Subgroups — #36572957

Order 22784.

808 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029170	C₂	1	1	yes	maximal, characteristic
2	#26029168	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151719	C₂	1	1	yes	maximal, characteristic
4	#26028862	C₂	1	2	yes	maximal, complemented
5	#26021043	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020495	C₂	1	2	yes	maximal, complemented
7	#26020489	C₂	1	2	yes	maximal, complemented
8	#26020425	C₂	1	1	yes	maximal, characteristic, complemented
9	#26020305	C₂	1	2	yes	maximal, complemented
10	#26020137	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₂²	1	1	yes	characteristic
13	#19152364	C₂²	1	2	yes	—
14	#19152364	C₂²	1	2	yes	—
15	C₂×#15668020	C₂²	1	2	yes	—
16	#19151721	—	2	2	no	—
17	#19151719	—	2	2	no	—
18	#19151719	C₄	1	4	yes	complemented
19	#19151719	C₄	1	4	yes	complemented
20	#19151716	—	2	2	no	—
21	#19151714	—	2	2	no	—
22	#19151647	C₂²	1	2	yes	—
23	#19151645	—	2	4	no	—
24	C₂×#15667948	C₂²	1	1	yes	characteristic
25	C₂×#15667947	C₂²	1	2	yes	—
26	C₂×#15667947	C₂²	1	1	yes	characteristic
27	#108×D₈₉	—	2	4	no	—
28	#19151634	C₂²	1	1	yes	characteristic
29	#19151621	C₂²	1	2	yes	—
30	#19151616	C₂²	1	2	yes	—
31	#19151614	C₂²	1	2	yes	—
32	C₂×#15667938	C₂²	1	2	yes	—
33	#19151553	C₂²	1	1	yes	characteristic
34	C₂×#15667922	C₂²	1	1	yes	characteristic
35	#19151370	C₂²	1	2	yes	—
36	#19151359	C₂²	1	1	yes	characteristic
37	#19151357	C₂²	1	1	yes	characteristic
38	#430×C₈₉	C₂²	1	2	yes	—
39	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
41	#405×C₈₉	C₂²	1	1	yes	characteristic
42	#344×C₈₉	C₂²	1	2	yes	—
43	C₂×#13083574	C₂³	1	2	yes	—
44	#15668034	C₂×C₄	1	2	yes	—
45	#15668034	C₂×C₄	1	2	yes	—
46	#15668034	D₄	1	2	yes	—
47	#15668034	D₄	1	2	yes	—
48	C₂×Dic₃₅₆	Q₈	1	1	yes	characteristic
49	C₂×#13083563	Q₈	1	2	yes	—
50	C₂×D₇₁₂	Q₈	1	1	yes	characteristic
51	#15668020	—	2	2	no	—
52	#15668020	—	2	2	no	—
53	#15668020	—	2	4	no	—
54	#15668020	—	4	2	no	—
55	#15668020	C₂×C₄	1	4	yes	—
56	#15668020	C₂×C₄	1	4	yes	—
57	#15668018	—	2	4	no	—
58	#15668018	—	4	2	no	—
59	C₂×#13083558	C₂³	1	2	yes	—
60	C₂×#13083558	C₂³	1	1	yes	characteristic
61	C₂×#13083557	—	2	2	no	—
62	M₁₆×D₈₉	—	2	4	no	—
63	M₁₆×D₈₉	—	2	4	no	—
64	C₂²×#10623494	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₈₉	—	2	2	no	—
66	#15667971	C₂³	1	1	yes	characteristic
67	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
68	#15667948	—	2	4	no	—
69	#15667948	—	2	1	no	—
70	#15667948	—	2	1	no	—
71	#15667947	—	2	2	no	—
72	#15667947	—	2	2	no	—
73	#15667947	D₄	1	4	yes	—
74	#15667945	—	2	1	no	—
75	#15667943	—	2	1	no	—
76	#15667941	—	4	2	no	—
77	#15667940	—	2	4	no	—
78	#15667938	—	2	4	no	—
79	#15667938	C₂×C₄	1	8	yes	—
80	#15667922	—	2	1	no	—
81	#15667922	C₂×C₄	1	2	yes	—
82	#15667922	C₂×C₄	1	2	yes	—
83	C₈×Dic₈₉	—	2	1	no	—
84	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
85	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
86	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	#117×C₈₉	C₂³	1	1	yes	characteristic
89	#108×C₈₉	—	2	4	no	—
90	#108×C₈₉	—	2	1	no	—
91	#108×C₈₉	—	2	1	no	—
92	#108×C₈₉	C₂×C₄	1	2	yes	—
