Subgroups — #36707640

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₈₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26022083	C₂	1	2	yes	maximal, complemented
4	#26022076	C₂	1	4	yes	maximal, complemented
5	#26022071	C₂	1	2	yes	maximal, complemented
6	#26021862	C₂	1	2	yes	maximal, complemented
7	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₈₉	C₂²	1	2	yes	—
10	C₂×C₄×D₃₅₆	C₄	1	1	yes	characteristic
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667951	C₂²	1	4	yes	complemented
14	M₃₂×D₈₉	—	2	8	no	—
15	C₂×C₁₆×D₈₉	C₂²	1	4	yes	complemented
16	C₈×D₃₅₆	C₄	1	2	yes	—
17	C₈×D₃₅₆	C₄	1	1	yes	characteristic
18	C₈×D₃₅₆	C₄	1	1	yes	characteristic
19	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
20	C₈×D₃₅₆	C₂²	1	2	yes	—
21	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
23	C₂×#15667924	C₄	1	2	yes	—
24	#19151377	C₂²	1	4	yes	complemented
25	#19151377	C₂²	1	4	yes	complemented
26	#19151374	C₂²	1	2	yes	complemented
27	C₄×#13083549	C₂²	1	2	yes	complemented
28	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
31	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
32	C₂²×D₃₅₆	C₈	1	1	yes	characteristic
33	C₂²×C₄×D₈₉	C₂×C₄	1	2	yes	—
34	C₂²×#10623494	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₈₉	—	2	4	no	—
36	C₂×C₈×D₈₉	—	2	4	no	—
37	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₈₉	C₂³	1	2	yes	—
40	C₂×C₈×D₈₉	C₂³	1	2	yes	—
41	C₄×D₃₅₆	C₈	1	2	yes	—
42	C₄×D₃₅₆	C₈	1	1	yes	characteristic
43	C₄×D₃₅₆	C₈	1	1	yes	characteristic
44	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
45	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
46	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
48	C₂×#13083551	C₈	1	2	yes	—
49	#15667952	—	2	2	no	—
50	#15667952	D₄	1	4	yes	complemented
51	#15667951	—	2	8	no	—
52	C₂×#13083549	C₂³	1	2	yes	—
53	C₂×#13083549	C₂³	1	2	yes	—
54	C₁₆×D₈₉	—	2	8	no	—
55	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#10623494	C₂×C₄	1	2	yes	—
57	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
58	#15667924	C₂×C₄	1	2	yes	—
59	#15667924	C₂×C₄	1	2	yes	—
60	#15667924	C₂×C₄	1	2	yes	—
61	#15667924	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₈₉	—	2	4	no	—
66	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
67	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂³×D₈₉	C₂×C₈	1	2	yes	—
72	C₂×D₃₅₆	—	2	4	no	—
73	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
74	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
75	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
76	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₈₉	—	2	4	no	—
78	C₂×C₄×D₈₉	—	2	4	no	—
79	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
83	C₂×#10623494	—	2	4	no	—
84	C₂×#10623494	C₂²×C₄	1	2	yes	—
85	C₂×#10623494	C₂²×C₄	1	2	yes	—
86	C₈×D₈₉	—	2	4	no	—
87	C₈×D₈₉	—	2	4	no	—
88	C₈×D₈₉	—	2	2	no	—
89	C₈×D₈₉	—	2	2	no	—
90	C₈×D₈₉	—	2	2	no	—
91	C₈×D₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
94	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
95	#13083551	C₂×C₈	1	2	yes	—
96	#13083551	C₂×C₈	1	2	yes	—
97	#13083551	C₂×C₈	1	2	yes	—
98	#13083551	C₂×C₈	1	2	yes	—
99	#13083549	—	2	2	no	—
100	#13083549	C₂×D₄	1	4	yes	complemented
101	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₉	#41	1	2	yes	—
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₁₆×C₈₉	—	2	4	no	—
116	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₈₉	—	2	4	no	—
118	C₂²×D₈₉	—	2	4	no	—
119	C₂²×D₈₉	—	4	2	no	—
120	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
121	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#131	1	2	yes	—
