Subgroups — #36709274

Order 22784.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029901	C₂	1	1	yes	maximal, characteristic
2	#26028848	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022446	C₂	1	2	yes	maximal, complemented
4	#26022445	C₂	1	2	yes	maximal, complemented
5	#26022439	C₂	1	2	yes	maximal, complemented
6	#26022434	C₂	1	2	yes	maximal, complemented
7	#26022034	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19151398	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021867	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
13	C₂×#15668018	C₂²	1	1	yes	characteristic
14	#19152167	—	2	1	no	—
15	#19152079	—	2	1	no	—
16	C₂²×#13083549	C₂²	1	1	yes	characteristic
17	#19151852	—	2	2	no	—
18	#19151835	C₂²	1	2	yes	—
19	#19151835	C₂²	1	2	yes	—
20	#19151639	—	2	2	no	—
21	#19151638	—	2	1	no	—
22	#19151636	C₂²	1	1	yes	characteristic
23	#19151635	C₂²	1	2	yes	complemented
24	#107×D₈₉	C₂²	1	2	yes	complemented
25	#19151602	—	2	1	no	—
26	#19151425	C₂²	1	1	yes	characteristic, complemented
27	#19151425	C₂²	1	1	yes	characteristic, complemented
28	#19151423	C₂²	1	1	yes	characteristic, complemented
29	#19151423	C₂²	1	1	yes	characteristic, complemented
30	#19151400	C₂²	1	2	yes	—
31	#19151400	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	—
33	#19151398	C₂²	1	2	yes	—
34	#19151377	C₂²	1	2	yes	complemented
35	#19151377	C₂²	1	2	yes	complemented
36	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
37	#427×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
39	#367×C₈₉	C₂²	1	2	yes	complemented
40	#366×C₈₉	C₂²	1	2	yes	complemented
41	C₂×#13083574	D₄	1	1	yes	characteristic
42	C₂²×D₃₅₆	—	2	1	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	—	2	2	no	—
45	C₂×D₇₁₂	—	2	2	no	—
46	C₂×D₇₁₂	C₂³	1	2	yes	—
47	#15668018	—	2	2	no	—
48	#15668018	—	2	2	no	—
49	C₂×#13083558	D₄	1	1	yes	characteristic
50	C₂×#13083557	D₄	1	2	yes	—
51	C₂×C₈×D₈₉	C₂³	1	2	yes	—
52	#15668009	—	4	1	no	—
53	#15668000	—	4	1	no	—
54	#15667987	—	2	2	no	—
55	#15667987	—	2	2	no	—
56	#15667982	—	4	2	no	—
57	#15667977	—	4	1	no	—
58	#15667974	—	4	1	no	—
59	C₄×C₂²⋊D₈₉	—	4	2	no	—
60	#15667971	—	2	1	no	—
61	#15667970	—	2	2	no	—
62	C₄⋊C₄×D₈₉	—	2	2	no	—
63	C₂×#13083549	—	2	2	no	—
64	C₂×#13083549	C₂³	1	1	yes	characteristic
65	C₂×#13083549	C₂³	1	1	yes	characteristic
66	#15667946	C₂³	1	1	yes	characteristic
67	#15667946	C₂³	1	1	yes	characteristic
68	#15667945	—	2	2	no	—
69	#15667944	—	2	1	no	—
70	#15667943	—	2	2	no	—
71	#15667932	—	2	2	no	—
72	#15667932	—	2	2	no	—
73	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	#117×C₈₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
76	M₃₂×C₈₉	—	2	2	no	—
77	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
78	#107×C₈₉	—	2	1	no	—
79	#107×C₈₉	C₂³	1	2	yes	—
80	#107×C₈₉	C₂³	1	1	yes	characteristic
81	#107×C₈₉	C₂³	1	1	yes	characteristic
82	#106×C₈₉	—	2	1	no	—
83	C₂³×D₈₉	—	4	1	no	—