93	#108×C₈₉	C₂×C₄	1	2	yes	—
94	#97×C₈₉	—	2	1	no	—
95	C₄×C₈×C₈₉	—	2	1	no	—
96	#13083574	—	2	2	no	—
97	#13083574	—	2	2	no	—
98	#13083574	C₂²×C₄	1	4	yes	—
99	#13083574	C₂×D₄	1	4	yes	—
100	C₂×Dic₁₇₈	C₂×Q₈	1	2	yes	—
101	C₂×D₃₅₆	C₂×Q₈	1	2	yes	—
102	C₂×C₂²⋊D₈₉	—	2	2	no	—
103	C₂²×Dic₈₉	—	2	1	no	—
104	C₂×C₄×D₈₉	—	2	2	no	—
105	Dic₃₅₆	C₄⋊C₄	1	2	yes	—
106	Dic₃₅₆	C₄⋊C₄	1	2	yes	—
107	#13083563	C₄⋊C₄	1	4	yes	—
108	#13083563	C₄⋊C₄	1	4	yes	—
109	D₇₁₂	C₄⋊C₄	1	2	yes	—
110	D₇₁₂	C₄⋊C₄	1	2	yes	—
111	#13083558	—	2	1	no	—
112	#13083558	—	2	2	no	—
113	#13083558	—	2	2	no	—
114	#13083558	C₂×D₄	1	2	yes	—
115	#13083557	—	2	4	no	—
116	#13083557	—	2	4	no	—
117	#13083557	—	4	4	no	—
118	#13083557	—	4	4	no	—
119	C₂×#10623494	—	2	1	no	—
120	C₂×#10623494	—	2	2	no	—
121	C₂×#10623494	C₂²×C₄	1	2	yes	—
122	C₂×#10623494	C₂×D₄	1	2	yes	—
123	C₂×#10623494	#41	1	1	yes	characteristic
124	C₂×#10623494	#41	1	1	yes	characteristic
125	C₈×D₈₉	—	2	4	no	—
126	C₈×D₈₉	—	2	4	no	—
127	C₈×D₈₉	—	4	4	no	—
128	C₈×D₈₉	—	4	4	no	—
129	C₄⋊Dic₈₉	—	2	1	no	—
130	C₂²⋊Dic₈₉	—	2	1	no	—
131	C₄×Dic₈₉	—	2	1	no	—
132	C₄×Dic₈₉	—	2	1	no	—
133	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
135	M₁₆×C₈₉	—	2	2	no	—
136	M₁₆×C₈₉	—	2	1	no	—
137	M₁₆×C₈₉	—	2	2	no	—
138	M₁₆×C₈₉	—	2	2	no	—
139	M₁₆×C₈₉	C₂²×C₄	1	4	yes	—
140	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₈₉	—	2	2	no	—
142	C₂×C₈×C₈₉	—	2	1	no	—
143	C₂×C₈×C₈₉	—	2	2	no	—
144	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
149	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
150	C₄⋊C₄×C₈₉	—	2	1	no	—
151	C₂²⋊C₄×C₈₉	—	2	1	no	—
152	C₄²×C₈₉	—	2	1	no	—
153	C₂³×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂²×D₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂²×Q₈	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	C₂²⋊D₈₉	—	2	4	no	—
163	C₂²⋊D₈₉	—	2	4	no	—
164	C₂×Dic₈₉	—	2	2	no	—
165	C₂×Dic₈₉	—	2	2	no	—
166	C₂×Dic₈₉	—	4	1	no	—
167	C₂×Dic₈₉	—	8	1	no	—
168	D₃₅₆	—	2	2	no	—
169	D₃₅₆	C₂×C₄⋊C₄	1	4	yes	—
170	C₄×D₈₉	—	2	4	no	—
171	C₄×D₈₉	—	2	4	no	—
172	Dic₁₇₈	—	2	2	no	—
173	Dic₁₇₈	C₂×C₄⋊C₄	1	4	yes	—
174	C₈×C₈₉	—	2	4	no	—
175	C₈×C₈₉	—	2	2	no	—
176	C₈×C₈₉	—	4	2	no	—
177	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
181	#10623494	—	2	2	no	—
182	#10623494	—	4	2	no	—
183	#10623494	C₄×D₄	1	2	yes	—
184	#10623494	C₄×D₄	1	2	yes	—
185	C₂²×C₈₉	—	4	1	no	—
186	C₂²×C₈₉	#520	1	1	yes	characteristic
187	C₂²×C₈₉	#547	1	1	yes	characteristic
188	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
189	C₂×D₈₉	—	2	4	no	—
190	C₂×D₈₉	—	8	2	no	—
191	C₄×C₈₉	—	8	1	no	—
192	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
194	C₄×C₈₉	D₄²	1	1	yes	characteristic
195	C₄×C₈₉	#554	1	1	yes	characteristic
196	Dic₈₉	—	2	4	no	—
197	Dic₈₉	—	4	2	no	—
198	Dic₈₉	—	8	1	no	—
199	#22695	—	89	1	no	maximal, Sylow
200	C₂×C₈₉	—	2	1	no	—
201	C₂×C₈₉	—	2	1	no	—
202	C₂×C₈₉	—	2	1	no	—
203	C₂×C₈₉	D₄×C₄⋊C₄	1	1	yes	characteristic
204	D₈₉	—	4	4	no	—
205	#2957	—	89	1	no	—
206	#2957	—	89	1	no	—
207	C₂×#444	—	89	1	no	—
208	#2916	—	89	2	no	—
209	#1953	—	89	1	no	—
210	#1953	D₈₉	1	1	yes	characteristic, complemented
211	#1861	—	89	2	no	—
212	#1861	—	89	2	no	—
