Subgroups — #36546030

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	#26033227	C₂	1	1	yes	maximal, characteristic
2	#26033120	C₂	1	1	yes	maximal, characteristic, complemented
3	#26032592	C₂	1	2	yes	maximal, complemented
4	#26019844	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019786	C₂	1	2	yes	maximal, complemented
6	#26019656	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019612	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019381	C₂	1	2	yes	maximal, complemented
9	#26019327	C₂	1	2	yes	maximal, complemented
10	#26019138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152308	C₂²	1	2	yes	—
13	#19152270	C₂²	1	2	yes	—
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152000	—	2	1	no	—
16	#19151975	—	2	1	no	—
17	#19151975	C₂²	1	1	yes	characteristic
18	#19151971	C₂²	1	2	yes	—
19	#19151945	C₂²	1	2	yes	—
20	#19151913	—	2	2	no	—
21	#19151899	—	2	2	no	—
22	#19151893	C₂²	1	2	yes	—
23	#19151889	C₂²	1	2	yes	—
24	#19151847	—	2	2	no	—
25	#19151798	C₂²	1	2	yes	—
26	#19151794	—	2	2	no	—
27	#19151794	C₂²	1	2	yes	—
28	#19151759	—	2	2	no	—
29	C₄×D₇₁₂	—	2	1	no	—
30	C₄×D₇₁₂	—	2	1	no	—
31	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
32	#19151677	C₂²	1	2	yes	—
33	#19151598	—	2	2	no	—
34	#19151351	C₂²	1	1	yes	characteristic
35	#19151346	C₂²	1	2	yes	—
36	#19151341	C₂²	1	2	yes	—
37	#19151327	C₂²	1	2	yes	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	#335×C₈₉	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#13083575	—	2	2	no	—
46	C₂×#13083563	—	4	2	no	—
47	C₂×#13083561	—	2	2	no	—
48	C₂×D₇₁₂	—	2	2	no	—
49	C₂×D₇₁₂	—	2	1	no	—
50	C₂×D₇₁₂	—	2	1	no	—
51	C₂×D₇₁₂	D₄	1	1	yes	characteristic
52	C₂×#13083557	—	2	2	no	—
53	#15668012	C₂³	1	2	yes	—
54	#15668009	—	2	2	no	—
55	#15668009	C₂³	1	2	yes	—
56	#15668004	—	2	2	no	—
57	#15667990	—	4	2	no	—
58	#15667988	—	2	2	no	—
59	#15667987	—	2	2	no	—
60	#15667987	—	4	2	no	—
61	C₄×D₃₅₆	—	2	2	no	—
62	C₄×D₃₅₆	—	2	2	no	—
63	C₄×D₃₅₆	C₂³	1	2	yes	—
64	C₄⋊C₄×D₈₉	—	2	2	no	—
65	#15667946	—	2	1	no	—
66	#15667946	—	2	1	no	—
67	#15667946	D₄	1	1	yes	characteristic
68	#15667945	—	2	2	no	—
69	#15667944	—	2	2	no	—
70	C₄⋊#10623494	C₂³	1	2	yes	—
71	#15667936	—	2	2	no	—
72	#15667934	—	2	2	no	—
73	#15667934	—	2	2	no	—
74	#15667934	—	2	2	no	—
75	#15667934	—	2	2	no	—
76	#15667934	D₄	1	2	yes	—
77	#15667932	—	2	2	no	—
78	#15667924	—	2	2	no	—
79	C₂×Q₁₆×C₈₉	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
82	#107×C₈₉	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	—	2	1	no	—
84	C₄⋊C₈×C₈₉	—	2	1	no	—
85	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
87	#103×C₈₉	D₄	1	1	yes	characteristic
88	#103×C₈₉	D₄	1	1	yes	characteristic
89	C₄×C₈×C₈₉	—	2	1	no	—
