Subgroups — #36711669

Order 22784.

608 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	#26032683	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022475	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022440	C₂	1	2	yes	maximal, complemented
4	#26022389	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022363	C₂	1	2	yes	maximal, complemented
6	#26022289	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022181	C₂	1	2	yes	maximal, complemented
8	C₄×#15667956	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021968	C₂	1	2	yes	maximal, complemented
11	D₄×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19152318	—	2	1	no	—
13	C₂×#15667956	C₂²	1	1	yes	characteristic
14	C₂×#15667956	C₂²	1	2	yes	complemented
15	C₂×#15667952	C₂²	1	2	yes	complemented
16	#19152003	C₂²	1	1	yes	characteristic, complemented
17	#19151940	—	2	2	no	—
18	#19151884	—	2	1	no	—
19	#19151839	C₂²	1	2	yes	complemented
20	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic, complemented
21	#19151636	C₂²	1	2	yes	complemented
22	#19151425	C₂²	1	1	yes	characteristic, complemented
23	#19151423	C₂²	1	2	yes	complemented
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic, complemented
25	#19151401	C₂²	1	1	yes	characteristic, complemented
26	#19151399	C₂²	1	2	yes	—
27	#19151399	C₂²	1	1	yes	characteristic
28	#19151399	C₂²	1	2	yes	complemented
29	#19151399	C₂²	1	2	yes	complemented
30	#19151399	C₂²	1	2	yes	complemented
31	C₂²⋊#13083549	C₂²	1	2	yes	complemented
32	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
33	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
34	D₄²×C₈₉	—	2	1	no	—
35	#493×C₈₉	C₂²	1	1	yes	characteristic
36	#466×C₈₉	C₂²	1	2	yes	—
37	#459×C₈₉	—	2	1	no	—
38	#447×C₈₉	—	2	2	no	—
39	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668005	—	4	2	no	—
44	#15667999	—	4	1	no	—
45	#15667991	—	2	2	no	—
46	C₄×Dic₁₇₈	—	2	1	no	—
47	#15667977	—	4	1	no	—
48	#15667971	—	2	2	no	—
49	#15667956	—	2	2	no	—
50	#15667956	—	4	2	no	—
51	#15667952	—	4	2	no	—
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	C₂×#13083549	C₂³	1	2	yes	—
55	#15667946	—	2	1	no	—
56	#15667946	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	2	yes	—
58	#15667944	—	2	2	no	—
59	#15667937	—	2	2	no	—
60	#15667937	—	2	1	no	—
61	C₂²×D₄×C₈₉	—	2	2	no	—
62	C₂²×D₄×C₈₉	—	4	1	no	—
63	C₂×D₈×C₈₉	—	2	4	no	—
64	C₂×D₈×C₈₉	—	2	2	no	—
65	C₂×D₈×C₈₉	—	2	1	no	—
66	C₂×D₈×C₈₉	—	4	2	no	—
67	C₂×D₈×C₈₉	C₂³	1	2	yes	—
68	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
70	#127×C₈₉	—	2	1	no	—
71	#121×C₈₉	—	2	2	no	—
72	#121×C₈₉	—	4	1	no	—
73	#120×C₈₉	—	4	2	no	—
74	C₄×D₄×C₈₉	—	2	1	no	—
75	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
78	#102×C₈₉	—	2	1	no	—
79	#102×C₈₉	—	2	2	no	—
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	1	no	—
83	C₂×Dic₁₇₈	—	4	1	no	—
84	Dic₃₅₆	—	2	1	no	—
85	Dic₃₅₆	—	2	1	no	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	4	1	no	—
89	#13083553	—	4	2	no	—
90	#13083553	—	4	1	no	—
91	C₂²⋊Dic₈₉	—	4	2	no	—
92	C₂²⋊Dic₈₉	—	4	2	no	—
93	C₄×Dic₈₉	—	4	2	no	—
94	C₄×Dic₈₉	—	4	1	no	—
95	#13083549	—	2	1	no	—
96	#13083549	—	2	1	no	—
97	#13083549	—	2	2	no	—
98	C₂⁴×C₈₉	—	4	2	no	—
99	C₂×D₄×C₈₉	—	2	1	no	—
100	C₂×D₄×C₈₉	—	2	2	no	—
101	C₂×D₄×C₈₉	—	4	4	no	—
102	C₂×D₄×C₈₉	—	4	2	no	—
