Subgroups — #36538252

Order 22784.

694 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030351	C₂	1	2	yes	maximal
3	#26019675	C₂	1	4	yes	maximal, complemented
4	#26019641	C₂	1	2	yes	maximal, complemented
5	#26019617	C₂	1	2	yes	maximal, complemented
6	#26019015	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018950	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152155	C₂²	1	2	yes	—
10	#19152003	C₂²	1	1	yes	characteristic, complemented
11	#19151978	C₂²	1	1	yes	characteristic, complemented
12	#19151949	C₂²	1	2	yes	complemented
13	#19151933	—	2	2	no	—
14	#19151878	C₂²	1	4	yes	—
15	#19151839	—	2	1	no	—
16	#19151839	—	2	1	no	—
17	#19151803	C₂²	1	4	yes	—
18	#19151749	—	2	2	no	—
19	#19151698	—	2	4	no	—
20	C₄×Dic₃₅₆	—	2	1	no	—
21	C₄×Dic₃₅₆	—	2	1	no	—
22	#19151655	C₂²	1	2	yes	—
23	#19151351	C₂²	1	2	yes	complemented
24	#19151349	C₂²	1	2	yes	complemented
25	#19151345	C₂²	1	4	yes	complemented
26	#19151343	C₂²	1	4	yes	complemented
27	C₂³×C₈×C₈₉	—	2	1	no	—
28	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₈×D₄×C₈₉	—	2	1	no	—
30	C₈×D₄×C₈₉	—	2	1	no	—
31	C₈×D₄×C₈₉	C₂²	1	2	yes	—
32	C₈×D₄×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×Dic₃₅₆	—	2	1	no	—
37	C₂×Dic₃₅₆	—	2	1	no	—
38	C₂×#13083562	—	2	4	no	—
39	#15668012	C₂³	1	2	yes	—
40	#15668012	C₂³	1	2	yes	—
41	#15668004	—	2	2	no	—
42	#15667991	—	2	2	no	—
43	#15667991	—	2	2	no	—
44	C₄×Dic₁₇₈	—	2	2	no	—
45	C₄×Dic₁₇₈	—	2	2	no	—
46	#15667971	—	2	2	no	—
47	#15667946	—	2	1	no	—
48	#15667946	—	2	1	no	—
49	#15667945	—	2	4	no	—
50	C₄⋊#10623494	C₂³	1	4	yes	—
51	#15667937	—	2	2	no	—
52	#15667937	—	2	2	no	—
53	#15667935	—	2	4	no	—
54	#15667933	—	2	4	no	—
55	C₂²⋊#10623494	—	2	2	no	—
56	#15667923	—	2	2	no	—
57	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₈₉	—	2	1	no	—
59	C₂²×C₈×C₈₉	—	2	1	no	—
60	C₂²×C₈×C₈₉	—	2	2	no	—
61	C₂²×C₈×C₈₉	—	2	2	no	—
62	C₂²×C₈×C₈₉	—	4	4	no	—
63	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
65	C₄×D₄×C₈₉	—	2	1	no	—
66	C₄×D₄×C₈₉	—	2	1	no	—
67	C₄×D₄×C₈₉	D₄	1	2	yes	—
68	C₄×D₄×C₈₉	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
71	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₈₉	—	2	1	no	—
73	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
74	C₂²⋊C₈×C₈₉	—	2	2	no	—
75	C₂²⋊C₈×C₈₉	—	2	2	no	—
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₇₈	—	2	2	no	—
81	C₂×Dic₁₇₈	—	2	2	no	—
82	Dic₃₅₆	—	4	2	no	—
83	Dic₃₅₆	—	4	2	no	—
84	#13083562	—	4	8	no	—
85	C₂×#10623494	—	2	4	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	#13083553	—	4	2	no	—
91	#13083553	—	4	2	no	—
92	C₂²⋊Dic₈₉	—	4	2	no	—
93	C₄×Dic₈₉	—	4	2	no	—
94	C₂⁴×C₈₉	—	2	1	no	—
95	C₂×D₄×C₈₉	—	2	4	no	—
96	C₂×D₄×C₈₉	—	2	1	no	—
97	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	—	2	2	no	—
100	C₂²×C₄×C₈₉	—	2	2	no	—
101	C₂²×C₄×C₈₉	—	4	2	no	—
102	C₂²×C₄×C₈₉	—	4	2	no	—
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	—	4	4	no	—
110	C₂×C₈×C₈₉	—	4	2	no	—
111	C₂×C₈×C₈₉	—	4	4	no	—
112	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
118	C₂²⋊C₄×C₈₉	—	2	2	no	—
119	C₂²⋊C₄×C₈₉	—	2	2	no	—
120	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₈₉	#41	1	1	yes	characteristic