213	#1849	—	89	1	no	—
214	#1814	—	89	2	no	—
215	#1773	—	89	1	no	—
216	C₈₉	#22695	1	1	yes	characteristic, complemented, Sylow
217	C₂×#135	—	89	1	no	—
218	#568	—	89	2	no	—
219	#568	—	89	2	no	—
220	C₂×#131	—	89	2	no	—
221	#444	—	89	4	no	—
222	#444	—	89	4	no	—
223	#444	—	178	2	no	—
224	#444	—	178	2	no	—
225	#443	—	178	2	no	—
226	#443	—	178	2	no	—
227	#430	—	89	2	no	—
228	#430	—	178	4	no	—
229	#430	C₂×D₈₉	1	2	yes	—
230	C₂×#108	—	89	1	no	—
231	C₂×#108	—	89	2	no	—
232	C₂×#108	—	89	1	no	—
233	C₂×#108	—	178	4	no	—
234	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
235	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
236	#427	—	89	1	no	—
237	#423	—	89	2	no	—
238	#421	—	89	2	no	—
239	#421	—	89	2	no	—
240	C₂×#104	—	89	2	no	—
241	#405	—	89	1	no	—
242	#405	C₂×D₈₉	1	1	yes	characteristic
243	C₂×#97	—	89	1	no	—
244	#344	—	89	2	no	—
245	#344	C₂×D₈₉	1	2	yes	—
246	#339	—	89	1	no	—
247	#339	—	89	1	no	—
248	C₂×#41	—	89	2	no	—
249	#135	—	89	2	no	—
250	#135	—	89	2	no	—
251	#135	—	89	2	no	—
252	#135	—	89	2	no	—
253	C₂×Q₁₆	—	89	1	no	—
254	C₂×SD₁₆	—	89	2	no	—
255	C₂×D₈	—	89	1	no	—
256	#131	—	89	4	no	—
257	#131	—	89	4	no	—
258	#131	—	356	2	no	—
259	#131	—	356	2	no	—
260	#131	—	178	4	no	—
261	#131	—	178	2	no	—
262	#131	—	178	2	no	—
263	#131	—	178	4	no	—
264	C₂×M₁₆	—	89	1	no	—
265	C₂×M₁₆	—	89	2	no	—
266	C₂×M₁₆	—	178	2	no	—
267	C₂×M₁₆	—	178	4	no	—
268	C₂×M₁₆	—	178	4	no	—
269	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
270	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
271	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
272	C₂²×C₈	—	89	1	no	—
273	C₂²×C₈	—	178	2	no	—
274	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	#117	—	89	1	no	—
276	#117	C₂²×D₈₉	1	1	yes	characteristic
277	C₂×C₄²	—	89	1	no	—
278	#108	—	2	4	no	—
279	#108	—	2	1	no	—
280	#108	—	2	1	no	—
281	#108	—	89	4	no	—
282	#108	—	178	4	no	—
283	#108	—	178	2	no	—
284	#108	—	178	2	no	—
285	#108	—	178	1	no	—
286	#108	—	178	1	no	—
287	#108	C₄×D₈₉	1	2	yes	—
288	#108	C₄×D₈₉	1	2	yes	—
289	#107	—	178	1	no	—
290	#106	—	178	1	no	—
291	C₄⋊C₈	—	356	2	no	—
292	#104	—	89	8	no	—
293	#104	—	178	4	no	—
294	#104	—	178	4	no	—
295	#97	—	2	1	no	—
296	#97	—	89	2	no	—
297	#97	—	89	2	no	—
298	#97	—	178	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	178	1	no	—
301	#41	—	89	4	no	—
302	#41	—	89	4	no	—
303	#41	—	178	2	no	—
304	#41	—	178	2	no	—
305	C₂×Q₈	—	89	2	no	—
306	C₂×D₄	—	89	2	no	—
307	C₂×D₄	—	178	2	no	—
308	C₂²×C₄	—	178	2	no	—
309	C₂²×C₄	—	178	1	no	—
310	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
311	Q₁₆	—	89	2	no	—
312	Q₁₆	—	89	2	no	—
313	SD₁₆	—	89	4	no	—
314	SD₁₆	—	89	4	no	—
315	D₈	—	89	2	no	—
316	D₈	—	89	2	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	1	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	89	2	no	—
322	M₁₆	—	356	4	no	—
323	M₁₆	—	356	4	no	—
324	M₁₆	—	178	4	no	—
325	M₁₆	—	178	4	no	—
326	M₁₆	—	178	2	no	—
327	M₁₆	—	178	2	no	—
328	M₁₆	—	178	1	no	—
329	M₁₆	C₂×C₄×D₈₉	1	4	yes	—
330	M₁₆	D₄×D₈₉	1	2	yes	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	89	2	no	—
335	C₂×C₈	—	89	2	no	—
336	C₂×C₈	—	89	1	no	—
