Subgroups — #36709301

Order 22784.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029918	C₂	1	1	yes	maximal, characteristic
2	#26028849	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022458	C₂	1	2	yes	maximal, complemented
4	#26022448	C₂	1	2	yes	maximal, complemented
5	#26022442	C₂	1	2	yes	maximal, complemented
6	#26022436	C₂	1	2	yes	maximal, complemented
7	#26022041	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021964	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021865	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668018	C₂²	1	1	yes	characteristic
13	#19152176	—	2	1	no	—
14	#19152078	—	2	1	no	—
15	C₂×#15667952	C₂²	1	1	yes	characteristic
16	#19151841	—	2	2	no	—
17	#19151836	C₂²	1	2	yes	—
18	#19151831	C₂²	1	2	yes	—
19	#19151641	—	2	1	no	—
20	#19151637	—	2	2	no	—
21	#19151635	C₂²	1	2	yes	—
22	C₂×#15667946	C₂²	1	1	yes	characteristic
23	C₂×#15667946	C₂²	1	1	yes	characteristic
24	#107×D₈₉	C₂²	1	2	yes	—
25	#19151599	—	2	1	no	—
26	#19151426	C₂²	1	2	yes	—
27	#19151426	C₂²	1	2	yes	—
28	#19151425	C₂²	1	2	yes	—
29	#19151423	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	#19151399	C₂²	1	2	yes	—
32	#19151377	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	#427×C₈₉	C₂²	1	1	yes	characteristic
37	#358×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	1	yes	characteristic
40	#357×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	D₄	1	1	yes	characteristic
42	#15668018	—	2	2	no	—
43	#15668018	—	2	2	no	—
44	C₂×#13083558	D₄	1	1	yes	characteristic
45	C₂×#13083557	D₄	1	2	yes	—
46	C₂×C₈×D₈₉	C₂³	1	2	yes	—
47	#15667991	—	4	1	no	—
48	#15667988	—	2	2	no	—
49	#15667986	—	4	1	no	—
50	#15667984	—	4	2	no	—
51	#15667983	—	2	2	no	—
52	C₄×Dic₁₇₈	—	4	1	no	—
53	C₄×D₃₅₆	—	4	1	no	—
54	#15667972	—	4	2	no	—
55	#15667970	—	2	2	no	—
56	C₂×C₄⋊Dic₈₉	—	2	1	no	—
57	C₂×C₄⋊Dic₈₉	—	2	1	no	—
58	C₄⋊C₄×D₈₉	—	2	2	no	—
59	#15667952	—	2	2	no	—
60	C₂×#13083549	C₂³	1	2	yes	—
61	#15667946	—	2	1	no	—
62	#15667946	—	2	1	no	—
63	#15667946	C₂³	1	2	yes	—
64	#15667946	C₂³	1	2	yes	—
65	#15667945	—	2	2	no	—
66	#15667943	—	2	2	no	—
67	#15667936	—	2	2	no	—
68	#15667933	—	2	2	no	—
69	C₂×#41×C₈₉	—	2	1	no	—
70	#135×C₈₉	—	2	2	no	—
71	#135×C₈₉	—	2	2	no	—
72	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×SD₁₆×C₈₉	—	2	1	no	—
74	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	#117×C₈₉	—	2	1	no	—
77	C₂×C₁₆×C₈₉	—	2	2	no	—
78	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
80	#107×C₈₉	C₂³	1	1	yes	characteristic
81	#107×C₈₉	C₂³	1	1	yes	characteristic
82	#106×C₈₉	—	2	1	no	—
83	#13083574	—	2	2	no	—
84	#13083574	—	2	2	no	—
85	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
86	C₂×D₃₅₆	D₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₈₉	—	4	2	no	—
88	C₂×C₂²⋊D₈₉	D₈	1	2	yes	—
89	C₂²×Dic₈₉	—	4	1	no	—
