Subgroups — #36538448

Order 22784.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030045	C₂	1	2	yes	maximal
2	C₂×#19151677	C₂	1	1	yes	maximal, characteristic
3	#26019824	C₂	1	4	yes	maximal, complemented
4	#26019772	C₂	1	2	yes	maximal, complemented
5	#26019738	C₂	1	2	yes	maximal, complemented
6	#26019008	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018942	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13083557	—	2	1	no	—
10	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
11	#19151895	—	2	4	no	—
12	#19151885	C₂²	1	4	yes	—
13	#19151875	C₂²	1	4	yes	—
14	#19151677	—	2	1	no	—
15	#19151677	—	2	1	no	—
16	#19151677	C₂²	1	2	yes	—
17	#19151677	C₂²	1	2	yes	—
18	#19151641	—	2	2	no	—
19	#19151622	C₂²	1	2	yes	—
20	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
21	#19151602	—	2	2	no	—
22	C₂×#15667924	—	2	1	no	—
23	#19151349	C₂²	1	2	yes	complemented
24	#19151349	C₂²	1	2	yes	complemented
25	#19151347	C₂²	1	4	yes	complemented
26	#19151344	C₂²	1	4	yes	complemented
27	#533×C₈₉	C₂²	1	2	yes	—
28	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#494×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	#487×C₈₉	C₂²	1	2	yes	complemented
31	#467×C₈₉	—	2	1	no	—
32	#467×C₈₉	—	2	1	no	—
33	C₄×Q₁₆×C₈₉	—	2	1	no	—
34	C₄×Q₁₆×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₈₉	—	2	1	no	—
38	C₂×#13083558	—	2	2	no	—
39	C₂×#13083557	—	2	2	no	—
40	C₂×#13083557	—	2	2	no	—
41	C₂×#13083557	—	4	4	no	—
42	C₂²×#10623494	—	2	1	no	—
43	C₄×D₃₅₆	—	2	1	no	—
44	C₄×D₃₅₆	—	2	1	no	—
45	C₄×D₃₅₆	D₄	1	2	yes	—
46	C₄×D₃₅₆	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
48	C₂×#13083551	—	2	1	no	—
49	#15667945	—	2	4	no	—
50	#15667943	—	2	4	no	—
51	C₄⋊#10623494	—	2	1	no	—
52	C₄⋊#10623494	—	2	2	no	—
53	C₄⋊#10623494	C₂³	1	2	yes	—
54	C₄⋊#10623494	C₂³	1	4	yes	—
55	#15667932	—	2	4	no	—
56	#15667932	—	2	4	no	—
57	#15667924	—	2	2	no	—
58	#15667924	—	2	2	no	—
59	C₂×Q₁₆×C₈₉	—	2	1	no	—
60	C₂×Q₁₆×C₈₉	—	2	1	no	—
61	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
65	#125×C₈₉	—	2	2	no	—
66	C₂²⋊Q₈×C₈₉	—	2	2	no	—
67	C₂²⋊Q₈×C₈₉	—	2	2	no	—
68	C₄×Q₈×C₈₉	—	2	2	no	—
69	C₄×Q₈×C₈₉	—	2	2	no	—
70	#117×C₈₉	—	2	2	no	—
71	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	—	2	1	no	—
74	#103×C₈₉	—	2	2	no	—
75	#103×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	C₂³×D₈₉	—	2	1	no	—
81	C₂×D₃₅₆	—	2	4	no	—
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
84	C₂²×Dic₈₉	—	2	1	no	—
85	C₂×C₄×D₈₉	—	2	2	no	—
86	C₂×C₄×D₈₉	—	2	2	no	—
87	C₂×C₄×D₈₉	—	4	2	no	—
88	C₂×C₄×D₈₉	—	4	2	no	—
89	#13083558	—	4	4	no	—
90	#13083557	—	4	4	no	—
91	#13083557	—	4	4	no	—
92	C₂×#10623494	—	2	2	no	—
93	C₂×#10623494	—	2	4	no	—
94	C₂×#10623494	—	4	2	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
97	#13083551	—	2	2	no	—
98	#13083551	—	2	2	no	—
99	C₂×Q₈×C₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	2	2	no	—
101	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
104	Q₁₆×C₈₉	—	4	2	no	—
105	Q₁₆×C₈₉	—	4	2	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	4	1	no	—
108	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	4	2	no	—
117	C₄⋊C₄×C₈₉	—	4	2	no	—
118	C₂²⋊C₄×C₈₉	—	4	2	no	—