123	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₂³×C₈₉	—	2	4	no	—
125	C₂³×C₈₉	—	2	2	no	—
126	C₂³×C₈₉	—	4	2	no	—
127	C₂³×C₈₉	—	4	2	no	—
128	C₂³×C₈₉	#121	1	1	yes	characteristic
129	D₄×C₈₉	—	2	4	no	—
130	D₄×C₈₉	—	2	2	no	—
131	D₄×C₈₉	#135	1	4	yes	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	4	4	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₈₉	—	4	2	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	Dic₁₇₈	—	4	4	no	—
150	Dic₁₇₈	—	4	2	no	—
151	Dic₁₇₈	—	4	2	no	—
152	C₈×C₈₉	—	2	1	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	4	2	no	—
156	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
157	#10623494	—	4	4	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	4	no	—
161	C₂²×C₈₉	—	4	8	no	—
162	C₂²×C₈₉	—	4	2	no	—
163	C₂²×C₈₉	—	4	2	no	—
164	C₂²×C₈₉	#473	1	2	yes	—
165	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	4	2	no	—
171	C₄×C₈₉	C₂×#135	1	2	yes	—
172	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₄×C₈₉	#575	1	1	yes	characteristic
174	Dic₈₉	—	8	2	no	—
175	Dic₈₉	—	8	2	no	—
176	#19259	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	2	no	—
178	C₂×C₈₉	—	4	4	no	—
179	C₂×C₈₉	#3036	1	1	yes	characteristic
180	C₂×C₈₉	#3050	1	1	yes	characteristic
181	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
182	#3170	—	89	1	no	—
183	#3110	—	89	2	no	—
184	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
185	#1679	—	89	4	no	—
186	#1673	—	89	2	no	—
187	#1669	—	89	2	no	—
188	#1568	—	89	1	no	—
189	#1552	—	89	2	no	—
190	C₈₉	#19259	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₈	—	2	1	no	—
192	#533	—	89	2	no	—
193	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
194	#500	—	89	1	no	—
195	#494	—	89	1	no	—
196	#487	—	89	2	no	—
197	#484	—	178	2	no	—
198	#474	—	89	4	no	—
199	#467	—	178	1	no	—
200	#467	—	178	1	no	—
201	#461	—	89	4	no	—
202	#450	—	178	2	no	—
203	#440	—	178	4	no	—
204	C₄×Q₁₆	—	178	1	no	—
205	C₄×Q₁₆	—	178	1	no	—
206	C₈×D₄	—	2	1	no	—
207	C₈×D₄	—	2	1	no	—
208	C₈×D₄	C₂×D₈₉	1	2	yes	—
209	C₈×D₄	C₂×D₈₉	1	2	yes	—
210	#432	—	89	2	no	—
211	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
212	C₂×C₂²⋊C₈	—	2	1	no	—
213	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
214	#335	—	89	2	no	—
215	#334	—	89	2	no	—
216	#332	—	89	4	no	—
217	#332	—	89	4	no	—
218	C₂²×D₄	D₃₅₆	1	1	yes	characteristic
219	C₂³×C₄	—	2	1	no	—
220	C₂×Q₁₆	—	178	1	no	—
221	C₂×Q₁₆	—	178	1	no	—
222	C₂×SD₁₆	—	178	4	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	—	4	4	no	—
227	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
229	C₄⋊Q₈	—	89	2	no	—
230	C₄⋊Q₈	—	89	2	no	—
231	#125	—	178	2	no	—
232	C₂²⋊Q₈	—	178	2	no	—
233	C₂²⋊Q₈	—	178	2	no	—
234	C₄×Q₈	—	178	2	no	—
235	C₄×Q₈	—	178	2	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	D₃₅₆	1	2	yes	—
239	C₄×D₄	C₂²×D₈₉	1	2	yes	—
240	#117	—	178	2	no	—
241	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	—	2	1	no	—
243	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
244	#107	—	178	1	no	—
245	#107	—	178	1	no	—
246	#107	—	178	4	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	89	4	no	—