337	C₂×C₈	—	89	1	no	—
338	C₂×C₈	—	356	4	no	—
339	C₂×C₈	—	356	4	no	—
340	C₂×C₈	—	178	4	no	—
341	C₂×C₈	—	178	4	no	—
342	C₂×C₈	—	178	2	no	—
343	C₂×C₈	—	178	1	no	—
344	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
346	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
347	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
348	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
349	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	178	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	178	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	89	2	no	—
356	C₄²	—	178	1	no	—
357	C₄²	—	178	1	no	—
358	C₂³	—	178	2	no	—
359	C₂³	#15668060	1	1	yes	characteristic
360	Q₈	—	89	4	no	—
361	Q₈	—	178	2	no	—
362	D₄	—	89	4	no	—
363	D₄	—	178	4	no	—
364	D₄	—	178	4	no	—
365	D₄	—	178	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	356	1	no	—
368	C₂×C₄	—	178	2	no	—
369	C₂×C₄	—	178	2	no	—
370	C₂×C₄	—	178	4	no	—
371	C₂×C₄	—	178	4	no	—
372	C₂×C₄	—	712	1	no	—
373	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
375	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
376	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
377	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
378	C₂×C₄	#15668058	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	2	4	no	—
381	C₈	—	4	2	no	—
382	C₈	—	89	2	no	—
383	C₈	—	89	2	no	—
384	C₈	—	356	2	no	—
385	C₈	—	178	2	no	—
386	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
387	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
388	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
389	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
390	C₂²	—	4	1	no	—
391	C₂²	—	178	4	no	—
392	C₂²	—	712	2	no	—
393	C₂²	#19152080	1	1	yes	characteristic
394	C₂²	#19152250	1	1	yes	characteristic
395	C₂²	#19152263	1	1	yes	characteristic
396	C₄	—	8	1	no	—
397	C₄	—	356	2	no	—
398	C₄	—	178	4	no	—
399	C₄	—	712	1	no	—
400	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
401	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
402	C₄	#19152232	1	1	yes	characteristic
403	C₄	#19152304	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	356	4	no	—
408	C₂	#26023153	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19152232
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1953	C₂
socle	C₂×C₈₉	D₄×C₄⋊C₄
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1953	D₈₉
hypocenter	C₈₉	#22695
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1953

Complements

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

N	H ≅ G/N	type
4. #26028862	C₂	semidirect
5. #26021043	C₂	semidirect
6. #26020495	C₂	semidirect
7. #26020489	C₂	semidirect
8. #26020425	C₂	semidirect
9. #26020305	C₂	semidirect
10. #26020137	C₂	semidirect
11. C₈₉×#1953	C₂	semidirect
18. #19151719	C₄	semidirect
19. #19151719	C₄	semidirect
210. #1953	D₈₉	semidirect
216. C₈₉	#22695	semidirect

Maximal subgroups

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

subgroup	index
#26029170	2
#26029168	2
C₂×#19151719	2
#26028862	2
#26021043	2
#26020495	2
#26020489	2
#26020425	2
#26020305	2
#26020137	2
C₈₉×#1953	2
#22695	89