119	C₄²×C₈₉	—	4	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	#121	1	1	yes	characteristic
125	C₂²×D₈₉	—	2	2	no	—
126	C₂²×D₈₉	—	4	2	no	—
127	C₂²×D₈₉	—	4	2	no	—
128	C₂²×D₈₉	—	4	2	no	—
129	Q₈×C₈₉	—	4	2	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	Q₈×C₈₉	—	8	2	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₈₉	#137	1	4	yes	—
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	D₃₅₆	—	2	4	no	—
147	D₃₅₆	—	4	2	no	—
148	C₄×D₈₉	—	4	4	no	—
149	C₄×D₈₉	—	4	2	no	—
150	C₄×D₈₉	—	4	2	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	2	1	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
155	#10623494	—	4	2	no	—
156	#10623494	—	4	4	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	#470	1	2	yes	—
160	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
161	C₂×D₈₉	—	4	2	no	—
162	C₂×D₈₉	—	4	2	no	—
163	C₂×D₈₉	—	4	2	no	—
164	C₂×D₈₉	—	8	4	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	8	2	no	—
169	C₄×C₈₉	—	8	2	no	—
170	C₄×C₈₉	#470	1	2	yes	—
171	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₄×C₈₉	C₂×#137	1	2	yes	—
173	C₄×C₈₉	#575	1	1	yes	characteristic
174	Dic₈₉	—	4	2	no	—
175	#19281	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	2	no	—
177	C₂×C₈₉	#3036	1	1	yes	characteristic
178	C₂×C₈₉	C₂×#470	1	1	yes	characteristic
179	C₂×C₈₉	#3067	1	1	yes	characteristic
180	D₈₉	—	8	2	no	—
181	#3170	D₈₉	1	1	yes	characteristic, complemented
182	#3077	—	89	2	no	—
183	C₂×#436	—	89	1	no	—
184	#1703	—	89	4	no	—
185	#1695	—	89	2	no	—
186	#1689	—	89	2	no	—
187	#1566	—	89	1	no	—
188	#1550	—	89	2	no	—
189	C₈₉	#19281	1	1	yes	characteristic, complemented, Sylow
190	C₂²×M₁₆	—	178	1	no	—
191	#533	C₂×D₈₉	1	2	yes	—
192	C₂×C₄×D₄	—	89	1	no	—
193	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
194	#494	C₂×D₈₉	1	1	yes	characteristic, complemented
195	#487	C₂×D₈₉	1	2	yes	complemented
196	#478	—	178	4	no	—
197	#476	—	89	4	no	—
198	#474	—	89	4	no	—
199	#467	—	2	1	no	—
200	#467	—	2	1	no	—
201	C₄×Q₁₆	—	2	1	no	—
202	C₄×Q₁₆	—	2	1	no	—
203	#436	—	89	2	no	—
204	#436	—	89	2	no	—
205	#436	—	178	1	no	—
206	#436	—	178	1	no	—
207	#428	—	178	2	no	—
208	#423	—	89	2	no	—
209	C₂×C₄⋊C₈	—	89	1	no	—
210	#418	—	178	2	no	—
211	C₂×C₂²⋊C₈	—	178	1	no	—
212	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
213	#334	—	89	2	no	—
214	#334	—	89	2	no	—
215	#333	—	89	4	no	—
216	#332	—	89	4	no	—
217	C₂²×D₄	—	89	1	no	—
218	C₂³×C₄	—	178	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×M₁₆	—	356	4	no	—
222	C₂×M₁₆	—	178	2	no	—
223	C₂×M₁₆	—	178	2	no	—
224	C₂×M₁₆	—	178	2	no	—
225	C₂²×C₈	—	178	1	no	—
226	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
227	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
229	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
230	#125	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×D₄	—	89	2	no	—
236	C₄×D₄	—	89	2	no	—
237	C₄×D₄	—	178	1	no	—
238	C₄×D₄	—	178	1	no	—
239	#117	—	2	2	no	—
240	C₂×C₄⋊C₄	—	89	1	no	—
241	C₂×C₂²⋊C₄	—	178	1	no	—
242	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
243	#107	—	2	1	no	—
244	#107	—	2	1	no	—
245	#107	—	178	4	no	—
246	#106	—	178	4	no	—
247	C₄⋊C₈	—	89	2	no	—
248	C₄⋊C₈	—	89	4	no	—
249	C₄⋊C₈	—	178	2	no	—
250	C₄⋊C₈	—	178	1	no	—
