Subgroups — #36546029

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033226	C₂	1	1	yes	maximal, characteristic
2	#26032584	C₂	1	2	yes	maximal, complemented
3	#26019845	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019787	C₂	1	2	yes	maximal, complemented
5	#26019653	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019612	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019378	C₂	1	2	yes	maximal, complemented
8	#26019324	C₂	1	2	yes	maximal, complemented
9	#26019137	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152269	C₂²	1	2	yes	—
13	#19151975	C₂²	1	1	yes	characteristic
14	#19151973	—	2	1	no	—
15	#19151972	C₂²	1	2	yes	—
16	#19151946	C₂²	1	2	yes	—
17	#19151911	—	2	2	no	—
18	#19151890	C₂²	1	2	yes	—
19	#19151846	—	2	2	no	—
20	#19151798	C₂²	1	2	yes	—
21	#19151793	C₂²	1	2	yes	—
22	#19151792	—	2	2	no	—
23	#19151758	—	2	2	no	—
24	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
25	C₄×#13083564	C₂²	1	1	yes	characteristic
26	#19151680	—	2	1	no	—
27	#19151676	C₂²	1	2	yes	—
28	#19151597	—	2	2	no	—
29	#19151351	C₂²	1	1	yes	characteristic
30	#19151350	C₂²	1	1	yes	characteristic
31	#19151347	C₂²	1	2	yes	—
32	#19151341	C₂²	1	2	yes	—
33	Q₈⋊#10623494	C₂²	1	1	yes	characteristic
34	#19151328	C₂²	1	1	yes	characteristic
35	D₄⋊#10623494	C₂²	1	2	yes	—
36	C₈×#10623494	C₂²	1	1	yes	characteristic
37	#554×C₈₉	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#499×C₈₉	—	2	1	no	—
40	#478×C₈₉	—	2	2	no	—
41	#476×C₈₉	C₂²	1	2	yes	—
42	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₈₉	—	2	1	no	—
44	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#13083574	—	2	2	no	—
46	C₂×#13083564	D₄	1	1	yes	characteristic
47	C₂×#13083561	—	2	2	no	—
48	C₂×#13083561	—	4	2	no	—
49	C₂×D₇₁₂	—	2	1	no	—
50	C₂×D₇₁₂	—	2	1	no	—
51	C₂×#13083559	—	2	2	no	—
52	C₂×#13083558	—	2	2	no	—
53	#15668009	C₂³	1	2	yes	—
54	#15668008	—	2	2	no	—
55	#15667986	—	2	2	no	—
56	#15667985	—	4	2	no	—
57	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
58	#15667974	—	2	2	no	—
59	#15667945	—	2	2	no	—
60	#15667945	D₄	1	1	yes	characteristic
61	C₄⋊#10623494	C₂³	1	2	yes	—
62	C₄⋊#10623494	C₂³	1	1	yes	characteristic
63	#15667941	—	2	1	no	—
64	#15667936	D₄	1	1	yes	characteristic
65	#15667935	—	2	2	no	—
66	#15667935	D₄	1	1	yes	characteristic
67	#15667932	—	2	2	no	—
68	#15667932	—	2	2	no	—
69	#15667932	—	2	2	no	—
70	#15667932	—	2	2	no	—
71	C₂²⋊#10623494	—	2	2	no	—
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₈×Dic₈₉	—	2	1	no	—
75	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
77	#125×C₈₉	—	2	2	no	—
78	C₂²⋊Q₈×C₈₉	—	2	2	no	—
79	C₂²⋊Q₈×C₈₉	—	4	2	no	—
80	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₈₉	—	2	2	no	—
82	C₄×D₄×C₈₉	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
84	#107×C₈₉	—	2	1	no	—
85	#107×C₈₉	—	2	1	no	—
86	#107×C₈₉	D₄	1	1	yes	characteristic
87	#106×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	—	2	1	no	—
