Subgroups — #36711253

Order 22784.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032601	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022434	C₂	1	2	yes	maximal, complemented
5	#26022389	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022337	C₂	1	2	yes	maximal, complemented
7	#26022287	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022162	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₈₉	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₈₉	C₂²	1	2	yes	—
13	#19152253	—	2	1	no	—
14	D₄×D₃₅₆	—	2	1	no	—
15	C₂×#15667953	C₂²	1	2	yes	complemented
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	complemented
17	#19152003	C₂²	1	1	yes	characteristic, complemented
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151910	—	2	2	no	—
20	#19151878	—	2	1	no	—
21	#19151835	C₂²	1	2	yes	—
22	#19151831	C₂²	1	2	yes	complemented
23	#19151793	—	2	1	no	—
24	#19151733	—	2	2	no	—
25	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
26	#107×D₈₉	C₂²	1	2	yes	complemented
27	#19151423	C₂²	1	2	yes	complemented
28	C₄⋊#13083549	C₂²	1	1	yes	characteristic, complemented
29	#19151400	C₂²	1	2	yes	complemented
30	#19151400	C₂²	1	2	yes	complemented
31	#19151399	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	complemented
33	#19151377	C₂²	1	2	yes	complemented
34	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	#366×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#357×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	C₂×#13083574	—	4	1	no	—
42	C₂²×D₃₅₆	—	4	1	no	—
43	C₂×D₄×D₈₉	—	2	2	no	—
44	C₂×D₇₁₂	—	2	1	no	—
45	C₂×D₇₁₂	C₂³	1	2	yes	—
46	C₂×#13083559	—	2	2	no	—
47	D₈×D₈₉	—	2	4	no	—
48	D₈×D₈₉	—	4	2	no	—
49	C₂×C₈×D₈₉	C₂³	1	2	yes	—
50	#15668012	—	2	1	no	—
51	#15668009	—	2	1	no	—
52	#15667994	—	4	2	no	—
53	#15667991	—	4	1	no	—
54	#15667987	—	2	2	no	—
55	#15667986	—	4	1	no	—
56	#15667983	—	2	2	no	—
57	#15667982	—	4	2	no	—
58	C₄×D₃₅₆	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₈₉	—	2	2	no	—
60	#15667953	—	4	2	no	—
61	C₂×#13083549	C₂³	1	2	yes	—
62	C₁₆×D₈₉	—	4	2	no	—
63	#15667946	—	2	1	no	—
64	#15667946	C₂³	1	2	yes	—
65	#15667945	—	2	2	no	—
66	C₄⋊#10623494	D₄	1	1	yes	characteristic
67	#15667933	—	2	2	no	—
68	#15667932	—	2	2	no	—
69	#15667924	D₄	1	2	yes	—
70	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	D₁₆×C₈₉	—	2	2	no	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	C₂³	1	1	yes	characteristic
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#102×C₈₉	—	2	1	no	—
80	#102×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×D₈₉	—	4	2	no	—
83	#13083574	—	8	4	no	—
84	C₂×Dic₁₇₈	—	4	1	no	—
85	C₂×D₃₅₆	—	2	2	no	—
86	C₂×D₃₅₆	—	4	2	no	—
87	C₂×D₃₅₆	—	4	1	no	—
88	C₂×D₃₅₆	—	4	1	no	—
89	C₂×D₃₅₆	—	8	4	no	—
90	C₂×D₃₅₆	D₈	1	1	yes	characteristic
91	C₂×C₂²⋊D₈₉	—	4	2	no	—
92	C₂×C₂²⋊D₈₉	—	4	2	no	—
93	D₄×D₈₉	—	4	4	no	—
94	D₄×D₈₉	—	8	2	no	—
95	C₂×C₄×D₈₉	—	4	2	no	—
96	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
