Subgroups — #36709525

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030822	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022455	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022438	C₂	1	1	yes	maximal, characteristic
5	#26022417	C₂	1	2	yes	maximal, complemented
6	#26022315	C₂	1	2	yes	maximal, complemented
7	#26022188	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022171	C₂	1	1	yes	maximal, characteristic
9	#26022124	C₂	1	2	yes	maximal, complemented
10	#26022118	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
13	#19152132	—	2	1	no	—
14	C₂×C₄×Dic₁₇₈	—	2	1	no	—
15	C₂×Dic₇₁₂	C₂²	1	2	yes	—
16	C₂×#15667957	C₂²	1	2	yes	—
17	#19152000	C₂²	1	1	yes	characteristic
18	#19151982	C₂²	1	1	yes	characteristic
19	#19151953	—	2	1	no	—
20	#19151839	C₂²	1	1	yes	characteristic
21	#19151834	C₂²	1	1	yes	characteristic
22	#19151827	—	2	1	no	—
23	C₄×Dic₃₅₆	—	2	1	no	—
24	C₄×Dic₃₅₆	—	2	1	no	—
25	C₄×Dic₃₅₆	C₂²	1	2	yes	—
26	C₄×Dic₃₅₆	C₂²	1	2	yes	—
27	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
28	C₄×#13083563	—	2	1	no	—
29	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
30	C₂×#15667946	C₂²	1	1	yes	characteristic
31	C₂×#15667937	—	2	1	no	—
32	#19151426	C₂²	1	2	yes	—
33	#19151424	C₂²	1	2	yes	—
34	#19151403	C₂²	1	2	yes	—
35	#19151403	C₂²	1	2	yes	—
36	#19151403	C₂²	1	2	yes	—
37	#19151400	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#346×C₈₉	C₂²	1	2	yes	—
43	C₂²×Dic₁₇₈	—	2	1	no	—
44	C₂×Dic₃₅₆	—	2	4	no	—
45	C₂×Dic₃₅₆	—	2	1	no	—
46	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₅₆	C₂³	1	2	yes	—
48	C₂×#13083563	—	2	1	no	—
49	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
50	#15668006	—	4	1	no	—
51	#15668004	—	2	1	no	—
52	#15668000	—	2	1	no	—
53	#15667995	—	4	1	no	—
54	#15667991	—	2	1	no	—
55	#15667986	—	2	1	no	—
56	C₄×Dic₁₇₈	—	2	1	no	—
57	C₄×Dic₁₇₈	—	2	2	no	—
58	C₄×Dic₁₇₈	—	2	2	no	—
59	C₄×Dic₁₇₈	—	4	1	no	—
60	C₄×Dic₁₇₈	—	4	1	no	—
61	C₄×D₃₅₆	—	2	1	no	—
62	C₄×C₂²⋊D₈₉	—	4	1	no	—
63	#15667969	—	4	1	no	—
64	C₂×C₄⋊Dic₈₉	—	2	1	no	—
65	C₂×#13083553	—	4	1	no	—
66	C₂×C₄×Dic₈₉	—	4	1	no	—
67	Dic₇₁₂	—	2	4	no	—
68	#15667957	—	2	4	no	—
69	#15667946	—	2	1	no	—
70	#15667946	C₂³	1	2	yes	—
71	#15667946	C₂³	1	1	yes	characteristic
72	#15667946	C₂³	1	1	yes	characteristic
73	#15667944	—	2	1	no	—
74	#15667937	—	2	1	no	—
75	#15667937	—	2	2	no	—
76	#15667937	—	2	2	no	—
77	#15667937	—	2	1	no	—
78	#15667934	—	2	1	no	—
79	#15667934	—	2	1	no	—
80	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
83	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
84	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
85	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×Dic₁₇₈	—	2	4	no	—
90	C₂×Dic₁₇₈	—	2	2	no	—
91	C₂×Dic₁₇₈	—	2	1	no	—