103	C₂×D₄×C₈₉	—	4	2	no	—
104	C₂×D₄×C₈₉	—	4	1	no	—
105	C₂×D₄×C₈₉	—	4	1	no	—
106	C₂×D₄×C₈₉	—	8	2	no	—
107	C₂×D₄×C₈₉	—	8	4	no	—
108	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	—	4	1	no	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
111	D₈×C₈₉	—	2	2	no	—
112	D₈×C₈₉	—	2	1	no	—
113	D₈×C₈₉	—	2	2	no	—
114	D₈×C₈₉	—	4	4	no	—
115	D₈×C₈₉	—	8	1	no	—
116	D₈×C₈₉	#41	1	2	yes	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
124	C₂²⋊C₄×C₈₉	—	4	1	no	—
125	C₂²⋊C₄×C₈₉	—	4	2	no	—
126	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₈₉	—	4	1	no	—
129	C₂³×C₈₉	—	4	2	no	—
130	C₂³×C₈₉	—	8	4	no	—
131	C₂³×C₈₉	—	8	4	no	—
132	C₂³×C₈₉	—	16	2	no	—
133	C₂³×C₈₉	C₂×D₈	1	2	yes	—
134	D₄×C₈₉	—	2	2	no	—
135	D₄×C₈₉	—	2	2	no	—
136	D₄×C₈₉	—	4	2	no	—
137	D₄×C₈₉	—	4	1	no	—
138	D₄×C₈₉	—	4	2	no	—
139	D₄×C₈₉	—	8	4	no	—
140	D₄×C₈₉	—	8	4	no	—
141	D₄×C₈₉	—	8	2	no	—
142	D₄×C₈₉	—	16	1	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	8	2	no	—
147	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×Dic₈₉	—	4	1	no	—
152	C₂×Dic₈₉	—	4	1	no	—
153	C₂×Dic₈₉	—	4	2	no	—
154	Dic₁₇₈	—	4	1	no	—
155	Dic₁₇₈	—	4	1	no	—
156	Dic₁₇₈	—	8	2	no	—
157	C₈×C₈₉	—	2	2	no	—
158	C₈×C₈₉	—	2	1	no	—
159	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₈₉	—	2	2	no	—
162	C₂²×C₈₉	—	2	2	no	—
163	C₂²×C₈₉	—	2	2	no	—
164	C₂²×C₈₉	—	4	2	no	—
165	C₂²×C₈₉	—	8	1	no	—
166	C₂²×C₈₉	—	8	1	no	—
167	C₂²×C₈₉	—	8	2	no	—
168	C₂²×C₈₉	—	8	2	no	—
169	C₂²×C₈₉	—	8	2	no	—
170	C₂²×C₈₉	—	16	8	no	—
171	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	2	2	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	#509	1	2	yes	—
176	C₄×C₈₉	#546	1	1	yes	characteristic
177	C₄×C₈₉	#577	1	1	yes	characteristic
178	Dic₈₉	—	8	2	no	—
179	Dic₈₉	—	8	1	no	—
180	Dic₈₉	—	8	1	no	—
181	#33891	—	89	1	no	maximal, Sylow
182	C₂×C₈₉	—	4	2	no	—
183	C₂×C₈₉	—	8	2	no	—
184	C₂×C₈₉	—	16	2	no	—
185	C₂×C₈₉	#3294	1	1	yes	characteristic
186	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
187	C₂×C₈₉	#3416	1	1	yes	characteristic
188	#3342	—	89	1	no	—
189	D₄×D₈	D₈₉	1	1	yes	characteristic
190	#2240	—	89	1	no	—
191	#2234	—	89	2	no	—
192	#2224	—	89	1	no	—
193	#2220	—	89	2	no	—
194	#2207	—	89	1	no	—
195	#2187	—	89	2	no	—
196	C₄×SD₃₂	—	89	1	no	—
197	#2169	—	89	1	no	—
198	#2140	—	89	2	no	—
199	C₈₉	#33891	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₈	C₂×D₈₉	1	2	yes	—
201	#556	—	178	1	no	—
202	D₄²	—	2	1	no	—
203	C₂×SD₃₂	—	89	2	no	—
204	C₂×SD₃₂	—	89	1	no	—
205	C₂×M₃₂	—	89	2	no	—
206	#500	—	89	1	no	—
207	#493	C₂×D₈₉	1	1	yes	characteristic
208	#485	—	178	2	no	—
209	#475	—	178	1	no	—
210	#467	—	89	2	no	—
211	#466	C₂×D₈₉	1	2	yes	—
212	#459	—	2	1	no	—
213	#447	—	2	2	no	—
214	C₄×Q₁₆	—	89	1	no	—
215	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
216	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
217	#427	—	89	2	no	—
218	#367	—	89	1	no	—
219	#366	—	89	2	no	—
220	C₄⋊C₁₆	—	89	1	no	—
221	#358	—	89	1	no	—
222	#357	—	89	1	no	—
223	#357	—	89	2	no	—
224	#357	—	89	2	no	—
225	#357	—	89	2	no	—
226	#357	—	89	2	no	—
227	C₂²⋊C₁₆	—	89	2	no	—
228	C₄×C₁₆	—	89	1	no	—
229	C₂²×D₄	—	2	2	no	—
230	C₂²×D₄	—	4	1	no	—
231	C₂×Q₁₆	—	89	1	no	—
232	C₂×D₈	—	2	4	no	—
233	C₂×D₈	—	2	2	no	—
234	C₂×D₈	—	2	1	no	—
235	C₂×D₈	—	4	2	no	—
236	C₂×D₈	C₂²×D₈₉	1	2	yes	—