97	D₇₁₂	—	2	2	no	—
98	D₇₁₂	—	2	2	no	—
99	D₇₁₂	—	8	1	no	—
100	#13083559	—	4	4	no	—
101	C₂×#10623494	C₂×D₄	1	2	yes	—
102	C₈×D₈₉	—	2	2	no	—
103	C₈×D₈₉	—	2	2	no	—
104	C₄⋊Dic₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	4	1	no	—
106	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
107	#13083553	—	4	2	no	—
108	C₂²⋊Dic₈₉	—	4	2	no	—
109	#13083551	—	4	2	no	—
110	#13083551	D₈	1	2	yes	—
111	#13083549	—	2	2	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂×D₄×C₈₉	—	2	1	no	—
114	C₂²×C₄×C₈₉	—	4	1	no	—
115	C₂²×C₄×C₈₉	—	4	1	no	—
116	D₈×C₈₉	—	2	1	no	—
117	D₈×C₈₉	—	2	1	no	—
118	D₈×C₈₉	—	2	1	no	—
119	D₈×C₈₉	#41	1	2	yes	complemented
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₄⋊C₄×C₈₉	—	2	1	no	—
124	C₂²⋊C₄×C₈₉	—	4	1	no	—
125	C₂²⋊C₄×C₈₉	—	4	1	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₈₉	—	2	1	no	—
128	C₁₆×C₈₉	—	2	1	no	—
129	C₂³×C₈₉	—	4	1	no	—
130	C₂³×C₈₉	—	4	1	no	—
131	C₂²×D₈₉	—	4	2	no	—
132	C₂²×D₈₉	—	8	4	no	—
133	C₂²×D₈₉	—	8	4	no	—
134	C₂²×D₈₉	—	16	2	no	—
135	C₂²×D₈₉	C₂×D₈	1	2	yes	—
136	D₄×C₈₉	—	2	2	no	—
137	D₄×C₈₉	—	4	1	no	—
138	D₄×C₈₉	—	4	1	no	—
139	D₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	—	4	1	no	—
141	C₂×C₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	—	8	2	no	—
144	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₈₉	—	8	4	no	—
148	C₂²⋊D₈₉	—	8	4	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
151	D₃₅₆	—	4	2	no	—
152	D₃₅₆	—	4	2	no	—
153	D₃₅₆	—	8	4	no	—
154	D₃₅₆	—	8	2	no	—
155	D₃₅₆	—	16	1	no	—
156	D₃₅₆	D₁₆	1	4	yes	complemented
157	C₄×D₈₉	—	2	2	no	—
158	C₄×D₈₉	—	2	2	no	—
159	C₄×D₈₉	—	8	4	no	—
160	Dic₁₇₈	—	8	2	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
164	#10623494	—	2	2	no	—
165	C₂²×C₈₉	—	4	2	no	—
166	C₂²×C₈₉	—	8	1	no	—
167	C₂²×C₈₉	—	8	1	no	—
168	C₂²×C₈₉	—	8	1	no	—
169	C₂²×C₈₉	—	8	1	no	—
170	C₂²×C₈₉	—	8	1	no	—
171	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₈₉	—	2	2	no	—
173	C₂×D₈₉	—	2	2	no	—
174	C₂×D₈₉	—	8	2	no	—
175	C₂×D₈₉	—	8	2	no	—
176	C₂×D₈₉	—	8	2	no	—
177	C₂×D₈₉	—	16	8	no	—
178	C₂×D₈₉	C₂×D₁₆	1	4	yes	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	8	1	no	—
181	C₄×C₈₉	—	8	1	no	—
182	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
183	C₄×C₈₉	#546	1	1	yes	characteristic
184	C₄×C₈₉	#577	1	1	yes	characteristic
185	Dic₈₉	—	2	2	no	—
186	Dic₈₉	—	8	2	no	—
187	#33861	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	8	2	no	—
189	C₂×C₈₉	—	16	1	no	—
190	C₂×C₈₉	#3294	1	1	yes	characteristic
191	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
192	C₂×C₈₉	#3416	1	1	yes	characteristic
193	D₈₉	—	2	4	no	—
194	D₈₉	—	16	2	no	—
195	#3334	—	89	1	no	—
196	D₄×D₈	—	89	1	no	—
197	#2240	—	89	1	no	—
198	#2232	—	89	2	no	—
199	#2224	—	89	1	no	—
200	#2215	—	89	2	no	—
201	#2206	—	89	1	no	—
202	#2184	—	89	2	no	—
203	C₄×D₁₆	D₈₉	1	1	yes	characteristic, complemented
204	D₄×C₁₆	—	89	1	no	—
205	C₂×#357	—	89	2	no	—