251	#103	—	2	2	no	—
252	#103	—	2	2	no	—
253	#102	—	178	4	no	—
254	#102	—	178	4	no	—
255	C₂²⋊C₈	—	178	2	no	—
256	C₂²⋊C₈	—	178	2	no	—
257	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
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	C₂⁴	—	178	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	89	2	no	—
265	C₂×D₄	—	178	1	no	—
266	C₂×D₄	—	178	4	no	—
267	C₂²×C₄	—	356	2	no	—
268	C₂²×C₄	—	356	2	no	—
269	C₂²×C₄	—	178	1	no	—
270	C₂²×C₄	—	178	2	no	—
271	C₂²×C₄	—	178	2	no	—
272	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₂²×C₄	#13083573	1	1	yes	characteristic
275	Q₁₆	—	4	2	no	—
276	Q₁₆	—	4	2	no	—
277	M₁₆	—	356	4	no	—
278	M₁₆	—	356	4	no	—
279	M₁₆	—	356	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	356	2	no	—
283	C₂×C₈	—	178	4	no	—
284	C₂×C₈	—	178	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	89	2	no	—
296	C₄⋊C₄	—	178	1	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	178	2	no	—
299	C₂²⋊C₄	—	178	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	D₄×D₈₉	1	1	yes	characteristic
302	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
303	C₄²	#13083573	1	1	yes	characteristic
304	C₄²	C₂³×D₈₉	1	1	yes	characteristic
305	C₂³	—	356	2	no	—
306	C₂³	—	356	2	no	—
307	C₂³	—	356	2	no	—
308	C₂³	—	178	2	no	—
309	C₂³	#15667983	1	1	yes	characteristic
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	356	2	no	—
314	D₄	—	178	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	356	4	no	—
321	C₂×C₄	—	356	2	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	#15667983	1	1	yes	characteristic
326	C₂×C₄	#15667983	1	1	yes	characteristic
327	C₂×C₄	#15667983	1	1	yes	characteristic
328	C₂×C₄	#15668044	1	4	yes	—
329	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	356	2	no	—
336	C₈	—	356	4	no	—
337	C₈	#15668037	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	356	2	no	—
341	C₂²	—	356	2	no	—
342	C₂²	—	356	2	no	—
343	C₂²	—	712	4	no	—
344	C₂²	#19151853	1	2	yes	—
345	C₂²	C₂×#15667983	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	356	2	no	—
352	C₄	#19151853	1	2	yes	—
353	C₄	C₂×#15668037	1	1	yes	characteristic
354	C₄	C₂×#15668044	1	2	yes	—
355	C₄	#19152396	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	712	2	no	—
358	C₂	#26029712	1	1	yes	characteristic
359	C₂	C₂×#19151853	1	1	yes	characteristic
360	C₂	#26029968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3170	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	#3170	D₈₉
hypocenter	C₈₉	#19281
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #26019824	C₂	semidirect
4. #26019772	C₂	semidirect
5. #26019738	C₂	semidirect
6. #26019008	C₂	semidirect
7. #26018942	C₂	semidirect
8. C₈₉×#3170	C₂	semidirect
23. #19151349	C₂²	semidirect
24. #19151349	C₂²	semidirect
25. #19151347	C₂²	semidirect
26. #19151344	C₂²	semidirect
28. #500×C₈₉	C₂²	semidirect
29. #494×C₈₉	C₂²	semidirect
30. #487×C₈₉	C₂²	semidirect
181. #3170	D₈₉	semidirect
189. C₈₉	#19281	semidirect
193. #500	C₂×D₈₉	semidirect
194. #494	C₂×D₈₉	semidirect
195. #487	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030045	2
C₂×#19151677	2
#26019824	2
#26019772	2
#26019738	2
#26019008	2
#26018942	2
C₈₉×#3170	2
#19281	89