89	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
90	#102×C₈₉	—	2	2	no	—
91	#102×C₈₉	D₄	1	2	yes	—
92	C₄×C₈×C₈₉	—	2	1	no	—
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	#13083574	—	4	4	no	—
95	#13083574	—	8	2	no	—
96	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
97	C₂×D₃₅₆	—	2	2	no	—
98	C₂×D₃₅₆	—	2	2	no	—
99	C₂×D₃₅₆	—	2	2	no	—
100	C₂×C₂²⋊D₈₉	—	4	2	no	—
101	C₂×C₂²⋊D₈₉	—	4	2	no	—
102	C₂×C₄×D₈₉	—	4	2	no	—
103	#13083564	—	2	2	no	—
104	#13083561	—	8	4	no	—
105	#13083561	—	8	2	no	—
106	D₇₁₂	—	4	2	no	—
107	D₇₁₂	—	8	1	no	—
108	#13083559	—	8	2	no	—
109	#13083558	—	8	2	no	—
110	C₂×#10623494	—	2	1	no	—
111	C₂×#10623494	—	2	2	no	—
112	C₂×#10623494	—	2	1	no	—
113	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
114	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
115	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	#13083553	—	2	1	no	—
117	#13083551	—	4	2	no	—
118	C₄×Dic₈₉	—	2	1	no	—
119	C₂×Q₈×C₈₉	—	2	2	no	—
120	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₈₉	—	2	2	no	—
123	C₂²×C₄×C₈₉	—	4	2	no	—
124	D₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	1	no	—
126	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₈₉	—	2	1	no	—
129	C₄⋊C₄×C₈₉	—	2	2	no	—
130	C₄⋊C₄×C₈₉	—	2	2	no	—
131	C₄⋊C₄×C₈₉	—	4	2	no	—
132	C₄⋊C₄×C₈₉	—	4	2	no	—
133	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
135	C₂²⋊C₄×C₈₉	—	2	2	no	—
136	C₂²⋊C₄×C₈₉	—	4	2	no	—
137	C₄²×C₈₉	—	2	1	no	—
138	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
139	C₂³×C₈₉	—	2	2	no	—
140	C₂²×D₈₉	—	4	2	no	—
141	Q₈×C₈₉	—	2	1	no	—
142	Q₈×C₈₉	—	2	1	no	—
143	Q₈×C₈₉	—	8	2	no	—
144	D₄×C₈₉	—	2	2	no	—
145	D₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	2	1	no	—
147	C₂×C₄×C₈₉	—	2	1	no	—
148	C₂×C₄×C₈₉	—	2	2	no	—
149	C₂×C₄×C₈₉	—	4	2	no	—
150	C₂×C₄×C₈₉	—	4	2	no	—
151	C₂×C₄×C₈₉	—	4	2	no	—
152	C₂×C₄×C₈₉	—	16	2	no	—
153	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₈₉	—	8	4	no	—
157	C₂²⋊D₈₉	—	16	2	no	—
158	C₂×Dic₈₉	—	2	1	no	—
159	C₂×Dic₈₉	—	2	1	no	—
160	D₃₅₆	—	4	2	no	—
161	D₃₅₆	—	4	2	no	—
162	D₃₅₆	—	4	2	no	—
163	D₃₅₆	—	4	2	no	—
164	D₃₅₆	—	8	2	no	—
165	C₄×D₈₉	—	8	4	no	—
166	Dic₁₇₈	—	2	1	no	—
167	Dic₁₇₈	#135	1	2	yes	—
168	C₈×C₈₉	—	2	1	no	—
169	C₈×C₈₉	—	4	1	no	—
170	C₈×C₈₉	#137	1	2	yes	—
171	#10623494	—	2	2	no	—
172	#10623494	—	2	1	no	—
173	#10623494	—	2	1	no	—
174	#10623494	—	4	2	no	—
175	#10623494	—	4	1	no	—
176	C₂²×C₈₉	—	4	2	no	—
177	C₂²×C₈₉	—	4	2	no	—
178	C₂²×C₈₉	—	4	2	no	—
179	C₂²×C₈₉	D₄²	1	1	yes	characteristic
180	C₂×D₈₉	—	8	2	no	—
181	C₂×D₈₉	—	8	2	no	—
182	C₂×D₈₉	—	8	2	no	—
183	C₄×C₈₉	—	2	1	no	—
184	C₄×C₈₉	—	4	2	no	—
185	C₄×C₈₉	—	4	1	no	—
186	C₄×C₈₉	—	4	1	no	—
187	C₄×C₈₉	—	8	2	no	—
188	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
189	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
190	C₄×C₈₉	#576	1	1	yes	characteristic
191	C₄×C₈₉	#577	1	1	yes	characteristic
192	Dic₈₉	—	2	2	no	—
193	Dic₈₉	—	4	1	no	—
194	#19909	—	89	1	no	maximal, Sylow