206	C₈₉	#33861	1	1	yes	characteristic, complemented, Sylow
207	C₂²×D₈	—	89	2	no	—
208	#547	—	178	1	no	—
209	D₄²	—	178	1	no	—
210	C₂×D₁₆	—	89	2	no	—
211	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
212	C₂²×C₁₆	—	89	2	no	—
213	#500	—	89	1	no	—
214	#493	—	89	1	no	—
215	#480	—	178	2	no	—
216	#474	—	178	1	no	—
217	#466	—	89	2	no	—
218	#466	—	89	2	no	—
219	#459	—	178	1	no	—
220	#447	—	178	2	no	—
221	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
222	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic, complemented
223	C₈×D₄	—	89	1	no	—
224	C₂×#107	—	89	2	no	—
225	#366	—	89	2	no	—
226	#366	C₂×D₈₉	1	1	yes	characteristic, complemented
227	C₄⋊C₁₆	—	89	1	no	—
228	#357	—	89	2	no	—
229	#357	—	89	2	no	—
230	#357	—	89	2	no	—
231	#357	—	89	2	no	—
232	#357	C₂×D₈₉	1	1	yes	characteristic
233	#357	C₂×D₈₉	1	1	yes	characteristic, complemented
234	C₂²⋊C₁₆	—	89	2	no	—
235	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
236	C₂×#41	—	356	1	no	—
237	C₂²×D₄	—	356	1	no	—
238	C₂²×D₄	—	178	2	no	—
239	C₂×D₈	—	89	2	no	—
240	C₂×D₈	—	356	2	no	—
241	C₂×D₈	—	178	2	no	—
242	C₂×D₈	—	178	4	no	—
243	C₂×D₈	—	178	1	no	—
244	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂²×C₈	—	89	2	no	—
247	C₄⋊Q₈	—	178	1	no	—
248	#127	—	178	1	no	—
249	#123	—	356	2	no	—
250	C₂²⋊Q₈	—	356	1	no	—
251	#121	—	356	1	no	—
252	#121	—	178	2	no	—
253	#121	—	178	2	no	—
254	#120	—	356	2	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	89	1	no	—
258	C₂×C₄⋊C₄	—	178	2	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	356	2	no	—
261	C₂×C₁₆	—	89	2	no	—
262	C₂×C₁₆	—	356	2	no	—
263	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
265	#107	—	89	2	no	—
266	#107	—	178	2	no	—
267	#107	—	178	1	no	—
268	#107	C₂²×D₈₉	1	1	yes	characteristic
269	#107	C₂²×D₈₉	1	1	yes	characteristic
270	C₄⋊C₈	—	89	1	no	—
271	#102	—	2	1	no	—
272	#102	—	2	1	no	—
273	#102	—	178	2	no	—
274	#102	—	178	2	no	—
275	C₂²⋊C₈	—	89	2	no	—
276	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
277	C₂⁴	—	356	2	no	—
278	#41	—	712	4	no	—
279	C₂×Q₈	—	356	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	89	1	no	—
283	C₂×D₄	—	356	2	no	—
284	C₂×D₄	—	356	2	no	—
285	C₂×D₄	—	356	2	no	—
286	C₂×D₄	—	356	4	no	—
287	C₂×D₄	—	356	1	no	—
288	C₂×D₄	—	356	1	no	—
289	C₂×D₄	—	178	2	no	—
290	C₂×D₄	—	712	4	no	—
291	C₂×D₄	—	712	2	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	89	2	no	—
295	C₂²×C₄	—	356	2	no	—
296	D₈	—	2	1	no	—
297	D₈	—	2	1	no	—
298	D₈	—	2	1	no	—
299	D₈	—	356	4	no	—
300	D₈	—	178	2	no	—
301	D₈	—	178	2	no	—
302	D₈	—	712	1	no	—
303	D₈	#13083574	1	2	yes	complemented
304	C₂×C₈	—	89	2	no	—
305	C₂×C₈	—	178	2	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	89	1	no	—
312	C₄⋊C₄	—	356	2	no	—
313	C₄⋊C₄	—	356	1	no	—
314	C₄⋊C₄	—	178	2	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	89	2	no	—
318	C₂²⋊C₄	—	356	2	no	—
319	C₂²⋊C₄	—	356	2	no	—
320	C₄²	D₄×D₈₉	1	1	yes	characteristic