237	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
239	C₄⋊Q₈	—	178	1	no	—
240	#127	—	2	1	no	—
241	#126	—	356	2	no	—
242	#124	—	356	1	no	—
243	C₂²⋊Q₈	—	178	2	no	—
244	#121	—	2	2	no	—
245	#121	—	4	1	no	—
246	#120	—	4	2	no	—
247	C₄×Q₈	—	356	1	no	—
248	C₄×Q₈	—	178	1	no	—
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
251	#117	—	178	2	no	—
252	SD₃₂	—	356	2	no	—
253	SD₃₂	—	178	2	no	—
254	M₃₂	—	356	2	no	—
255	C₂×C₁₆	—	89	1	no	—
256	C₂×C₁₆	—	89	1	no	—
257	C₂×C₁₆	—	89	2	no	—
258	#107	—	89	2	no	—
259	#107	—	89	1	no	—
260	#107	—	178	1	no	—
261	#107	C₂²×D₈₉	1	1	yes	characteristic
262	#106	—	178	2	no	—
263	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
264	#103	—	178	1	no	—
265	#103	—	178	2	no	—
266	#102	—	2	1	no	—
267	#102	—	2	2	no	—
268	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
269	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂⁴	—	4	2	no	—
271	C₂×Q₈	—	356	1	no	—
272	C₂×Q₈	—	178	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	4	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	8	2	no	—
281	C₂×D₄	—	8	4	no	—
282	C₂×D₄	#13083559	1	1	yes	characteristic
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
285	Q₁₆	—	178	1	no	—
286	Q₁₆	—	178	1	no	—
287	D₈	—	2	2	no	—
288	D₈	—	2	1	no	—
289	D₈	—	2	2	no	—
290	D₈	—	4	4	no	—
291	D₈	—	8	1	no	—
292	D₈	#13083573	1	2	yes	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
297	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	356	2	no	—
300	C₄⋊C₄	—	356	1	no	—
301	C₄⋊C₄	—	356	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₄⋊C₄	#13083559	1	1	yes	characteristic
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	356	2	no	—
308	C₂²⋊C₄	—	356	2	no	—
309	C₂²⋊C₄	#13083559	1	2	yes	—
310	C₄²	—	356	2	no	—
311	C₄²	—	356	1	no	—
312	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₁₆	—	178	1	no	—
314	C₁₆	—	178	1	no	—
315	C₁₆	—	178	2	no	—
316	C₂³	—	4	1	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	8	4	no	—
319	C₂³	—	8	4	no	—
320	C₂³	—	16	2	no	—
321	C₂³	C₂×#13083559	1	2	yes	—
322	Q₈	—	356	1	no	—
323	Q₈	—	356	1	no	—
324	Q₈	—	712	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	4	no	—
331	D₄	—	8	4	no	—
332	D₄	—	8	2	no	—
333	D₄	—	16	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	356	1	no	—
339	C₂×C₄	—	356	1	no	—
340	C₂×C₄	—	356	2	no	—
341	C₂×C₄	C₂×#13083559	1	2	yes	—
342	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
343	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
344	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	C₂×#13083573	1	1	yes	characteristic
348	C₈	#15668057	1	1	yes	characteristic
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	16	8	no	—
359	C₂²	C₂²×#13083559	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	712	1	no	—
364	C₄	—	712	2	no	—
365	C₄	—	712	1	no	—
366	C₄	#19152034	1	2	yes	—
367	C₄	#19152242	1	1	yes	characteristic
368	C₄	#19152409	1	1	yes	characteristic
369	C₂	—	4	2	no	—
370	C₂	—	8	2	no	—
371	C₂	—	16	2	no	—
372	C₂	#26032065	1	1	yes	characteristic
373	C₂	C₂×#19152034	1	1	yes	characteristic
374	C₂	#26033399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	D₄×D₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	D₄×D₈	D₈₉
hypocenter	C₈₉	#33891
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×D₈

