Subgroups — #36599558

Order 22784.

620 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151870	C₂	1	1	yes	maximal, characteristic
2	#26021720	C₂	1	2	yes	maximal, complemented
3	#26021530	C₂	1	2	yes	maximal
4	#26021168	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021042	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020746	C₂	1	2	yes	maximal
7	#26020375	C₂	1	1	yes	maximal, characteristic, complemented
8	#26020366	C₂	1	2	yes	maximal, complemented
9	#26020303	C₂	1	1	yes	maximal, characteristic
10	#26020053	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668043	C₂²	1	2	yes	—
13	C₂×#15668018	C₂²	1	1	yes	characteristic
14	#19151870	—	2	2	no	—
15	#19151870	—	2	2	no	—
16	#19151646	C₂²	1	2	yes	—
17	C₂×#15667948	C₂²	1	1	yes	characteristic
18	C₂×#15667948	C₂²	1	1	yes	characteristic
19	#19151616	C₂²	1	2	yes	—
20	#19151607	C₂²	1	2	yes	—
21	#19151580	C₂²	1	1	yes	characteristic
22	C₂×#15667931	C₂²	1	2	yes	—
23	C₂×#15667931	C₂²	1	2	yes	—
24	#19151571	C₂²	1	1	yes	characteristic
25	#19151553	C₂²	1	1	yes	characteristic
26	#19151370	C₂²	1	1	yes	characteristic
27	#19151368	C₂²	1	1	yes	characteristic
28	#19151368	C₂²	1	2	yes	—
29	#19151368	C₂²	1	2	yes	—
30	#19151368	C₂²	1	2	yes	—
31	#19151354	C₂²	1	2	yes	—
32	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	#519×C₈₉	C₂²	1	2	yes	—
34	#441×C₈₉	—	2	2	no	—
35	C₄×Q₁₆×C₈₉	—	2	2	no	—
36	#427×C₈₉	C₂²	1	1	yes	characteristic
37	#419×C₈₉	C₂²	1	2	yes	—
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#13083574	C₂³	1	1	yes	characteristic
40	#15668043	—	2	4	no	—
41	#15668043	—	4	2	no	—
42	C₂×#13083564	—	2	2	no	—
43	C₂×#13083561	—	2	2	no	—
44	#15668018	—	2	1	no	—
45	#15668018	—	2	1	no	—
46	#15668018	—	2	1	no	—
47	#15668018	—	2	1	no	—
48	C₂×#13083558	C₂³	1	2	yes	—
49	C₂×#13083558	C₂³	1	2	yes	—
50	C₂×#13083558	C₂³	1	1	yes	characteristic
51	C₂×#13083558	C₂³	1	1	yes	characteristic
52	C₂×#13083557	—	2	1	no	—
53	C₂×C₈×D₈₉	—	2	1	no	—
54	#15667971	C₂³	1	1	yes	characteristic
55	#15667948	—	2	1	no	—
56	#15667948	—	2	1	no	—
57	#15667948	—	2	1	no	—
58	#15667948	—	2	1	no	—
59	#15667947	—	4	2	no	—
60	#15667940	—	4	2	no	—
61	#15667936	—	4	2	no	—
62	#15667931	—	2	4	no	—
63	#15667931	—	4	2	no	—
64	#15667930	—	2	1	no	—
65	#15667928	—	2	1	no	—
66	#15667926	—	2	2	no	—
67	#15667922	—	2	1	no	—
68	C₈×Dic₈₉	—	2	1	no	—
69	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₈₉	—	2	1	no	—
71	C₂×Q₁₆×C₈₉	—	2	1	no	—
72	C₂×Q₁₆×C₈₉	—	2	1	no	—
73	C₂×Q₁₆×C₈₉	—	2	1	no	—
74	C₂×Q₁₆×C₈₉	D₄	1	2	yes	—
75	C₂×Q₁₆×C₈₉	D₄	1	2	yes	—
76	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₈₉	—	2	4	no	—
79	C₄×Q₈×C₈₉	—	4	2	no	—
80	#117×C₈₉	—	2	4	no	—
81	#117×C₈₉	C₂³	1	2	yes	—
82	#117×C₈₉	C₂³	1	1	yes	characteristic
83	#107×C₈₉	—	2	1	no	—
84	#106×C₈₉	—	2	1	no	—
85	#103×C₈₉	—	2	4	no	—
86	#97×C₈₉	—	2	1	no	—
87	C₄×C₈×C₈₉	—	2	1	no	—
88	#13083574	—	2	1	no	—
89	#13083574	—	2	1	no	—
90	#13083574	—	2	1	no	—
91	#13083574	—	2	1	no	—
92	C₂×Dic₁₇₈	#41	1	1	yes	characteristic