92	C₂×Dic₁₇₈	—	4	2	no	—
93	C₂×Dic₁₇₈	—	4	1	no	—
94	C₂×D₃₅₆	—	2	1	no	—
95	C₂×C₂²⋊D₈₉	—	4	1	no	—
96	C₂²×Dic₈₉	—	4	1	no	—
97	C₂²×Dic₈₉	—	4	1	no	—
98	C₂×C₄×D₈₉	—	4	1	no	—
99	Dic₃₅₆	—	2	4	no	—
100	Dic₃₅₆	—	2	2	no	—
101	Dic₃₅₆	—	2	2	no	—
102	Dic₃₅₆	#41	1	4	yes	—
103	#13083563	—	4	2	no	—
104	D₇₁₂	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	1	no	—
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	—	2	1	no	—
108	C₄⋊Dic₈₉	—	4	1	no	—
109	#13083553	—	4	1	no	—
110	#13083553	—	4	1	no	—
111	#13083553	—	4	1	no	—
112	#13083553	—	4	2	no	—
113	#13083553	—	4	2	no	—
114	#13083553	—	4	1	no	—
115	C₂²⋊Dic₈₉	—	4	1	no	—
116	#13083551	—	4	1	no	—
117	#13083551	—	4	1	no	—
118	#13083551	—	4	1	no	—
119	#13083551	—	4	1	no	—
120	C₄×Dic₈₉	—	4	1	no	—
121	C₄×Dic₈₉	—	4	2	no	—
122	C₄×Dic₈₉	—	4	2	no	—
123	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
124	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
125	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	—	4	1	no	—
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₄²×C₈₉	D₈	1	1	yes	characteristic
133	C₄²×C₈₉	D₈	1	1	yes	characteristic
134	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₁₆×C₈₉	—	2	2	no	—
137	C₁₆×C₈₉	—	2	2	no	—
138	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	1	no	—
142	C₂×C₄×C₈₉	—	2	1	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₈₉	—	8	2	no	—
151	C₂×Dic₈₉	—	4	4	no	—
152	C₂×Dic₈₉	—	4	2	no	—
153	C₂×Dic₈₉	—	4	2	no	—
154	C₂×Dic₈₉	—	4	1	no	—
155	C₂×Dic₈₉	—	4	1	no	—
156	C₂×Dic₈₉	—	4	1	no	—
157	C₂×Dic₈₉	—	8	2	no	—
158	D₃₅₆	—	4	1	no	—
159	D₃₅₆	—	4	1	no	—
160	C₄×D₈₉	—	8	2	no	—
161	Dic₁₇₈	—	2	4	no	—
162	Dic₁₇₈	—	4	4	no	—
163	Dic₁₇₈	—	4	2	no	—
164	Dic₁₇₈	—	4	1	no	—
165	Dic₁₇₈	—	4	1	no	—
166	C₈×C₈₉	—	2	1	no	—
167	C₈×C₈₉	—	2	1	no	—
168	C₈×C₈₉	—	2	1	no	—
169	C₈×C₈₉	C₂×#41	1	2	yes	—
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	#510	1	2	yes	—
173	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₈₉	—	8	1	no	—
175	C₂×D₈₉	—	8	1	no	—
176	C₂×D₈₉	—	8	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	#525	1	1	yes	characteristic
183	C₄×C₈₉	#575	1	1	yes	characteristic
184	Dic₈₉	—	4	4	no	—
185	Dic₈₉	—	8	1	no	—
186	Dic₈₉	—	8	1	no	—
187	Dic₈₉	—	8	1	no	—
188	Dic₈₉	—	8	2	no	—
189	#33677	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	—	2	2	no	—
191	C₂×C₈₉	#3100	1	1	yes	characteristic
192	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
193	C₂×C₈₉	#3414	1	1	yes	characteristic
194	D₈₉	—	16	1	no	—
195	#3168	—	89	1	no	—
196	C₂×C₄×Q₁₆	—	89	1	no	—
197	#2236	—	89	1	no	—
198	#2233	—	89	1	no	—
199	#2229	—	89	2	no	—
200	#2211	—	89	2	no	—
201	#2188	—	89	1	no	—
202	#2185	—	89	1	no	—
203	#2176	—	89	2	no	—
204	#2175	—	89	2	no	—
205	#2163	D₈₉	1	1	yes	characteristic, complemented