Complements

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

N	H ≅ G/N	type
1. #26032683	C₂	semidirect
2. #26022475	C₂	semidirect
3. #26022440	C₂	semidirect
4. #26022389	C₂	semidirect
5. #26022363	C₂	semidirect
6. #26022289	C₂	semidirect
7. #26022181	C₂	semidirect
8. C₄×#15667956	C₂	semidirect
9. #26022084	C₂	semidirect
10. #26021968	C₂	semidirect
14. C₂×#15667956	C₂²	semidirect
15. C₂×#15667952	C₂²	semidirect
16. #19152003	C₂²	semidirect
19. #19151839	C₂²	semidirect
20. C₄×Dic₃₅₆	C₂²	semidirect
21. #19151636	C₂²	semidirect
22. #19151425	C₂²	semidirect
23. #19151423	C₂²	semidirect
24. C₄⋊#13083549	C₂²	semidirect
25. #19151401	C₂²	semidirect
28. #19151399	C₂²	semidirect
29. #19151399	C₂²	semidirect
30. #19151399	C₂²	semidirect
31. C₂²⋊#13083549	C₂²	semidirect
32. C₄×#13083549	C₂²	semidirect
199. C₈₉	#33891	semidirect

Maximal subgroups

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

subgroup	index
#26032683	2
#26022475	2
#26022440	2
#26022389	2
#26022363	2
#26022289	2
#26022181	2
C₄×#15667956	2
#26022084	2
#26021968	2
D₄×D₈×C₈₉	2
#33891	89