93	C₂×D₃₅₆	#41	1	1	yes	characteristic
94	C₂×C₂²⋊D₈₉	—	2	1	no	—
95	C₂×C₄×D₈₉	—	2	1	no	—
96	#13083564	—	4	4	no	—
97	#13083561	—	4	4	no	—
98	#13083558	—	2	1	no	—
99	#13083558	—	2	2	no	—
100	#13083558	—	2	1	no	—
101	#13083558	—	2	1	no	—
102	#13083558	—	2	2	no	—
103	#13083558	—	2	2	no	—
104	#13083558	—	2	1	no	—
105	#13083558	—	2	2	no	—
106	#13083557	—	4	1	no	—
107	#13083557	—	4	1	no	—
108	C₂×#10623494	—	2	1	no	—
109	C₂×#10623494	—	2	2	no	—
110	C₂×#10623494	—	2	1	no	—
111	C₂×#10623494	—	2	2	no	—
112	C₈×D₈₉	—	4	1	no	—
113	C₈×D₈₉	—	4	1	no	—
114	C₄⋊Dic₈₉	—	2	1	no	—
115	C₂²⋊Dic₈₉	—	2	1	no	—
116	C₄×Dic₈₉	—	2	1	no	—
117	C₂×Q₈×C₈₉	—	2	2	no	—
118	C₂×Q₈×C₈₉	—	2	2	no	—
119	C₂×Q₈×C₈₉	—	2	2	no	—
120	C₂×Q₈×C₈₉	—	2	2	no	—
121	C₂×Q₈×C₈₉	—	2	2	no	—
122	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₈₉	—	2	2	no	—
124	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₈₉	—	2	2	no	—
126	Q₁₆×C₈₉	—	2	2	no	—
127	Q₁₆×C₈₉	—	2	2	no	—
128	Q₁₆×C₈₉	—	2	2	no	—
129	M₁₆×C₈₉	—	2	1	no	—
130	M₁₆×C₈₉	—	2	1	no	—
131	C₂×C₈×C₈₉	—	2	1	no	—
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
135	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
136	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
137	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
138	C₄⋊C₄×C₈₉	—	2	1	no	—
139	C₄⋊C₄×C₈₉	—	2	2	no	—
140	C₄⋊C₄×C₈₉	—	4	4	no	—
141	C₂²⋊C₄×C₈₉	—	2	2	no	—
142	C₂²⋊C₄×C₈₉	#41	1	2	yes	—
143	C₄²×C₈₉	—	2	1	no	—
144	C₄²×C₈₉	—	2	2	no	—
145	C₄²×C₈₉	—	4	4	no	—
146	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₂²×D₈₉	—	2	1	no	—
148	Q₈×C₈₉	—	2	4	no	—
149	Q₈×C₈₉	—	2	4	no	—
150	Q₈×C₈₉	—	4	2	no	—
151	Q₈×C₈₉	—	4	2	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	4	no	—
154	C₂×C₄×C₈₉	—	4	2	no	—
155	C₂×C₄×C₈₉	—	4	2	no	—
156	C₂×C₄×C₈₉	—	4	2	no	—
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²⋊D₈₉	—	4	1	no	—
164	C₂²⋊D₈₉	—	4	1	no	—
165	C₂×Dic₈₉	—	2	1	no	—
166	C₂×Dic₈₉	—	4	1	no	—
167	D₃₅₆	—	2	1	no	—
168	D₃₅₆	—	2	1	no	—
169	C₄×D₈₉	—	4	1	no	—
170	C₄×D₈₉	—	4	1	no	—
171	Dic₁₇₈	—	2	1	no	—
172	Dic₁₇₈	—	2	1	no	—
173	C₈×C₈₉	—	4	1	no	—
174	C₈×C₈₉	#123	1	1	yes	characteristic
175	C₈×C₈₉	#123	1	1	yes	characteristic
176	C₈×C₈₉	#123	1	1	yes	characteristic
177	C₈×C₈₉	#123	1	1	yes	characteristic
178	#10623494	—	4	1	no	—
179	#10623494	—	4	2	no	—
180	#10623494	—	4	2	no	—
181	#10623494	—	4	1	no	—
182	C₂²×C₈₉	—	4	1	no	—
183	C₂²×C₈₉	#529	1	1	yes	characteristic
184	C₂²×C₈₉	#546	1	1	yes	characteristic
185	C₂²×C₈₉	#555	1	1	yes	characteristic
186	C₂×D₈₉	—	4	1	no	—
187	C₂×D₈₉	—	8	1	no	—
188	C₄×C₈₉	—	4	2	no	—
189	C₄×C₈₉	—	4	4	no	—
190	C₄×C₈₉	—	8	2	no	—
191	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
192	C₄×C₈₉	#540	1	1	yes	characteristic
193	C₄×C₈₉	#546	1	1	yes	characteristic
194	C₄×C₈₉	#553	1	1	yes	characteristic
195	Dic₈₉	—	4	1	no	—
196	Dic₈₉	—	8	1	no	—
197	#24989	—	89	1	no	maximal, Sylow
198	C₂×C₈₉	—	2	1	no	—
199	C₂×C₈₉	—	2	1	no	—
200	C₂×C₈₉	—	2	1	no	—
201	C₂×C₈₉	#2420	1	1	yes	characteristic