195	C₂×C₈₉	—	8	2	no	—
196	C₂×C₈₉	#3278	1	1	yes	characteristic
197	C₂×C₈₉	#3280	1	1	yes	characteristic
198	C₂×C₈₉	#3292	1	1	yes	characteristic
199	D₈₉	—	16	2	no	—
200	#3391	—	89	1	no	—
201	#3380	D₈₉	1	1	yes	characteristic, complemented
202	#3333	—	89	2	no	—
203	#1706	—	89	1	no	—
204	#1697	—	89	2	no	—
205	#1676	—	89	1	no	—
206	#1668	—	89	1	no	—
207	#1631	—	89	2	no	—
208	#1622	—	89	2	no	—
209	#1589	—	89	1	no	—
210	C₈×Q₁₆	—	89	1	no	—
211	C₈₉	#19909	1	1	yes	characteristic, complemented, Sylow
212	#554	C₂×D₈₉	1	2	yes	—
213	#550	—	89	2	no	—
214	#500	C₂×D₈₉	1	1	yes	characteristic
215	#499	—	2	1	no	—
216	#493	—	89	1	no	—
217	#493	—	178	1	no	—
218	#492	—	89	2	no	—
219	#486	—	89	2	no	—
220	#480	—	178	2	no	—
221	#478	—	2	2	no	—
222	#477	—	89	2	no	—
223	#476	C₂×D₈₉	1	2	yes	—
224	#468	—	178	2	no	—
225	#460	—	89	2	no	—
226	#459	—	89	2	no	—
227	#459	—	178	2	no	—
228	#452	—	178	2	no	—
229	C₈×Q₈	—	89	1	no	—
230	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
231	C₄×Q₁₆	—	89	1	no	—
232	C₄×D₈	—	2	1	no	—
233	C₄×D₈	—	178	1	no	—
234	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
235	#436	—	89	2	no	—
236	#417	—	178	2	no	—
237	#335	—	89	1	no	—
238	#335	—	89	1	no	—
239	#333	—	89	2	no	—
240	#331	—	89	2	no	—
241	Q₈⋊C₈	—	89	1	no	—
242	Q₈⋊C₈	—	89	1	no	—
243	D₄⋊C₈	—	89	2	no	—
244	C₈²	—	89	1	no	—
245	C₂×#41	—	178	2	no	—
246	C₂×Q₁₆	—	89	1	no	—
247	C₂×SD₁₆	—	356	2	no	—
248	C₂×SD₁₆	—	178	2	no	—
249	C₂×D₈	—	178	2	no	—
250	C₂×D₈	—	178	1	no	—
251	C₂×D₈	—	178	1	no	—
252	C₂×D₈	C₂²⋊D₈₉	1	1	yes	characteristic
253	C₂×M₁₆	—	178	2	no	—
254	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
255	#127	—	89	2	no	—
256	#127	—	178	2	no	—
257	#125	—	2	2	no	—
258	C₂²⋊Q₈	—	2	2	no	—
259	C₂²⋊Q₈	—	4	2	no	—
260	#121	—	356	2	no	—
261	#121	—	178	2	no	—
262	C₄×Q₈	—	89	1	no	—
263	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₄×D₄	—	2	2	no	—
265	C₄×D₄	—	178	2	no	—
266	C₄×D₄	C₂²×D₈₉	1	2	yes	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	#107	—	2	1	no	—
269	#107	—	2	1	no	—
270	#107	—	89	1	no	—
271	#107	—	178	2	no	—
272	#107	C₂²⋊D₈₉	1	1	yes	characteristic
273	#106	—	2	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	89	2	no	—
276	C₄⋊C₈	—	89	1	no	—
277	C₄⋊C₈	—	178	1	no	—
278	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
279	#103	—	89	1	no	—
280	#103	—	89	1	no	—
281	#103	—	178	2	no	—
282	#102	—	2	2	no	—
283	#102	—	178	2	no	—
284	#102	—	178	2	no	—
285	#102	—	178	2	no	—
286	#102	—	178	2	no	—
287	#102	C₂²⋊D₈₉	1	2	yes	—
288	C₂²⋊C₈	—	178	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	89	1	no	—
291	C₄×C₈	—	89	1	no	—
292	C₄×C₈	—	178	1	no	—
293	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
294	#41	—	356	4	no	—
295	#41	—	712	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	89	1	no	—
298	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
299	C₂×D₄	—	356	2	no	—
300	C₂×D₄	—	356	2	no	—
301	C₂×D₄	—	178	2	no	—
302	C₂×D₄	—	178	2	no	—