90	C₂²×Dic₈₉	—	4	1	no	—
91	C₂×C₄×D₈₉	—	4	2	no	—
92	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
93	#13083558	—	2	2	no	—
94	#13083557	—	2	4	no	—
95	C₂×#10623494	C₂×D₄	1	2	yes	—
96	C₈×D₈₉	—	2	2	no	—
97	C₈×D₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	4	1	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	#13083553	—	4	2	no	—
103	#13083553	—	4	2	no	—
104	C₂²⋊Dic₈₉	—	4	2	no	—
105	#13083551	—	4	2	no	—
106	#13083551	—	4	2	no	—
107	C₄×Dic₈₉	—	4	2	no	—
108	#13083549	—	2	2	no	—
109	#41×C₈₉	—	4	2	no	—
110	#41×C₈₉	—	4	2	no	—
111	C₂×Q₈×C₈₉	—	2	1	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂×D₄×C₈₉	—	4	1	no	—
114	C₂²×C₄×C₈₉	—	4	1	no	—
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	Q₁₆×C₈₉	—	2	1	no	—
117	Q₁₆×C₈₉	—	2	1	no	—
118	SD₁₆×C₈₉	—	4	2	no	—
119	D₈×C₈₉	—	2	1	no	—
120	D₈×C₈₉	—	2	1	no	—
121	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	#41	1	2	yes	—
123	C₂×C₈×C₈₉	#41	1	2	yes	—
124	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₄⋊C₄×C₈₉	—	2	1	no	—
127	C₂²⋊C₄×C₈₉	—	4	1	no	—
128	C₄²×C₈₉	—	4	1	no	—
129	C₁₆×C₈₉	—	2	1	no	—
130	C₁₆×C₈₉	—	2	1	no	—
131	C₂³×C₈₉	—	4	1	no	—
132	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂²×D₈₉	C₂×D₈	1	2	yes	—
134	Q₈×C₈₉	—	4	1	no	—
135	Q₈×C₈₉	—	4	1	no	—
136	D₄×C₈₉	—	4	1	no	—
137	D₄×C₈₉	—	4	1	no	—
138	D₄×C₈₉	—	8	2	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	—	4	1	no	—
141	C₂×C₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	#123	1	2	yes	—
144	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#137	1	2	yes	—
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₈₉	—	2	4	no	—
148	C₂²⋊D₈₉	—	16	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	16	1	no	—
152	C₂×Dic₈₉	—	16	1	no	—
153	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
154	D₃₅₆	—	2	2	no	—
155	C₄×D₈₉	—	2	2	no	—
156	C₄×D₈₉	—	2	2	no	—
157	C₄×D₈₉	—	16	2	no	—
158	Dic₁₇₈	—	2	2	no	—
159	C₈×C₈₉	#123	1	2	yes	—
160	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
162	#10623494	—	2	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	8	1	no	—
166	C₂²×C₈₉	—	8	1	no	—
167	C₂²×C₈₉	—	8	1	no	—
168	C₂²×C₈₉	#480	1	2	yes	—
169	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₈₉	—	2	2	no	—
171	C₂×D₈₉	—	2	2	no	—
172	C₂×D₈₉	—	2	2	no	—
173	C₄×C₈₉	—	8	1	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	—	8	1	no	—
176	C₄×C₈₉	#480	1	2	yes	—
177	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
178	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
179	Dic₈₉	—	2	2	no	—
180	Dic₈₉	—	8	2	no	—
181	Dic₈₉	—	8	2	no	—
182	#33653	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	2	no	—
184	C₂×C₈₉	—	16	1	no	—
185	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
186	C₂×C₈₉	#3416	1	1	yes	characteristic
187	C₂×C₈₉	#3416	1	1	yes	characteristic
188	D₈₉	—	4	2	no	—