202	D₈₉	—	8	1	no	—
203	C₂×#473	—	89	1	no	—
204	#2950	D₈₉	1	1	yes	characteristic, complemented
205	#2082	—	89	2	no	—
206	#2049	—	89	2	no	—
207	#1979	—	89	1	no	—
208	#1953	—	89	1	no	—
209	#1905	—	89	2	no	—
210	#1833	—	89	1	no	—
211	#1830	—	89	2	no	—
212	#1814	—	89	1	no	—
213	#1757	—	89	1	no	—
214	C₈₉	#24989	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	89	2	no	—
216	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
217	C₂×#131	—	89	1	no	—
218	#519	C₂×D₈₉	1	2	yes	—
219	#473	—	178	2	no	—
220	#473	—	178	2	no	—
221	#441	—	2	2	no	—
222	C₄×Q₁₆	—	2	2	no	—
223	#430	—	89	2	no	—
224	C₂×#108	—	89	1	no	—
225	C₂×#108	—	89	1	no	—
226	#427	C₂×D₈₉	1	1	yes	characteristic
227	#421	—	89	2	no	—
228	#419	—	89	2	no	—
229	#419	C₂×D₈₉	1	2	yes	—
230	#413	—	89	1	no	—
231	C₂×#101	—	89	2	no	—
232	C₂×#101	—	89	2	no	—
233	#410	—	89	1	no	—
234	#405	—	89	1	no	—
235	#405	C₂×D₈₉	1	1	yes	characteristic
236	#344	—	89	1	no	—
237	#344	—	89	2	no	—
238	#344	—	89	2	no	—
239	#344	—	89	1	no	—
240	#344	—	89	2	no	—
241	H(3,ℤ/4)	—	89	2	no	—
242	C₂×#41	—	89	1	no	—
243	C₂²×Q₈	C₂²×D₈₉	1	2	yes	—
244	#137	—	356	2	no	—
245	#137	—	178	4	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	2	1	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	178	2	no	—
251	C₂×Q₁₆	C₂²⋊D₈₉	1	2	yes	—
252	C₂×Q₁₆	C₂²⋊D₈₉	1	2	yes	—
253	C₂×SD₁₆	—	178	2	no	—
254	#131	—	178	1	no	—
255	#131	—	178	1	no	—
256	#131	—	178	1	no	—
257	#131	—	178	1	no	—
258	C₂×M₁₆	—	89	1	no	—
259	C₂×M₁₆	—	89	2	no	—
260	C₂×M₁₆	—	89	1	no	—
261	C₂×M₁₆	—	89	2	no	—
262	C₂×M₁₆	—	178	1	no	—
263	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
264	C₂²×C₈	—	178	1	no	—
265	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	4	2	no	—
268	#117	—	2	4	no	—
269	#117	—	89	1	no	—
270	#117	C₂²×D₈₉	1	2	yes	—
271	#117	C₂²×D₈₉	1	1	yes	characteristic
272	#108	—	356	2	no	—
273	#108	—	178	1	no	—
274	#108	—	178	1	no	—
275	#108	—	178	1	no	—
276	#108	—	178	1	no	—
277	#107	—	2	1	no	—
278	#106	—	2	1	no	—
279	#104	—	356	2	no	—
280	#103	—	2	4	no	—
281	#103	—	356	2	no	—
282	#101	—	356	2	no	—
283	#101	—	178	1	no	—
284	#101	—	178	4	no	—
285	#100	—	178	1	no	—
286	#99	—	178	2	no	—
287	#97	—	2	1	no	—
288	#97	—	178	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	178	1	no	—
291	#41	—	178	1	no	—
292	#41	—	178	1	no	—
293	#41	—	178	1	no	—
294	#41	—	178	1	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	89	1	no	—
301	C₂×Q₈	C₂×C₂²⋊D₈₉	1	4	yes	—
302	C₂×D₄	—	89	1	no	—
303	C₂×D₄	—	178	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	178	1	no	—
306	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
307	Q₁₆	—	2	2	no	—
308	Q₁₆	—	2	2	no	—
309	Q₁₆	—	2	2	no	—
310	Q₁₆	—	2	2	no	—
311	Q₁₆	—	356	4	no	—
312	SD₁₆	—	356	4	no	—
313	M₁₆	—	2	1	no	—
314	M₁₆	—	2	1	no	—
315	M₁₆	—	356	1	no	—
316	M₁₆	—	356	1	no	—
317	M₁₆	—	178	1	no	—
318	M₁₆	—	178	2	no	—
319	M₁₆	—	178	1	no	—
320	M₁₆	—	178	2	no	—
321	M₁₆	—	178	1	no	—
322	M₁₆	—	178	1	no	—
323	M₁₆	—	178	2	no	—