90	C₄×C₈×C₈₉	—	2	1	no	—
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	#13083575	—	4	4	no	—
95	#13083575	—	8	2	no	—
96	C₂×Dic₁₇₈	—	2	2	no	—
97	C₂×D₃₅₆	—	2	2	no	—
98	C₂×D₃₅₆	—	2	2	no	—
99	C₂×D₃₅₆	—	2	2	no	—
100	C₂×D₃₅₆	—	4	2	no	—
101	C₂×D₃₅₆	—	4	2	no	—
102	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
103	C₂×C₄×D₈₉	—	2	2	no	—
104	C₂×C₄×D₈₉	—	4	2	no	—
105	C₂×C₄×D₈₉	—	4	2	no	—
106	#13083563	—	8	4	no	—
107	#13083561	—	8	2	no	—
108	D₇₁₂	—	2	2	no	—
109	D₇₁₂	—	4	2	no	—
110	D₇₁₂	—	8	1	no	—
111	D₇₁₂	—	8	2	no	—
112	#13083557	—	8	2	no	—
113	C₂×#10623494	—	2	2	no	—
114	C₄⋊Dic₈₉	—	2	2	no	—
115	C₄⋊Dic₈₉	—	2	2	no	—
116	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
117	#13083553	—	4	2	no	—
118	#13083553	—	4	2	no	—
119	#13083551	—	2	2	no	—
120	#13083551	—	4	2	no	—
121	#13083551	—	4	2	no	—
122	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
124	Q₁₆×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	1	no	—
126	C₂×C₈×C₈₉	—	2	1	no	—
127	C₂×C₈×C₈₉	—	2	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	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	2	1	no	—
134	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₈₉	—	2	1	no	—
137	C₄²×C₈₉	—	2	1	no	—
138	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
139	C₂²×D₈₉	—	2	2	no	—
140	C₂²×D₈₉	—	4	2	no	—
141	Q₈×C₈₉	—	2	1	no	—
142	Q₈×C₈₉	—	2	1	no	—
143	Q₈×C₈₉	—	2	1	no	—
144	Q₈×C₈₉	#135	1	2	yes	—
145	C₂×C₄×C₈₉	—	2	1	no	—
146	C₂×C₄×C₈₉	—	2	1	no	—
147	C₂×C₄×C₈₉	—	2	1	no	—
148	C₂×C₄×C₈₉	—	2	1	no	—
149	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	4	2	no	—
154	D₃₅₆	—	2	2	no	—
155	D₃₅₆	—	2	2	no	—
156	D₃₅₆	—	4	2	no	—
157	D₃₅₆	—	4	2	no	—
158	D₃₅₆	—	4	2	no	—
159	D₃₅₆	—	4	2	no	—
160	D₃₅₆	—	8	4	no	—
161	D₃₅₆	—	8	2	no	—
162	D₃₅₆	—	16	2	no	—
163	C₄×D₈₉	—	4	2	no	—
164	C₄×D₈₉	—	4	2	no	—
165	C₄×D₈₉	—	8	4	no	—
166	C₄×D₈₉	—	16	2	no	—
167	Dic₁₇₈	—	8	2	no	—
168	C₈×C₈₉	—	2	2	no	—
169	C₈×C₈₉	—	2	1	no	—
170	C₈×C₈₉	—	2	1	no	—
171	C₈×C₈₉	—	2	1	no	—
172	C₈×C₈₉	—	4	1	no	—
173	C₈×C₈₉	—	4	1	no	—
174	C₈×C₈₉	#137	1	2	yes	—
175	#10623494	—	4	2	no	—
176	C₂²×C₈₉	D₄²	1	1	yes	characteristic
177	C₂×D₈₉	—	4	2	no	—
178	C₂×D₈₉	—	4	2	no	—
179	C₂×D₈₉	—	4	2	no	—
180	C₂×D₈₉	—	8	2	no	—
181	C₂×D₈₉	—	8	2	no	—
182	C₂×D₈₉	—	8	2	no	—
183	C₄×C₈₉	—	2	2	no	—
184	C₄×C₈₉	—	2	1	no	—
185	C₄×C₈₉	—	4	1	no	—
186	C₄×C₈₉	—	4	1	no	—
187	C₄×C₈₉	—	4	1	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	—	4	2	no	—
193	Dic₈₉	—	8	2	no	—
194	#19909	—	89	1	no	maximal, Sylow
195	C₂×C₈₉	#3278	1	1	yes	characteristic
196	C₂×C₈₉	#3280	1	1	yes	characteristic
197	C₂×C₈₉	#3292	1	1	yes	characteristic
198	D₈₉	—	8	2	no	—
199	D₈₉	—	16	2	no	—
200	#3391	—	89	1	no	—
201	#3380	—	89	1	no	—