303	C₂×D₄	—	178	2	no	—
304	C₂×D₄	C₂×C₂²⋊D₈₉	1	2	yes	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	356	2	no	—
308	Q₁₆	—	178	2	no	—
309	SD₁₆	—	712	2	no	—
310	SD₁₆	—	712	4	no	—
311	D₈	—	2	2	no	—
312	D₈	—	356	2	no	—
313	D₈	—	712	1	no	—
314	D₈	—	712	2	no	—
315	M₁₆	—	712	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	89	1	no	—
318	C₂×C₈	—	89	1	no	—
319	C₂×C₈	—	178	2	no	—
320	C₂×C₈	—	178	1	no	—
321	C₂×C₈	—	178	1	no	—
322	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
323	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	89	1	no	—
330	C₄⋊C₄	—	178	1	no	—
331	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
332	C₄⋊C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	—	356	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	178	1	no	—
338	C₄²	C₂³×D₈₉	1	1	yes	characteristic
339	C₂³	—	2	2	no	—
340	C₂³	—	356	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	89	2	no	—
345	Q₈	—	178	1	no	—
346	D₄	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	356	2	no	—
349	D₄	—	356	2	no	—
350	D₄	—	356	2	no	—
351	D₄	—	356	2	no	—
352	D₄	—	1424	2	no	—
353	D₄	—	712	2	no	—
354	D₄	—	712	4	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	16	2	no	—
362	C₂×C₄	—	178	1	no	—
363	C₂×C₄	—	178	1	no	—
364	C₂×C₄	—	712	4	no	—
365	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
367	C₂×C₄	#15668057	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	4	1	no	—
370	C₈	—	356	2	no	—
371	C₈	—	356	1	no	—
372	C₈	—	178	2	no	—
373	C₈	—	178	1	no	—
374	C₈	—	178	1	no	—
375	C₈	#15668043	1	2	yes	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	712	2	no	—
380	C₂²	—	712	2	no	—
381	C₂²	—	712	2	no	—
382	C₂²	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	8	2	no	—
388	C₄	—	356	1	no	—
389	C₄	—	178	2	no	—
390	C₄	#135×D₈₉	1	1	yes	characteristic
391	C₄	C₂×#15668043	1	1	yes	characteristic
392	C₄	#19152400	1	1	yes	characteristic
393	C₄	#19152406	1	1	yes	characteristic
394	C₂	—	8	2	no	—
395	C₂	—	1424	2	no	—
396	C₂	#26031914	1	1	yes	characteristic
397	C₂	#26031931	1	1	yes	characteristic
398	C₂	#26032049	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3380	C₂
socle	C₂²×C₈₉	D₄²
prime radical	#26033226	C₂
hypercenter	#3380	D₈₉
hypocenter	C₈₉	#19909
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3380

Complements

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

N	H ≅ G/N	type
2. #26032584	C₂	semidirect
3. #26019845	C₂	semidirect
4. #26019787	C₂	semidirect
5. #26019653	C₂	semidirect
6. #26019612	C₂	semidirect
7. #26019378	C₂	semidirect
8. #26019324	C₂	semidirect
9. #26019137	C₂	semidirect
10. C₈×#13083564	C₂	semidirect
11. C₈₉×#3380	C₂	semidirect
201. #3380	D₈₉	semidirect
211. C₈₉	#19909	semidirect

Maximal subgroups

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

subgroup	index
#26033226	2
#26032584	2
#26019845	2
#26019787	2
#26019653	2
#26019612	2
#26019378	2
#26019324	2
#26019137	2
C₈×#13083564	2
C₈₉×#3380	2
#19909	89