324	M₁₆	—	178	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	356	1	no	—
327	C₂×C₈	—	356	1	no	—
328	C₂×C₈	—	178	2	no	—
329	C₂×C₈	—	178	1	no	—
330	C₂×C₈	—	178	1	no	—
331	C₂×C₈	—	178	2	no	—
332	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
333	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
334	C₂×C₈	#13083574	1	1	yes	characteristic
335	C₂×C₈	#13083574	1	1	yes	characteristic
336	C₂×C₈	#13083574	1	1	yes	characteristic
337	C₂×C₈	#13083574	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₄⋊C₄	—	178	1	no	—
342	C₂²⋊C₄	—	2	2	no	—
343	C₂²⋊C₄	—	178	1	no	—
344	C₂²⋊C₄	#13083573	1	2	yes	—
345	C₄²	—	2	1	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	4	4	no	—
348	C₄²	—	178	1	no	—
349	C₂³	—	178	1	no	—
350	C₂³	C₂×#13083573	1	1	yes	characteristic
351	Q₈	—	2	4	no	—
352	Q₈	—	2	4	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	178	1	no	—
356	Q₈	—	178	1	no	—
357	D₄	—	356	1	no	—
358	D₄	—	356	1	no	—
359	D₄	—	178	1	no	—
360	D₄	—	178	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	356	1	no	—
367	C₂×C₄	—	356	1	no	—
368	C₂×C₄	—	356	1	no	—
369	C₂×C₄	—	178	1	no	—
370	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
371	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
372	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
373	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
374	C₂×C₄	#15668057	1	1	yes	characteristic
375	C₂×C₄	#15668058	1	1	yes	characteristic
376	C₈	—	4	1	no	—
377	C₈	—	356	2	no	—
378	C₈	—	356	2	no	—
379	C₈	—	356	1	no	—
380	C₈	—	356	1	no	—
381	C₈	#15667995	1	1	yes	characteristic
382	C₈	#15667995	1	1	yes	characteristic
383	C₈	#15667995	1	1	yes	characteristic
384	C₈	#15667995	1	1	yes	characteristic
385	C₂²	—	4	1	no	—
386	C₂²	—	356	1	no	—
387	C₂²	—	712	1	no	—
388	C₂²	#19152130	1	1	yes	characteristic
389	C₂²	#19152242	1	1	yes	characteristic
390	C₂²	#19152313	1	1	yes	characteristic
391	C₄	—	4	2	no	—
392	C₄	—	4	4	no	—
393	C₄	—	8	2	no	—
394	C₄	—	356	1	no	—
395	C₄	—	712	1	no	—
396	C₄	C₂×#15667995	1	1	yes	characteristic
397	C₄	#19152199	1	1	yes	characteristic
398	C₄	#19152236	1	1	yes	characteristic
399	C₄	#19152298	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	712	1	no	—
404	C₂	#26023738	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26023738
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#2950	C₂
socle	C₂×C₈₉	#2420
prime radical	C₂×#13083574	C₂³
hypercenter	#2950	D₈₉
hypocenter	C₈₉	#24989
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2950

Complements

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

N	H ≅ G/N	type
2. #26021720	C₂	semidirect
4. #26021168	C₂	semidirect
5. #26021042	C₂	semidirect
7. #26020375	C₂	semidirect
8. #26020366	C₂	semidirect
11. C₈₉×#2950	C₂	semidirect
204. #2950	D₈₉	semidirect
214. C₈₉	#24989	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151870	2
#26021720	2
#26021530	2
#26021168	2
#26021042	2
#26020746	2
#26020375	2
#26020366	2
#26020303	2
#26020053	2
C₈₉×#2950	2
#24989	89