131	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₈₉	—	2	4	no	—
133	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
134	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
135	D₃₅₆	—	2	4	no	—
136	D₃₅₆	—	2	2	no	—
137	D₃₅₆	M₃₂	1	4	yes	complemented
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	2	2	no	—
142	C₄×D₈₉	—	2	4	no	—
143	C₄×D₈₉	—	2	4	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	C₄×D₄	1	2	yes	—
147	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	#10623494	—	2	2	no	—
150	#10623494	—	2	2	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	C₈×D₄	1	2	yes	—
154	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
155	C₂×D₈₉	—	2	2	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	—	2	2	no	—
158	C₂×D₈₉	—	2	2	no	—
159	C₂×D₈₉	—	2	2	no	—
160	C₂×D₈₉	—	4	8	no	—
161	C₂×D₈₉	C₂×M₃₂	1	4	yes	complemented
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	C₈×D₄	1	2	yes	—
166	C₄×C₈₉	C₂×M₃₂	1	2	yes	complemented
167	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	2	2	no	—
171	D₄×M₃₂	—	89	1	no	maximal, Sylow
172	C₂×C₈₉	—	2	2	no	—
173	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
174	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
175	C₂×C₈₉	#3407	1	1	yes	characteristic
176	D₈₉	—	2	4	no	—
177	D₈₉	—	4	2	no	—
178	C₂²×M₃₂	—	89	2	no	—
179	C₂×C₈×D₄	—	89	1	no	—
180	#2169	—	89	2	no	—
181	#2168	—	89	4	no	—
182	D₄×C₁₆	—	89	2	no	—
183	#2150	D₈₉	1	1	yes	characteristic, complemented
184	#2112	—	89	2	no	—
185	C₄×M₃₂	—	89	1	no	—
186	C₈₉	D₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₈	—	89	2	no	—
188	C₂×C₄×D₄	—	89	1	no	—
189	C₂×M₃₂	—	89	1	no	—
190	C₂×M₃₂	—	89	1	no	—
191	C₂×M₃₂	—	89	4	no	—
192	C₂×M₃₂	—	178	8	no	—
193	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
194	C₂²×C₁₆	—	89	4	no	—
195	C₈×D₄	—	89	2	no	—
196	C₈×D₄	—	89	1	no	—
197	C₈×D₄	—	89	1	no	—
198	C₈×D₄	—	89	1	no	—
199	C₈×D₄	—	89	2	no	—
200	C₈×D₄	—	89	1	no	—
201	C₂×C₄⋊C₈	—	89	1	no	—
202	C₂×C₂²⋊C₈	—	89	2	no	—
203	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
204	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
205	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
206	C₂²⋊C₁₆	—	89	4	no	—
207	C₂²⋊C₁₆	—	89	4	no	—
208	#346	—	89	2	no	—
209	C₄×C₁₆	—	89	2	no	—
210	C₂²×D₄	—	89	1	no	—
211	C₂³×C₄	—	89	2	no	—
212	C₂²×C₈	—	89	2	no	—
213	C₂²×C₈	—	89	2	no	—
214	C₂²×C₈	—	89	2	no	—
215	C₂²×C₈	—	89	2	no	—
216	C₂²×C₈	—	89	2	no	—
217	C₂²×C₈	—	178	4	no	—
218	C₂²×C₈	—	178	4	no	—
219	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
221	C₄×D₄	—	89	1	no	—
222	C₄×D₄	—	89	2	no	—
223	C₄×D₄	—	89	1	no	—
224	C₄×D₄	—	89	1	no	—
225	C₄×D₄	—	89	2	no	—
226	C₄×D₄	—	89	1	no	—
227	C₂×C₄⋊C₄	—	89	1	no	—
228	C₂×C₂²⋊C₄	—	89	2	no	—
229	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
230	M₃₂	—	2	4	no	—
231	M₃₂	—	89	4	no	—
232	M₃₂	—	178	8	no	—
233	M₃₂	—	178	2	no	—
234	C₂×C₁₆	—	89	2	no	—
235	C₂×C₁₆	—	89	2	no	—
236	C₂×C₁₆	—	178	8	no	—
237	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
238	C₄⋊C₈	—	89	1	no	—
239	C₄⋊C₈	—	89	2	no	—
240	C₄⋊C₈	—	89	1	no	—
241	C₂²⋊C₈	—	89	2	no	—
242	C₂²⋊C₈	—	89	2	no	—
243	C₂²⋊C₈	—	89	2	no	—
244	C₂²⋊C₈	—	89	2	no	—
245	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
246	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂⁴	—	89	2	no	—
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	89	2	no	—
252	C₂×D₄	—	89	1	no	—
253	C₂×D₄	—	178	4	no	—
254	C₂²×C₄	—	89	2	no	—
255	C₂²×C₄	—	89	2	no	—
256	C₂²×C₄	—	89	2	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	—	178	4	no	—
260	C₂²×C₄	—	178	4	no	—
