Subgroups — #36539618

Order 22784.

654 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	#26031197	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030045	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151879	C₂	1	1	yes	maximal, characteristic
4	#26029275	C₂	1	1	yes	maximal, characteristic
5	#26019706	C₂	1	4	yes	maximal, complemented
6	#26019616	C₂	1	4	yes	maximal, complemented
7	#26019026	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018936	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668012	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	#19151977	C₂²	1	2	yes	complemented
13	#19151950	C₂²	1	2	yes	complemented
14	#19151895	—	2	2	no	—
15	#19151885	C₂²	1	2	yes	—
16	#19151879	—	2	2	no	—
17	#19151879	C₂²	1	4	yes	—
18	#19151875	C₂²	1	2	yes	—
19	#19151808	—	2	2	no	—
20	#19151753	—	2	2	no	—
21	C₈×D₃₅₆	—	2	2	no	—
22	#19151663	C₂²	1	4	yes	—
23	#19151651	C₂²	1	1	yes	characteristic
24	#19151641	—	2	1	no	—
25	C₂×#15667946	—	2	1	no	—
26	#19151622	C₂²	1	1	yes	characteristic
27	#19151602	—	2	1	no	—
28	C₂×#15667934	—	2	1	no	—
29	C₂²⋊C₈×D₈₉	—	2	2	no	—
30	#19151350	C₂²	1	4	yes	complemented
31	#19151345	C₂²	1	4	yes	complemented
32	#533×C₈₉	C₂²	1	1	yes	characteristic
33	#431×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
35	#333×C₈₉	C₂²	1	2	yes	complemented
36	#332×C₈₉	C₂²	1	2	yes	complemented
37	C₂²×Dic₁₇₈	—	2	1	no	—
38	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
39	C₂²×C₄×D₈₉	—	2	1	no	—
40	C₂×#13083564	—	2	4	no	—
41	C₂×#13083558	—	2	1	no	—
42	C₂×C₈×D₈₉	—	2	4	no	—
43	#15668012	—	2	1	no	—
44	#15668012	—	2	1	no	—
45	#15668012	—	2	1	no	—
46	#15668012	C₂³	1	2	yes	—
47	C₄×D₃₅₆	—	2	1	no	—
48	C₄×D₃₅₆	—	2	1	no	—
49	C₄×D₃₅₆	D₄	1	2	yes	—
50	C₄×D₃₅₆	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₈₉	—	2	1	no	—
52	C₂×C₄⋊Dic₈₉	—	2	1	no	—
53	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
54	C₂×#13083551	—	2	1	no	—
55	#15667946	—	2	4	no	—
56	#15667945	—	2	2	no	—
57	#15667943	—	2	2	no	—
58	C₄⋊#10623494	—	2	1	no	—
59	C₄⋊#10623494	D₄	1	2	yes	—
60	C₄⋊#10623494	C₂³	1	2	yes	—
61	#15667935	—	2	4	no	—
62	#15667934	—	2	4	no	—
63	#15667932	—	2	2	no	—
64	#15667932	—	2	2	no	—
65	C₂²⋊#10623494	D₄	1	2	yes	—
66	#15667924	—	2	4	no	—
67	C₄×#10623494	C₂³	1	2	yes	—
68	C₂²×C₈×C₈₉	—	2	1	no	—
69	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
71	#125×C₈₉	—	2	1	no	—
72	C₂²⋊Q₈×C₈₉	—	2	1	no	—
73	C₂²⋊Q₈×C₈₉	—	2	1	no	—
74	C₄×Q₈×C₈₉	—	2	1	no	—
75	C₄×Q₈×C₈₉	—	2	1	no	—
76	#117×C₈₉	—	2	1	no	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₈₉	—	2	1	no	—
79	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
80	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₈₉	—	2	1	no	—
82	C₄×C₈×C₈₉	—	2	1	no	—
83	C₂³×D₈₉	—	2	1	no	—
84	C₂×Dic₁₇₈	—	2	2	no	—
85	C₂×Dic₁₇₈	—	4	1	no	—
86	C₂×Dic₁₇₈	—	8	2	no	—
87	C₂×D₃₅₆	—	2	1	no	—
88	C₂×D₃₅₆	—	2	2	no	—
89	C₂×D₃₅₆	Q₁₆	1	4	yes	complemented
90	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
91	C₂²×Dic₈₉	—	2	1	no	—
92	C₂²×Dic₈₉	—	4	1	no	—
93	C₂×C₄×D₈₉	—	2	2	no	—
94	C₂×C₄×D₈₉	—	2	2	no	—
95	C₂×C₄×D₈₉	—	8	2	no	—
96	#13083564	—	8	4	no	—
97	#13083558	—	4	2	no	—
98	C₂×#10623494	—	2	2	no	—
99	C₂×#10623494	C₂×D₄	1	4	yes	—
100	C₈×D₈₉	—	8	4	no	—
101	C₄⋊Dic₈₉	—	2	1	no	—
102	C₄⋊Dic₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	—	2	2	no	—
104	C₄⋊Dic₈₉	—	4	1	no	—
105	C₄⋊Dic₈₉	—	4	1	no	—
106	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
107	#13083551	—	2	2	no	—
108	#13083551	—	2	2	no	—
109	C₂×Q₈×C₈₉	—	2	1	no	—
110	C₂×Q₈×C₈₉	—	2	1	no	—
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	—	4	2	no	—
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₂²⋊C₄×C₈₉	—	4	1	no	—