189	#3061	—	89	1	no	—
190	#2913	—	89	1	no	—
191	#2237	—	89	2	no	—
192	#2235	—	89	2	no	—
193	#2234	—	89	2	no	—
194	#2233	—	89	2	no	—
195	#2159	—	89	1	no	—
196	C₂×#367	—	89	1	no	—
197	#2139	—	89	1	no	—
198	#2138	D₈₉	1	1	yes	characteristic, complemented
199	#2113	—	89	1	no	—
200	C₈₉	#33653	1	1	yes	characteristic, complemented, Sylow
201	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
202	C₂×#131	—	89	1	no	—
203	#537	—	178	1	no	—
204	#520	—	178	1	no	—
205	C₂×M₃₂	—	89	1	no	—
206	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
207	#468	—	178	2	no	—
208	#467	—	89	2	no	—
209	#466	—	89	2	no	—
210	#428	—	178	2	no	—
211	#428	—	178	1	no	—
212	#427	—	89	2	no	—
213	#427	C₂×D₈₉	1	1	yes	characteristic
214	C₂×#107	—	89	1	no	—
215	C₂×#107	—	89	1	no	—
216	C₂×#107	—	89	2	no	—
217	#417	—	178	1	no	—
218	#367	—	89	2	no	—
219	#367	—	89	2	no	—
220	#367	—	89	2	no	—
221	#366	—	89	2	no	—
222	#358	—	89	2	no	—
223	#358	C₂×D₈₉	1	1	yes	characteristic
224	#358	C₂×D₈₉	1	1	yes	characteristic
225	#357	—	89	2	no	—
226	#357	C₂×D₈₉	1	1	yes	characteristic
227	#357	C₂×D₈₉	1	1	yes	characteristic
228	C₂²⋊C₁₆	—	89	2	no	—
229	C₂²⋊C₁₆	—	89	2	no	—
230	C₂×#41	—	2	1	no	—
231	C₂×#41	—	89	1	no	—
232	#135	—	2	2	no	—
233	#135	—	2	2	no	—
234	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
235	C₂×SD₁₆	—	2	1	no	—
236	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
237	#131	—	178	2	no	—
238	#131	—	178	2	no	—
239	C₂×M₁₆	—	89	1	no	—
240	C₂×M₁₆	—	89	2	no	—
241	C₂²×C₈	—	89	2	no	—
242	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂²⋊Q₈	—	356	1	no	—
244	C₂²⋊Q₈	—	178	2	no	—
245	#121	—	356	1	no	—
246	#121	—	356	2	no	—
247	#121	—	178	2	no	—
248	C₄×Q₈	—	356	1	no	—
249	C₄×D₄	—	356	1	no	—
250	C₄×D₄	—	356	2	no	—
251	#117	—	2	1	no	—
252	#117	—	178	2	no	—
253	C₂×C₄⋊C₄	—	178	2	no	—
254	C₂×C₄⋊C₄	—	178	1	no	—
255	C₂×C₄⋊C₄	—	178	1	no	—
256	M₃₂	—	178	2	no	—
257	C₂×C₁₆	—	2	2	no	—
258	C₂×C₁₆	—	89	2	no	—
259	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
261	#107	—	89	2	no	—
262	#107	—	89	2	no	—
263	#107	—	178	2	no	—
264	#107	—	178	1	no	—
265	#107	—	178	1	no	—
266	#107	C₂²×D₈₉	1	1	yes	characteristic
267	#107	C₂²×D₈₉	1	1	yes	characteristic
268	#106	—	2	1	no	—
269	#106	—	178	2	no	—
270	#103	—	178	2	no	—
271	#102	—	178	2	no	—
272	#41	—	4	2	no	—
273	#41	—	4	2	no	—
274	#41	—	178	2	no	—
275	#41	—	178	2	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	89	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	89	2	no	—
281	C₂×D₄	—	89	1	no	—
282	C₂×D₄	—	356	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	89	2	no	—
285	C₂²×C₄	—	356	2	no	—
286	C₂²×C₄	—	356	1	no	—
287	C₂²×C₄	—	356	1	no	—
288	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
289	Q₁₆	—	2	1	no	—
290	Q₁₆	—	2	1	no	—
291	SD₁₆	—	4	2	no	—
292	D₈	—	2	1	no	—
293	D₈	—	2	1	no	—