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	178	2	no	—
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	89	2	no	—
327	C₂³	—	356	2	no	—
328	C₂³	—	1424	2	no	—
329	C₂³	—	712	4	no	—
330	C₂³	—	712	4	no	—
331	Q₈	—	712	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	89	4	no	—
337	D₄	—	356	2	no	—
338	D₄	—	356	2	no	—
339	D₄	—	1424	1	no	—
340	D₄	—	712	4	no	—
341	D₄	—	712	4	no	—
342	D₄	—	712	4	no	—
343	D₄	—	712	2	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	—	89	2	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	178	2	no	—
351	C₂×C₄	—	178	2	no	—
352	C₂×C₄	—	712	4	no	—
353	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
354	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
355	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	178	2	no	—
358	C₈	C₂×#13083574	1	1	yes	characteristic
359	C₈	#15668058	1	1	yes	characteristic
360	C₂²	—	4	2	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	89	4	no	—
367	C₂²	—	1424	8	no	—
368	C₂²	—	178	2	no	—
369	C₂²	—	178	2	no	—
370	C₂²	—	712	2	no	—
371	C₂²	—	712	2	no	—
372	C₂²	—	712	2	no	—
373	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	1	no	—
377	C₄	—	178	2	no	—
378	C₄	—	712	2	no	—
379	C₄	D₁₆×D₈₉	1	2	yes	—
380	C₄	#19152244	1	1	yes	characteristic
381	C₄	#19152405	1	1	yes	characteristic
382	C₂	—	8	2	no	—
383	C₂	—	16	1	no	—
384	C₂	—	1424	2	no	—
385	C₂	—	178	4	no	—
386	C₂	#26032067	1	1	yes	characteristic
387	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic
388	C₂	#26033395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×D₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₄×D₁₆	D₈₉
hypocenter	C₈₉	#33861
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #26032601	C₂	semidirect
2. D₈×D₃₅₆	C₂	semidirect
3. #26022476	C₂	semidirect
4. #26022434	C₂	semidirect
5. #26022389	C₂	semidirect
6. #26022337	C₂	semidirect
7. #26022287	C₂	semidirect
8. #26022162	C₂	semidirect
9. C₁₆×D₃₅₆	C₂	semidirect
10. #357×D₈₉	C₂	semidirect
11. C₄×D₁₆×C₈₉	C₂	semidirect
15. C₂×#15667953	C₂²	semidirect
16. C₂×C₁₆×D₈₉	C₂²	semidirect
17. #19152003	C₂²	semidirect
22. #19151831	C₂²	semidirect
26. #107×D₈₉	C₂²	semidirect
27. #19151423	C₂²	semidirect
28. C₄⋊#13083549	C₂²	semidirect
29. #19151400	C₂²	semidirect
30. #19151400	C₂²	semidirect
32. #19151398	C₂²	semidirect
33. #19151377	C₂²	semidirect
36. C₄×D₈×C₈₉	C₂²	semidirect
37. #366×C₈₉	C₂²	semidirect
39. #357×C₈₉	C₂²	semidirect
40. C₄×C₁₆×C₈₉	C₂²	semidirect
119. D₈×C₈₉	#41	semidirect
156. D₃₅₆	D₁₆	semidirect
203. C₄×D₁₆	D₈₉	semidirect
206. C₈₉	#33861	semidirect
222. C₄×D₈	C₂×D₈₉	semidirect
226. #366	C₂×D₈₉	semidirect
233. #357	C₂×D₈₉	semidirect
235. C₄×C₁₆	C₂×D₈₉	semidirect
303. D₈	#13083574	semidirect

Maximal subgroups

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

subgroup	index
#26032601	2
D₈×D₃₅₆	2
#26022476	2
#26022434	2
#26022389	2
#26022337	2
#26022287	2
#26022162	2
C₁₆×D₃₅₆	2
#357×D₈₉	2
C₄×D₁₆×C₈₉	2
#33861	89