84	#13083574	—	2	2	no	—
85	#13083574	—	2	2	no	—
86	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
87	C₂×D₃₅₆	—	2	2	no	—
88	C₂×D₃₅₆	—	4	2	no	—
89	C₂×D₃₅₆	—	4	2	no	—
90	C₂×D₃₅₆	—	4	1	no	—
91	C₂×D₃₅₆	—	4	2	no	—
92	C₂×D₃₅₆	—	4	2	no	—
93	C₂×D₃₅₆	D₈	1	1	yes	characteristic
94	C₂×C₂²⋊D₈₉	D₈	1	2	yes	—
95	C₂×C₄×D₈₉	—	4	2	no	—
96	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
97	D₇₁₂	—	2	2	no	—
98	D₇₁₂	—	2	2	no	—
99	D₇₁₂	—	4	2	no	—
100	#13083558	—	2	2	no	—
101	#13083557	—	2	4	no	—
102	C₂×#10623494	C₂×D₄	1	2	yes	—
103	C₈×D₈₉	—	2	2	no	—
104	C₈×D₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	1	no	—
106	C₄⋊Dic₈₉	—	2	1	no	—
107	#13083553	—	4	2	no	—
108	C₂²⋊Dic₈₉	—	4	1	no	—
109	#13083551	—	4	2	no	—
110	#13083551	—	4	2	no	—
111	#13083551	—	4	2	no	—
112	C₄×Dic₈₉	—	4	1	no	—
113	C₄×Dic₈₉	—	4	2	no	—
114	#13083549	—	2	1	no	—
115	#13083549	—	2	1	no	—
116	C₂²×C₄×C₈₉	—	4	1	no	—
117	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	#41	1	2	yes	—
120	C₂×C₈×C₈₉	#41	1	2	yes	—
121	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	—	2	1	no	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	4	1	no	—
126	C₂²⋊C₄×C₈₉	—	4	1	no	—
127	C₄²×C₈₉	—	4	1	no	—
128	C₁₆×C₈₉	—	2	2	no	—
129	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₈₉	—	4	2	no	—
131	C₂²×D₈₉	—	8	2	no	—
132	C₂²×D₈₉	—	16	1	no	—
133	C₂²×D₈₉	—	16	1	no	—
134	C₂²×D₈₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	—	4	2	no	—
138	C₂×C₄×C₈₉	—	16	1	no	—
139	C₂×C₄×C₈₉	#123	1	2	yes	—
140	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#137	1	2	yes	—
142	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₈₉	—	2	4	no	—
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	—	4	1	no	—
146	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
147	D₃₅₆	—	2	2	no	—
148	D₃₅₆	—	4	2	no	—
149	D₃₅₆	—	4	2	no	—
150	D₃₅₆	—	8	2	no	—
151	D₃₅₆	—	16	2	no	—
152	D₃₅₆	—	16	2	no	—
153	C₄×D₈₉	—	2	2	no	—
154	C₄×D₈₉	—	2	2	no	—
155	C₄×D₈₉	—	16	2	no	—
156	Dic₁₇₈	—	2	2	no	—
157	C₈×C₈₉	#123	1	2	yes	—
158	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
160	#10623494	—	2	2	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	#480	1	2	yes	—
164	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₈₉	—	2	2	no	—
166	C₂×D₈₉	—	2	2	no	—
167	C₂×D₈₉	—	2	2	no	—
168	C₂×D₈₉	—	8	2	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	8	2	no	—
171	C₂×D₈₉	—	16	4	no	—
172	C₄×C₈₉	—	8	2	no	—
173	C₄×C₈₉	—	8	1	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	#480	1	2	yes	—
176	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
177	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
178	Dic₈₉	—	2	2	no	—
179	Dic₈₉	—	8	1	no	—
180	Dic₈₉	—	8	1	no	—
181	#33650	—	89	1	no	maximal, Sylow
182	C₂×C₈₉	—	2	2	no	—
183	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
184	C₂×C₈₉	#3415	1	1	yes	characteristic
185	C₂×C₈₉	#3416	1	1	yes	characteristic
186	D₈₉	—	4	2	no	—
187	D₈₉	—	16	2	no	—