261	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	—	89	2	no	—
267	C₂×C₈	—	178	4	no	—
268	C₂×C₈	—	178	4	no	—
269	C₂×C₈	—	178	4	no	—
270	C₂×C₈	—	178	2	no	—
271	C₂×C₈	—	178	2	no	—
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	—	178	2	no	—
274	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
276	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
277	C₂×C₈	D₄×D₈₉	1	2	yes	—
278	C₂×C₈	#13083575	1	2	yes	—
279	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
280	C₄⋊C₄	—	89	2	no	—
281	C₄⋊C₄	—	89	1	no	—
282	C₄⋊C₄	—	89	1	no	—
283	C₂²⋊C₄	—	89	2	no	—
284	C₂²⋊C₄	—	89	2	no	—
285	C₂²⋊C₄	—	89	2	no	—
286	C₂²⋊C₄	—	89	2	no	—
287	C₄²	C₈×D₈₉	1	1	yes	characteristic
288	C₄²	C₈×D₈₉	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
290	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
291	C₁₆	—	2	4	no	—
292	C₁₆	—	89	4	no	—
293	C₁₆	—	178	2	no	—
294	C₂³	—	89	2	no	—
295	C₂³	—	89	2	no	—
296	C₂³	—	356	2	no	—
297	C₂³	—	178	4	no	—
298	C₂³	—	178	4	no	—
299	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
300	D₄	—	89	4	no	—
301	D₄	—	178	4	no	—
302	D₄	—	178	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	89	2	no	—
307	C₂×C₄	—	178	4	no	—
308	C₂×C₄	—	178	4	no	—
309	C₂×C₄	—	178	4	no	—
310	C₂×C₄	—	178	2	no	—
311	C₂×C₄	—	178	2	no	—
312	C₂×C₄	—	178	2	no	—
313	C₂×C₄	—	178	2	no	—
314	C₂×C₄	#15667974	1	2	yes	—
315	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
317	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
318	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
319	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
320	C₂×C₄	#15668020	1	2	yes	—
321	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	178	2	no	—
325	C₈	—	178	2	no	—
326	C₈	#15667974	1	2	yes	—
327	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
328	C₈	C₂×#13083575	1	1	yes	characteristic
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	89	4	no	—
332	C₂²	—	356	8	no	—
333	C₂²	—	178	2	no	—
334	C₂²	—	178	2	no	—
335	C₂²	—	178	2	no	—
336	C₂²	—	178	2	no	—
337	C₂²	—	178	2	no	—
338	C₂²	#19151663	1	2	yes	—
339	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	178	2	no	—
344	C₄	—	178	2	no	—
345	C₄	#19151663	1	2	yes	—
346	C₄	M₃₂×D₈₉	1	2	yes	complemented
347	C₄	C₂×#15667974	1	1	yes	characteristic
348	C₄	C₂×#15668020	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	356	2	no	—
351	C₂	—	178	4	no	—
352	C₂	C₂×#19151663	1	1	yes	characteristic
353	C₂	C₂×M₃₂×D₈₉	1	1	yes	characteristic
354	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2150	C₂
prime radical	C₂²×D₃₅₆	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2150	D₈₉
hypocenter	C₈₉	D₄×M₃₂
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×M₃₂×D₈₉	C₂	semidirect
3. #26022083	C₂	semidirect
4. #26022076	C₂	semidirect
5. #26022071	C₂	semidirect
6. #26021862	C₂	semidirect
7. C₄×#15667952	C₂	semidirect
8. C₈₉×#2150	C₂	semidirect
13. C₂×#15667951	C₂²	semidirect
15. C₂×C₁₆×D₈₉	C₂²	semidirect
24. #19151377	C₂²	semidirect
25. #19151377	C₂²	semidirect
26. #19151374	C₂²	semidirect
27. C₄×#13083549	C₂²	semidirect
30. C₄⋊C₁₆×C₈₉	C₂²	semidirect
31. C₄⋊C₁₆×C₈₉	C₂²	semidirect
50. #15667952	D₄	semidirect
100. #13083549	C₂×D₄	semidirect
137. D₃₅₆	M₃₂	semidirect
161. C₂×D₈₉	C₂×M₃₂	semidirect
166. C₄×C₈₉	C₂×M₃₂	semidirect
183. #2150	D₈₉	semidirect
186. C₈₉	D₄×M₃₂	semidirect
204. C₄⋊C₁₆	C₂×D₈₉	semidirect
205. C₄⋊C₁₆	C₂×D₈₉	semidirect
346. C₄	M₃₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₈₉	2
C₂×C₈×D₃₅₆	2
#26022083	2
#26022076	2
#26022071	2
#26021862	2
C₄×#15667952	2
C₈₉×#2150	2
D₄×M₃₂	89