206	C₈₉	#33677	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₁₆	—	89	1	no	—
208	#529	—	178	1	no	—
209	C₂×C₄×Q₈	—	178	1	no	—
210	C₂×Q₃₂	—	89	2	no	—
211	C₂×SD₃₂	—	89	2	no	—
212	#499	—	89	1	no	—
213	#495	—	89	1	no	—
214	#488	—	178	1	no	—
215	#467	—	89	1	no	—
216	#466	—	89	1	no	—
217	#465	—	178	1	no	—
218	C₄×Q₁₆	—	89	1	no	—
219	C₄×Q₁₆	—	89	2	no	—
220	C₄×Q₁₆	—	89	2	no	—
221	C₄×Q₁₆	—	178	1	no	—
222	C₄×Q₁₆	—	178	1	no	—
223	C₄×SD₁₆	—	178	1	no	—
224	C₄×D₈	—	89	1	no	—
225	C₂×#107	—	89	1	no	—
226	C₂×#103	—	178	1	no	—
227	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
228	#367	—	89	2	no	—
229	#366	—	89	2	no	—
230	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
231	#358	—	89	2	no	—
232	#358	—	89	2	no	—
233	#358	—	89	2	no	—
234	#357	—	89	2	no	—
235	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
236	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
237	#346	C₂×D₈₉	1	2	yes	—
238	C₂²×Q₈	—	178	1	no	—
239	C₂×Q₁₆	—	89	2	no	—
240	C₂×Q₁₆	—	89	1	no	—
241	C₂×Q₁₆	—	178	4	no	—
242	C₂×Q₁₆	—	178	1	no	—
243	C₂×SD₁₆	—	178	1	no	—
244	C₂×D₈	—	89	1	no	—
245	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
246	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	#126	—	356	1	no	—
248	#125	—	178	1	no	—
249	#124	—	178	1	no	—
250	#123	—	356	1	no	—
251	C₂²⋊Q₈	—	178	1	no	—
252	#121	—	178	1	no	—
253	C₄×Q₈	—	356	1	no	—
254	C₄×Q₈	—	356	1	no	—
255	C₄×Q₈	—	178	2	no	—
256	C₄×Q₈	—	178	2	no	—
257	C₄×Q₈	—	178	1	no	—
258	C₄×D₄	—	356	1	no	—
259	C₄×D₄	—	178	1	no	—
260	#117	—	356	1	no	—
261	C₂×C₄⋊C₄	—	356	1	no	—
262	C₂×C₄⋊C₄	—	178	1	no	—
263	C₂×C₄²	—	356	1	no	—
264	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
265	Q₃₂	—	178	4	no	—
266	SD₃₂	—	178	4	no	—
267	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
268	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
269	#107	—	89	2	no	—
270	#107	—	89	1	no	—
271	#107	—	89	1	no	—
272	#107	—	178	1	no	—
273	#106	—	178	1	no	—
274	#103	—	178	2	no	—
275	#103	—	178	2	no	—
276	#103	—	178	1	no	—
277	#103	—	178	1	no	—
278	#102	—	178	1	no	—
279	#102	—	178	1	no	—
280	C₄×C₈	D₃₅₆	1	1	yes	characteristic
281	C₄×C₈	D₃₅₆	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
284	C₂×Q₈	—	356	2	no	—
285	C₂×Q₈	—	356	1	no	—
286	C₂×Q₈	—	178	4	no	—
287	C₂×Q₈	—	178	1	no	—
288	C₂×Q₈	—	178	2	no	—
289	C₂×D₄	—	356	1	no	—
290	C₂×D₄	—	178	1	no	—
291	C₂²×C₄	—	356	1	no	—
292	C₂²×C₄	—	356	1	no	—
293	C₂²×C₄	—	356	1	no	—
294	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
295	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
297	Q₁₆	—	89	4	no	—
298	Q₁₆	—	178	4	no	—
299	Q₁₆	—	178	2	no	—
300	Q₁₆	—	178	2	no	—
301	SD₁₆	—	356	2	no	—
302	D₈	—	178	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