188	#3058	—	89	1	no	—
189	#2913	—	89	1	no	—
190	#2234	—	89	2	no	—
191	#2234	—	89	2	no	—
192	#2233	—	89	2	no	—
193	#2232	—	89	2	no	—
194	#2159	D₈₉	1	1	yes	characteristic, complemented
195	#2158	—	89	1	no	—
196	#2140	—	89	1	no	—
197	C₂×#357	—	89	1	no	—
198	#2113	—	89	1	no	—
199	C₈₉	#33650	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₈	—	89	1	no	—
201	C₂×#131	—	89	1	no	—
202	#535	—	178	1	no	—
203	#520	—	178	1	no	—
204	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
205	C₂²×C₁₆	—	89	1	no	—
206	#469	—	178	2	no	—
207	#466	—	89	2	no	—
208	#466	—	89	2	no	—
209	#428	—	178	2	no	—
210	#428	—	178	1	no	—
211	#427	—	89	1	no	—
212	#427	—	89	2	no	—
213	#427	C₂×D₈₉	1	1	yes	characteristic
214	C₂×#107	—	89	2	no	—
215	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
216	#418	—	178	1	no	—
217	#367	—	89	1	no	—
218	#367	—	89	1	no	—
219	#367	C₂×D₈₉	1	2	yes	complemented
220	#366	—	89	1	no	—
221	#366	—	89	1	no	—
222	#366	C₂×D₈₉	1	2	yes	complemented
223	#357	—	89	2	no	—
224	#357	—	89	2	no	—
225	#357	—	89	2	no	—
226	#357	—	89	2	no	—
227	C₂²⋊C₁₆	—	89	2	no	—
228	C₂²⋊C₁₆	—	89	2	no	—
229	C₂×#41	—	89	1	no	—
230	C₂²×D₄	—	178	1	no	—
231	C₂×D₈	—	89	2	no	—
232	C₂×D₈	—	178	1	no	—
233	C₂×D₈	—	178	2	no	—
234	C₂×D₈	—	178	2	no	—
235	#131	—	178	2	no	—
236	#131	—	178	2	no	—
237	C₂×M₁₆	—	89	1	no	—
238	C₂×M₁₆	—	89	2	no	—
239	C₂²×C₈	—	89	2	no	—
240	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	#127	—	356	1	no	—
242	#124	—	356	1	no	—
243	#121	—	178	2	no	—
244	#121	—	178	2	no	—
245	#120	—	356	2	no	—
246	C₄×Q₈	—	356	1	no	—
247	C₄×D₄	—	356	1	no	—
248	C₄×D₄	—	356	2	no	—
249	#117	—	2	1	no	—
250	#117	—	178	2	no	—
251	#117	—	178	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	178	2	no	—
254	M₃₂	—	2	2	no	—
255	C₂×C₁₆	—	89	1	no	—
256	C₂×C₁₆	—	89	1	no	—
257	C₂×C₁₆	—	178	2	no	—
258	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
259	#107	—	2	1	no	—
260	#107	—	89	1	no	—
261	#107	—	89	1	no	—
262	#107	—	178	2	no	—
263	#107	C₂²×D₈₉	1	2	yes	—
264	#107	C₂²×D₈₉	1	1	yes	characteristic
265	#107	C₂²×D₈₉	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	178	2	no	—
268	#106	—	178	1	no	—
269	#102	—	178	2	no	—
270	#102	—	178	2	no	—
271	C₂⁴	—	356	1	no	—
272	#41	—	178	2	no	—
273	#41	—	178	2	no	—
274	C₂×Q₈	—	89	1	no	—
275	C₂×D₄	—	89	2	no	—
276	C₂×D₄	—	89	1	no	—
277	C₂×D₄	—	356	2	no	—
278	C₂×D₄	—	356	2	no	—
279	C₂×D₄	—	356	2	no	—
280	C₂×D₄	—	356	1	no	—
281	C₂×D₄	—	356	2	no	—
282	C₂×D₄	—	178	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	89	2	no	—
285	C₂²×C₄	—	356	2	no	—
286	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
287	D₈	—	356	2	no	—
288	D₈	—	178	2	no	—
289	D₈	—	178	2	no	—
290	M₁₆	—	178	2	no	—
291	M₁₆	—	178	4	no	—
292	C₂×C₈	—	89	2	no	—
293	C₂×C₈	—	178	2	no	—
294	C₂×C₈	—	178	2	no	—
295	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
296	C₂×C₈	#13083574	1	2	yes	—
297	C₂×C₈	#13083575	1	2	yes	—