249	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
250	#103	—	178	2	no	—
251	#103	—	178	2	no	—
252	#103	—	178	4	no	—
253	#102	—	178	4	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	178	2	no	—
257	#97	—	178	2	no	—
258	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂⁴	—	2	1	no	—
263	C₂×Q₈	—	178	2	no	—
264	C₂×Q₈	—	178	2	no	—
265	C₂×D₄	—	2	4	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	C₂×D₃₅₆	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
275	C₂²×C₄	#13083574	1	1	yes	characteristic
276	Q₁₆	—	356	2	no	—
277	Q₁₆	—	356	2	no	—
278	SD₁₆	—	356	8	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	4	no	—
283	C₂×C₈	—	4	2	no	—
284	C₂×C₈	—	4	4	no	—
285	C₂×C₈	—	178	4	no	—
286	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	356	2	no	—
292	C₄⋊C₄	—	356	2	no	—
293	C₄⋊C₄	—	178	2	no	—
294	C₄⋊C₄	—	178	2	no	—
295	C₄⋊C₄	—	178	2	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	356	2	no	—
301	C₄²	—	356	2	no	—
302	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
303	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
304	C₄²	#13083574	1	1	yes	characteristic
305	C₄²	C₂³×D₈₉	1	1	yes	characteristic
306	C₂³	—	2	4	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	#15667986	1	1	yes	characteristic
311	Q₈	—	356	2	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	356	4	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	D₄	#15668034	1	4	yes	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	#15667986	1	1	yes	characteristic
329	C₂×C₄	#15667986	1	1	yes	characteristic
330	C₂×C₄	#15667986	1	1	yes	characteristic
331	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
333	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	356	4	no	—
339	C₈	#15668038	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	4	no	—
343	C₂²	—	4	8	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	#19151873	1	2	yes	—
347	C₂²	C₂×#15667986	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	712	2	no	—
354	C₄	—	712	2	no	—
355	C₄	C₂×#15668034	1	2	yes	—
356	C₄	C₂×#15668038	1	1	yes	characteristic
357	C₄	#19152396	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	4	4	no	—
360	C₂	#26029716	1	1	yes	characteristic
361	C₂	#26029833	1	1	yes	characteristic
362	C₂	C₂×#19151873	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667986
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈×D₄×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₄×C₈₉	D₄
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	#19259
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #26030838	C₂	semidirect
3. #26019675	C₂	semidirect
4. #26019641	C₂	semidirect
5. #26019617	C₂	semidirect
6. #26019015	C₂	semidirect
7. #26018950	C₂	semidirect
10. #19152003	C₂²	semidirect
11. #19151978	C₂²	semidirect
12. #19151949	C₂²	semidirect
23. #19151351	C₂²	semidirect
24. #19151349	C₂²	semidirect
25. #19151345	C₂²	semidirect
26. #19151343	C₂²	semidirect
190. C₈₉	#19259	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26030351	2
#26019675	2
#26019641	2
#26019617	2
#26019015	2
#26018950	2
C₂×C₈×D₄×C₈₉	2
#19259	89