306	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
307	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
309	C₄⋊C₄	—	356	2	no	—
310	C₄⋊C₄	—	356	2	no	—
311	C₄⋊C₄	—	356	1	no	—
312	C₄⋊C₄	—	356	1	no	—
313	C₄⋊C₄	—	356	1	no	—
314	C₄⋊C₄	—	356	1	no	—
315	C₄⋊C₄	—	356	1	no	—
316	C₄⋊C₄	—	178	1	no	—
317	C₄⋊C₄	—	178	1	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₂²⋊C₄	—	356	1	no	—
320	C₂²⋊C₄	—	356	1	no	—
321	C₂²⋊C₄	—	356	1	no	—
322	C₂²⋊C₄	—	356	1	no	—
323	C₂²⋊C₄	—	356	1	no	—
324	C₄²	—	356	2	no	—
325	C₄²	—	356	2	no	—
326	C₄²	—	356	1	no	—
327	C₄²	D₇₁₂	1	1	yes	characteristic
328	C₄²	D₇₁₂	1	1	yes	characteristic
329	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
330	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	2	2	no	—
333	C₂³	—	356	1	no	—
334	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
335	Q₈	—	356	4	no	—
336	Q₈	—	356	2	no	—
337	Q₈	—	356	1	no	—
338	Q₈	—	356	1	no	—
339	Q₈	—	178	4	no	—
340	D₄	—	356	1	no	—
341	D₄	—	356	1	no	—
342	D₄	—	712	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	356	4	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	356	1	no	—
351	C₂×C₄	—	356	1	no	—
352	C₂×C₄	—	356	1	no	—
353	C₂×C₄	—	712	2	no	—
354	C₂×C₄	—	712	2	no	—
355	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
356	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
357	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
358	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
359	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	#41×D₈₉	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	712	1	no	—
368	C₂²	—	712	1	no	—
369	C₂²	—	712	1	no	—
370	C₂²	#19152041	1	2	yes	—
371	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	356	4	no	—
378	C₄	—	712	1	no	—
379	C₄	—	712	1	no	—
380	C₄	—	712	2	no	—
381	C₄	—	712	1	no	—
382	C₄	#19152110	1	1	yes	characteristic
383	C₄	#19152398	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	1424	1	no	—
386	C₂	#26030253	1	1	yes	characteristic
387	C₂	C₂×#19152041	1	1	yes	characteristic
388	C₂	#26033388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2163	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#19151834	C₂²
hypercenter	#2163	D₈₉
hypocenter	C₈₉	#33677
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₅₆	C₂	semidirect
3. #26022455	C₂	semidirect
5. #26022417	C₂	semidirect
6. #26022315	C₂	semidirect
7. #26022188	C₂	semidirect
9. #26022124	C₂	semidirect
10. #26022118	C₂	semidirect
11. C₈₉×#2163	C₂	semidirect
205. #2163	D₈₉	semidirect
206. C₈₉	#33677	semidirect

Maximal subgroups

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

subgroup	index
#26030822	2
C₂×C₄×Dic₃₅₆	2
#26022455	2
#26022438	2
#26022417	2
#26022315	2
#26022188	2
#26022171	2
#26022124	2
#26022118	2
C₈₉×#2163	2
#33677	89