124	C₄²×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	#41	1	1	yes	characteristic
128	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₈₉	#121	1	1	yes	characteristic
130	C₂²×D₈₉	—	2	2	no	—
131	C₂²×D₈₉	—	2	4	no	—
132	C₂²×D₈₉	—	8	1	no	—
133	C₂²×D₈₉	—	8	1	no	—
134	Q₈×C₈₉	—	4	2	no	—
135	Q₈×C₈₉	—	8	1	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
144	C₂×C₄×C₈₉	#135	1	2	yes	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	8	1	no	—
151	C₂×Dic₈₉	—	8	2	no	—
152	D₃₅₆	—	2	2	no	—
153	D₃₅₆	—	2	4	no	—
154	D₃₅₆	C₂×Q₁₆	1	4	yes	—
155	C₄×D₈₉	—	4	4	no	—
156	C₄×D₈₉	—	8	2	no	—
157	Dic₁₇₈	—	4	4	no	—
158	Dic₁₇₈	—	8	2	no	—
159	C₈×C₈₉	—	4	2	no	—
160	C₈×C₈₉	—	4	2	no	—
161	#10623494	—	2	4	no	—
162	#10623494	—	4	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	#467	1	2	yes	—
166	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
167	C₂×D₈₉	—	2	4	no	—
168	C₂×D₈₉	—	4	2	no	—
169	C₂×D₈₉	—	4	2	no	—
170	C₂×D₈₉	—	8	4	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	—	8	1	no	—
176	C₄×C₈₉	#467	1	2	yes	—
177	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
178	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
179	C₄×C₈₉	#576	1	2	yes	—
180	Dic₈₉	—	4	2	no	—
181	Dic₈₉	—	8	2	no	—
182	#19387	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	2	no	—
184	C₂×C₈₉	#3041	1	1	yes	characteristic
185	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
186	C₂×C₈₉	#3069	1	1	yes	characteristic
187	D₈₉	—	4	4	no	—
188	#3209	—	89	1	no	—
189	#3077	—	89	1	no	—
190	C₂×#474	—	89	1	no	—
191	#2972	—	89	1	no	—
192	#1683	—	89	4	no	—
193	#1669	—	89	4	no	—
194	#1571	—	89	1	no	—
195	#1549	—	89	1	no	—
196	#1542	D₈₉	1	1	yes	characteristic, complemented
197	C₈₉	#19387	1	1	yes	characteristic, complemented, Sylow
198	#533	C₂×D₈₉	1	1	yes	characteristic
199	C₂×C₄⋊Q₈	—	89	1	no	—
200	C₂×C₄×D₄	—	89	1	no	—
201	#494	—	89	2	no	—
202	#487	—	89	2	no	—
203	#478	—	178	2	no	—
204	#476	—	89	2	no	—
205	#474	—	89	2	no	—
206	#474	—	89	4	no	—
207	#474	—	178	2	no	—
208	#462	—	178	2	no	—
209	#451	—	178	2	no	—
210	C₈×D₄	—	89	4	no	—
211	C₈×D₄	—	178	2	no	—
212	#431	—	89	1	no	—
213	#431	C₂×D₈₉	1	1	yes	characteristic
214	#428	—	178	1	no	—
215	C₂×#107	—	178	1	no	—
216	#423	—	89	1	no	—
217	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
218	#418	—	178	1	no	—
219	C₂×#102	—	178	1	no	—
220	C₂×C₂²⋊C₈	—	178	2	no	—
221	#335	—	89	4	no	—
222	#333	C₂×D₈₉	1	2	yes	complemented
223	#332	—	89	4	no	—
224	#332	C₂×D₈₉	1	2	yes	complemented
225	C₂²×Q₈	—	178	1	no	—
226	C₂²×D₄	—	89	1	no	—
227	C₂³×C₄	—	178	1	no	—
228	C₂×Q₁₆	—	178	4	no	—
229	C₂×M₁₆	—	178	1	no	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	—	178	4	no	—
232	C₄⋊Q₈	—	89	2	no	—
233	C₄⋊Q₈	—	178	1	no	—
234	C₄⋊Q₈	—	178	1	no	—
235	C₄⋊Q₈	—	178	1	no	—
236	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
238	#125	—	2	1	no	—
239	C₂²⋊Q₈	—	2	1	no	—
240	C₂²⋊Q₈	—	2	1	no	—
241	C₄×Q₈	—	2	1	no	—
242	C₄×Q₈	—	2	1	no	—
243	C₄×D₄	—	89	2	no	—
244	C₄×D₄	—	89	2	no	—
245	C₄×D₄	—	178	1	no	—
246	C₄×D₄	—	178	1	no	—
247	#117	—	2	1	no	—
248	C₂×C₄⋊C₄	—	89	1	no	—
249	C₂×C₄⋊C₄	—	178	1	no	—
250	C₂×C₄⋊C₄	—	178	1	no	—
251	C₂×C₂²⋊C₄	—	178	1	no	—
252	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
253	#107	—	178	2	no	—
254	#107	—	178	4	no	—
255	#106	—	178	2	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	89	2	no	—
258	C₄⋊C₈	—	89	2	no	—
259	C₄⋊C₈	—	178	1	no	—
260	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
261	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
262	#103	—	178	4	no	—
263	#102	—	178	2	no	—
264	#102	—	178	2	no	—
265	#102	—	178	4	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	89	2	no	—