294	M₁₆	—	178	2	no	—
295	M₁₆	—	178	4	no	—
296	C₂×C₈	—	89	2	no	—
297	C₂×C₈	—	178	2	no	—
298	C₂×C₈	—	178	2	no	—
299	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
300	C₂×C₈	#13083573	1	2	yes	—
301	C₂×C₈	#13083574	1	2	yes	—
302	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	356	2	no	—
306	C₄⋊C₄	—	356	2	no	—
307	C₄⋊C₄	—	356	1	no	—
308	C₄⋊C₄	—	356	1	no	—
309	C₄⋊C₄	—	178	2	no	—
310	C₄⋊C₄	—	178	2	no	—
311	C₂²⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	356	2	no	—
313	C₂²⋊C₄	—	356	2	no	—
314	C₂²⋊C₄	—	356	2	no	—
315	C₄²	—	4	1	no	—
316	C₄²	—	356	2	no	—
317	C₁₆	—	2	1	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	178	2	no	—
320	C₂³	—	4	1	no	—
321	C₂³	—	89	2	no	—
322	C₂³	D₈×D₈₉	1	1	yes	characteristic
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	178	2	no	—
326	D₄	—	4	1	no	—
327	D₄	—	4	1	no	—
328	D₄	—	8	2	no	—
329	D₄	—	1424	2	no	—
330	D₄	—	178	4	no	—
331	D₄	—	178	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	89	2	no	—
337	C₂×C₄	—	356	2	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	1424	2	no	—
340	C₂×C₄	—	1424	1	no	—
341	C₂×C₄	—	1424	1	no	—
342	C₂×C₄	—	178	2	no	—
343	C₂×C₄	—	178	2	no	—
344	C₂×C₄	#15667994	1	2	yes	—
345	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
346	C₂×C₄	#15668041	1	2	yes	—
347	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₈	—	178	2	no	—
349	C₈	#15667994	1	2	yes	—
350	C₈	C₂×#13083573	1	1	yes	characteristic
351	C₈	C₂×#13083574	1	1	yes	characteristic
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	178	2	no	—
358	C₂²	—	178	2	no	—
359	C₂²	—	178	2	no	—
360	C₂²	#19151910	1	2	yes	—
361	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	178	2	no	—
366	C₄	—	712	2	no	—
367	C₄	—	712	2	no	—
368	C₄	#19151910	1	2	yes	—
369	C₄	C₂×#15667994	1	1	yes	characteristic
370	C₄	C₂×#15668041	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	1	no	—
373	C₂	—	356	2	no	—
374	C₂	C₂×#19151910	1	1	yes	characteristic
375	C₂	#26033395	1	1	yes	characteristic
376	C₂	#26033395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2138	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26029918	C₂
hypercenter	#2138	D₈₉
hypocenter	C₈₉	#33653
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
2. #26028849	C₂	semidirect
3. #26022458	C₂	semidirect
4. #26022448	C₂	semidirect
5. #26022442	C₂	semidirect
6. #26022436	C₂	semidirect
7. #26022041	C₂	semidirect
8. C₂×#19151426	C₂	semidirect
9. #26021964	C₂	semidirect
10. #26021865	C₂	semidirect
11. C₈₉×#2138	C₂	semidirect
198. #2138	D₈₉	semidirect
200. C₈₉	#33653	semidirect

Maximal subgroups

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

subgroup	index
#26029918	2
#26028849	2
#26022458	2
#26022448	2
#26022442	2
#26022436	2
#26022041	2
C₂×#19151426	2
#26021964	2
#26021865	2
C₈₉×#2138	2
#33653	89