298	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	356	2	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	356	2	no	—
308	C₂²⋊C₄	—	356	2	no	—
309	C₂²⋊C₄	—	356	2	no	—
310	C₂²⋊C₄	—	356	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	356	2	no	—
313	C₄²	—	356	1	no	—
314	C₁₆	—	2	2	no	—
315	C₁₆	—	178	1	no	—
316	C₁₆	—	178	1	no	—
317	C₂³	—	89	2	no	—
318	C₂³	—	356	2	no	—
319	C₂³	—	1424	1	no	—
320	C₂³	—	1424	1	no	—
321	C₂³	—	712	2	no	—
322	C₂³	D₈×D₈₉	1	1	yes	characteristic
323	Q₈	—	178	2	no	—
324	D₄	—	356	2	no	—
325	D₄	—	356	2	no	—
326	D₄	—	1424	2	no	—
327	D₄	—	1424	2	no	—
328	D₄	—	178	4	no	—
329	D₄	—	178	2	no	—
330	D₄	—	712	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	16	1	no	—
335	C₂×C₄	—	89	2	no	—
336	C₂×C₄	—	356	1	no	—
337	C₂×C₄	—	356	1	no	—
338	C₂×C₄	—	1424	2	no	—
339	C₂×C₄	—	178	2	no	—
340	C₂×C₄	—	178	2	no	—
341	C₂×C₄	#15667997	1	2	yes	—
342	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
343	C₂×C₄	#15668044	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
345	C₈	—	178	2	no	—
346	C₈	#15667997	1	2	yes	—
347	C₈	C₂×#13083574	1	1	yes	characteristic
348	C₈	C₂×#13083575	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	1424	4	no	—
352	C₂²	—	178	2	no	—
353	C₂²	—	178	2	no	—
354	C₂²	—	178	2	no	—
355	C₂²	—	712	2	no	—
356	C₂²	—	712	2	no	—
357	C₂²	—	712	2	no	—
358	C₂²	#19151913	1	2	yes	—
359	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
360	C₄	—	8	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	178	2	no	—
364	C₄	—	712	1	no	—
365	C₄	—	712	1	no	—
366	C₄	#19151913	1	2	yes	—
367	C₄	C₂×#15667997	1	1	yes	characteristic
368	C₄	C₂×#15668044	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	356	2	no	—
371	C₂	—	1424	2	no	—
372	C₂	C₂×#19151913	1	1	yes	characteristic
373	C₂	#26033392	1	1	yes	characteristic
374	C₂	#26033397	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×#2159	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26029901	C₂
hypercenter	#2159	D₈₉
hypocenter	C₈₉	#33650
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
2. #26028848	C₂	semidirect
3. #26022446	C₂	semidirect
4. #26022445	C₂	semidirect
5. #26022439	C₂	semidirect
6. #26022434	C₂	semidirect
7. #26022034	C₂	semidirect
8. #26021969	C₂	semidirect
9. C₂×#19151398	C₂	semidirect
10. #26021867	C₂	semidirect
11. C₈₉×#2159	C₂	semidirect
23. #19151635	C₂²	semidirect
24. #107×D₈₉	C₂²	semidirect
26. #19151425	C₂²	semidirect
27. #19151425	C₂²	semidirect
28. #19151423	C₂²	semidirect
29. #19151423	C₂²	semidirect
34. #19151377	C₂²	semidirect
35. #19151377	C₂²	semidirect
39. #367×C₈₉	C₂²	semidirect
40. #366×C₈₉	C₂²	semidirect
194. #2159	D₈₉	semidirect
199. C₈₉	#33650	semidirect
219. #367	C₂×D₈₉	semidirect
222. #366	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029901	2
#26028848	2
#26022446	2
#26022445	2
#26022439	2
#26022434	2
#26022034	2
#26021969	2
C₂×#19151398	2
#26021867	2
C₈₉×#2159	2
#33650	89