268	C₂²⋊C₈	—	178	4	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	—	89	2	no	—
271	C₂⁴	—	178	1	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	356	1	no	—
275	C₂×Q₈	—	178	2	no	—
276	C₂×Q₈	—	712	2	no	—
277	C₂×D₄	—	89	4	no	—
278	C₂×D₄	—	89	2	no	—
279	C₂×D₄	—	178	1	no	—
280	C₂×D₄	—	178	2	no	—
281	C₂²×C₄	—	356	1	no	—
282	C₂²×C₄	—	178	2	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	—	178	2	no	—
285	C₂²×C₄	—	712	2	no	—
286	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
287	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
288	C₂²×C₄	#13083574	1	1	yes	characteristic
289	Q₁₆	—	712	4	no	—
290	M₁₆	—	356	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	—	89	4	no	—
295	C₂×C₈	—	178	2	no	—
296	C₂×C₈	—	712	4	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	89	2	no	—
304	C₄⋊C₄	—	356	1	no	—
305	C₄⋊C₄	—	356	1	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	—	178	2	no	—
308	C₄⋊C₄	—	178	1	no	—
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	178	2	no	—
311	C₂²⋊C₄	—	178	2	no	—
312	C₄²	—	4	1	no	—
313	C₄²	D₄×D₈₉	1	1	yes	characteristic
314	C₄²	D₄×D₈₉	1	1	yes	characteristic
315	C₄²	#13083574	1	1	yes	characteristic
316	C₄²	C₂³×D₈₉	1	1	yes	characteristic
317	C₂³	—	178	2	no	—
318	C₂³	—	178	4	no	—
319	C₂³	—	712	1	no	—
320	C₂³	—	712	1	no	—
321	C₂³	#15667985	1	1	yes	characteristic
322	Q₈	—	4	2	no	—
323	Q₈	—	8	1	no	—
324	Q₈	—	356	4	no	—
325	Q₈	—	712	2	no	—
326	D₄	—	89	4	no	—
327	D₄	—	178	4	no	—
328	D₄	—	178	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	356	4	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	178	2	no	—
336	C₂×C₄	—	712	2	no	—
337	C₂×C₄	—	712	1	no	—
338	C₂×C₄	—	712	2	no	—
339	C₂×C₄	#15667985	1	1	yes	characteristic
340	C₂×C₄	#15667985	1	1	yes	characteristic
341	C₂×C₄	#15667985	1	1	yes	characteristic
342	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
343	C₂×C₄	#15668033	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
345	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
346	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
347	C₈	—	4	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	356	2	no	—
350	C₈	—	178	4	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	356	2	no	—
354	C₂²	—	356	2	no	—
355	C₂²	—	178	4	no	—
356	C₂²	—	712	4	no	—
357	C₂²	#19151838	1	2	yes	—
358	C₂²	C₂×#15667985	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	356	2	no	—
365	C₄	—	712	2	no	—
366	C₄	#19151838	1	2	yes	—
367	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
368	C₄	C₂×#15668033	1	1	yes	characteristic
369	C₄	#19152400	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	356	4	no	—
372	C₂	#26029758	1	1	yes	characteristic
373	C₂	C₂×#19151838	1	1	yes	characteristic
374	C₂	#26029992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667985
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1542	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	#1542	D₈₉
hypocenter	C₈₉	#19387
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
1. #26031197	C₂	semidirect
5. #26019706	C₂	semidirect
6. #26019616	C₂	semidirect
7. #26019026	C₂	semidirect
8. #26018936	C₂	semidirect
9. C₈₉×#1542	C₂	semidirect
12. #19151977	C₂²	semidirect
13. #19151950	C₂²	semidirect
30. #19151350	C₂²	semidirect
31. #19151345	C₂²	semidirect
35. #333×C₈₉	C₂²	semidirect
36. #332×C₈₉	C₂²	semidirect
89. C₂×D₃₅₆	Q₁₆	semidirect
196. #1542	D₈₉	semidirect
197. C₈₉	#19387	semidirect
222. #333	C₂×D₈₉	semidirect
224. #332	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26031197	2
#26030045	2
C₂×#19151879	2
#26029275	2
#26019706	2
#26019616	2
#26019026	2
#26018936	2
C₈₉×#1542	2
#19387	89