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	D₈₉	1	1	yes	characteristic, complemented
211	C₈₉	#19909	1	1	yes	characteristic, complemented, Sylow
212	#554	—	89	2	no	—
213	#550	—	89	2	no	—
214	#500	—	89	1	no	—
215	#499	—	178	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	—	178	2	no	—
222	#477	—	89	2	no	—
223	#476	—	89	2	no	—
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	C₂×D₈₉	1	1	yes	characteristic
230	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
231	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
232	C₄×D₈	—	89	1	no	—
233	C₄×D₈	—	178	1	no	—
234	C₄×D₈	—	178	1	no	—
235	#436	—	89	2	no	—
236	#417	—	178	2	no	—
237	#335	—	89	1	no	—
238	#335	C₂×D₈₉	1	1	yes	characteristic
239	#333	—	89	2	no	—
240	#331	—	89	2	no	—
241	Q₈⋊C₈	C₂×D₈₉	1	1	yes	characteristic
242	Q₈⋊C₈	C₂×D₈₉	1	1	yes	characteristic
243	D₄⋊C₈	—	89	2	no	—
244	C₈²	C₂×D₈₉	1	1	yes	characteristic
245	C₂×#41	—	178	2	no	—
246	C₂×Q₁₆	D₃₅₆	1	1	yes	characteristic
247	C₂×SD₁₆	—	356	2	no	—
248	C₂×SD₁₆	—	178	2	no	—
249	C₂×D₈	—	89	1	no	—
250	C₂×D₈	—	178	2	no	—
251	C₂×D₈	—	178	1	no	—
252	C₂×D₈	—	178	1	no	—
253	C₂×M₁₆	—	178	2	no	—
254	C₄⋊Q₈	—	89	2	no	—
255	#127	—	89	2	no	—
256	#127	—	178	2	no	—
257	#125	—	178	2	no	—
258	C₂²⋊Q₈	—	356	2	no	—
259	C₂²⋊Q₈	—	178	2	no	—
260	#121	—	356	2	no	—
261	#121	—	178	2	no	—
262	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
263	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₄×D₄	—	89	2	no	—
265	C₄×D₄	—	178	2	no	—
266	C₄×D₄	—	178	2	no	—
267	C₂×C₄⋊C₄	—	178	2	no	—
268	#107	—	89	1	no	—
269	#107	—	178	1	no	—
270	#107	—	178	1	no	—
271	#107	—	178	2	no	—
272	#107	D₃₅₆	1	1	yes	characteristic
273	#106	—	178	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	89	2	no	—
277	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
278	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
279	#103	—	178	2	no	—
280	#103	D₃₅₆	1	1	yes	characteristic
281	#103	D₃₅₆	1	1	yes	characteristic
282	#102	—	89	2	no	—
283	#102	—	178	2	no	—
284	#102	—	178	2	no	—
285	#102	—	178	2	no	—
286	#102	—	178	2	no	—
287	#102	—	178	2	no	—
288	C₂²⋊C₈	—	178	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
294	#41	—	356	4	no	—
295	#41	—	712	2	no	—
296	C₂×Q₈	—	178	2	no	—
297	C₂×Q₈	C₂×D₃₅₆	1	1	yes	characteristic
298	C₂×Q₈	C₂×D₃₅₆	1	1	yes	characteristic
299	C₂×D₄	—	89	2	no	—
300	C₂×D₄	—	356	2	no	—
301	C₂×D₄	—	356	2	no	—
302	C₂×D₄	—	178	2	no	—
303	C₂×D₄	—	178	2	no	—
304	C₂×D₄	—	178	2	no	—
305	C₂²×C₄	—	356	2	no	—
306	C₂²×C₄	—	356	2	no	—
307	C₂²×C₄	—	178	2	no	—
308	Q₁₆	—	2	2	no	—
309	SD₁₆	—	712	4	no	—
310	SD₁₆	—	712	2	no	—
311	D₈	—	356	2	no	—
312	D₈	—	178	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	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	178	2	no	—
