₆	—	89	4	no	—
174	C₄⋊C₁₆	—	89	4	no	—
175	C₄⋊C₁₆	—	89	2	no	—
176	C₄⋊C₁₆	—	89	2	no	—
177	#346	—	89	4	no	—
178	#346	C₂×D₈₉	1	2	yes	—
179	C₄×C₁₆	—	89	4	no	—
180	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
181	C₂²×Q₈	—	89	1	no	—
182	C₂²×C₈	—	89	2	no	—
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₄×Q₈	—	89	1	no	—
186	C₄×Q₈	—	89	2	no	—
187	C₄×Q₈	—	89	1	no	—
188	C₄×Q₈	—	89	1	no	—
189	C₄×Q₈	—	89	2	no	—
190	C₄×Q₈	—	89	1	no	—
191	C₂×C₄⋊C₄	—	89	2	no	—
192	C₂×C₄⋊C₄	—	89	1	no	—
193	C₂×C₄²	—	89	2	no	—
194	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
195	M₃₂	—	2	2	no	—
196	M₃₂	—	178	4	no	—
197	M₃₂	Dic₁₇₈	1	4	yes	direct product factor
198	C₂×C₁₆	—	89	4	no	—
199	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
200	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
201	C₄⋊C₈	—	89	2	no	—
202	C₄⋊C₈	—	89	2	no	—
203	C₄⋊C₈	—	89	2	no	—
204	C₄⋊C₈	—	89	2	no	—
205	C₄⋊C₈	—	89	1	no	—
206	C₄⋊C₈	—	89	2	no	—
207	C₄⋊C₈	—	89	1	no	—
208	C₄×C₈	—	89	2	no	—
209	C₄×C₈	—	89	2	no	—
210	C₄×C₈	—	89	2	no	—
211	C₄×C₈	—	89	2	no	—
212	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
213	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
216	C₂×Q₈	—	89	1	no	—
217	C₂×Q₈	—	89	2	no	—
218	C₂×Q₈	—	89	1	no	—
219	C₂×Q₈	—	178	4	no	—
220	C₂²×C₄	—	89	2	no	—
221	C₂²×C₄	—	89	2	no	—
222	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
223	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	89	2	no	—
227	C₂×C₈	—	89	2	no	—
228	C₂×C₈	—	178	4	no	—
229	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
232	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
233	C₂×C₈	#13083574	1	2	yes	—
234	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
235	C₄⋊C₄	—	89	2	no	—
236	C₄⋊C₄	—	89	2	no	—
237	C₄⋊C₄	—	89	2	no	—
238	C₄⋊C₄	—	89	2	no	—
239	C₄⋊C₄	—	89	2	no	—
240	C₄⋊C₄	—	89	1	no	—
241	C₄⋊C₄	—	89	1	no	—
242	C₄²	—	89	2	no	—
243	C₄²	—	89	2	no	—
244	C₄²	—	89	2	no	—
245	C₄²	—	89	2	no	—
246	C₄²	C₈×D₈₉	1	1	yes	characteristic
247	C₄²	C₈×D₈₉	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
250	C₁₆	—	2	2	no	—
251	C₁₆	—	178	4	no	—
252	C₁₆	C₂×Dic₁₇₈	1	4	yes	complemented
253	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
254	Q₈	—	89	4	no	—
255	Q₈	—	178	4	no	—
256	Q₈	—	178	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	89	2	no	—
260	C₂×C₄	—	89	2	no	—
261	C₂×C₄	—	89	2	no	—
262	C₂×C₄	—	89	2	no	—
263	C₂×C₄	—	356	2	no	—
264	C₂×C₄	—	178	4	no	—
265	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
271	C₂×C₄	#15668018	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	178	2	no	—
276	C₈	—	178	2	no	—
277	C₈	C₄×Dic₁₇₈	1	2	yes	—
278	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
279	C₈	C₂×#13083574	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	C₈×Dic₁₇₈	1	2	yes	—
283	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	89	4	no	—
288	C₄	—	178	2	no	—
289	C₄	—	178	2	no	—
290	C₄	—	178	2	no	—
291	C₄	C₈×Dic₁₇₈	1	2	yes	—
292	C₄	M₃₂×D₈₉	1	2	yes	—
293	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
294	C₄	C₂×#15668018	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×C₈×Dic₁₇₈	1	1	yes	characteristic
297	C₂	C₂×M₃₂×D₈₉	1	1	yes	characteristic
298	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₄×M₃₂×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	Q₈×M₃₂
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #26022160	C₂	semidirect
3. #26022158	C₂	semidirect
4. C₁₆×Dic₁₇₈	C₂	semidirect
59. M₃₂×C₈₉	Q₈	semidirect
102. C₁₆×C₈₉	C₂×Q₈	semidirect
122. Dic₁₇₈	M₃₂	direct (N × H)
158. C₈₉	Q₈×M₃₂	semidirect
197. M₃₂	Dic₁₇₈	direct (N × H)
252. C₁₆	C₂×Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₇₈	2
#26022160	2
#26022158	2
C₁₆×Dic₁₇₈	2
#26022008	2
#26022007	2
M₃₂×Dic₈₉	2
C₄×M₃₂×C₈₉	2
Q₈×M₃₂	89

Subgroups — M32×Dic178 #36707941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×Dic₁₇₈ #36707941