320	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
321	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
322	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
323	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	89	2	no	—
327	C₄⋊C₄	—	356	2	no	—
328	C₄⋊C₄	—	356	2	no	—
329	C₄⋊C₄	—	178	2	no	—
330	C₄⋊C₄	—	178	2	no	—
331	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
332	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
333	C₂²⋊C₄	—	356	2	no	—
334	C₂²⋊C₄	—	356	2	no	—
335	C₂²⋊C₄	—	178	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	C₂³×D₈₉	1	1	yes	characteristic
339	C₂³	—	356	2	no	—
340	C₂³	—	178	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	2	1	no	—
344	Q₈	—	712	2	no	—
345	Q₈	#15668034	1	2	yes	—
346	D₄	—	356	2	no	—
347	D₄	—	356	2	no	—
348	D₄	—	356	2	no	—
349	D₄	—	356	2	no	—
350	D₄	—	1424	2	no	—
351	D₄	—	178	2	no	—
352	D₄	—	178	2	no	—
353	D₄	—	712	4	no	—
354	D₄	—	712	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	356	2	no	—
360	C₂×C₄	—	356	2	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	1424	2	no	—
363	C₂×C₄	—	178	2	no	—
364	C₂×C₄	—	712	4	no	—
365	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
367	C₂×C₄	#15668058	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	356	2	no	—
375	C₈	#15668044	1	2	yes	—
376	C₂²	—	356	2	no	—
377	C₂²	—	356	2	no	—
378	C₂²	—	356	2	no	—
379	C₂²	—	712	2	no	—
380	C₂²	—	712	2	no	—
381	C₂²	—	712	2	no	—
382	C₂²	D₄×D₃₅₆	1	1	yes	characteristic
383	C₄	—	2	2	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	356	2	no	—
389	C₄	—	712	2	no	—
390	C₄	C₂×#15668034	1	1	yes	characteristic
391	C₄	C₂×#15668044	1	1	yes	characteristic
392	C₄	#19152403	1	1	yes	characteristic
393	C₄	#19152405	1	1	yes	characteristic
394	C₂	—	1424	2	no	—
395	C₂	—	712	2	no	—
396	C₂	#26031919	1	1	yes	characteristic
397	C₂	#26031939	1	1	yes	characteristic
398	C₂	#26032052	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₆
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	D₄²
prime radical	#26033227	C₂
hypercenter	C₈×Q₁₆	D₈₉
hypocenter	C₈₉	#19909
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #26033120	C₂	semidirect
3. #26032592	C₂	semidirect
4. #26019844	C₂	semidirect
5. #26019786	C₂	semidirect
6. #26019656	C₂	semidirect
7. #26019612	C₂	semidirect
8. #26019381	C₂	semidirect
9. #26019327	C₂	semidirect
10. #26019138	C₂	semidirect
11. C₈×Q₁₆×C₈₉	C₂	semidirect
210. C₈×Q₁₆	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
#26033227	2
#26033120	2
#26032592	2
#26019844	2
#26019786	2
#26019656	2
#26019612	2
#26019381	2
#26019327	2
#26019138	2
C₈×Q₁₆×C₈₉	2
#19909	89