Subgroups — #36711663

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	D₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26022477	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022443	C₂	1	2	yes	maximal, complemented
4	#26022386	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022367	C₂	1	2	yes	maximal, complemented
6	#26022292	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022180	C₂	1	2	yes	maximal, complemented
8	C₄×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021967	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₁₂	C₂²	1	2	yes	—
13	D₄×D₃₅₆	—	2	1	no	—
14	C₂×#15667955	C₂²	1	1	yes	characteristic
15	C₂×#15667955	C₂²	1	2	yes	complemented
16	C₂×#15667952	C₂²	1	2	yes	complemented
17	#19151975	C₂²	1	1	yes	characteristic
18	#19151834	C₂²	1	2	yes	—
19	#19151794	—	2	1	no	—
20	#19151734	—	2	2	no	—
21	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
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	#19151402	C₂²	1	1	yes	characteristic, complemented
26	#19151398	C₂²	1	2	yes	—
27	#19151398	C₂²	1	1	yes	characteristic
28	#19151398	C₂²	1	2	yes	complemented
29	#19151398	C₂²	1	2	yes	complemented
30	#19151398	C₂²	1	2	yes	complemented
31	C₂²⋊#13083549	C₂²	1	2	yes	complemented
32	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
33	#556×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
35	#485×C₈₉	—	2	2	no	—
36	#475×C₈₉	—	2	1	no	—
37	#467×C₈₉	C₂²	1	2	yes	complemented
38	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	#427×C₈₉	C₂²	1	2	yes	complemented
41	C₂²×D₃₅₆	—	2	2	no	—
42	C₂×D₄×D₈₉	—	4	1	no	—
43	C₂×D₇₁₂	—	2	4	no	—
44	C₂×D₇₁₂	—	2	2	no	—
45	C₂×D₇₁₂	—	2	1	no	—
46	C₂×D₇₁₂	—	4	2	no	—
47	C₂×D₇₁₂	C₂³	1	2	yes	—
48	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
49	#15668009	—	2	1	no	—
50	#15667987	—	4	1	no	—
51	#15667986	—	2	2	no	—
52	#15667982	—	4	2	no	—
53	C₄×D₃₅₆	—	2	1	no	—
54	#15667955	—	2	2	no	—
55	#15667955	—	4	2	no	—
56	#15667952	—	4	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667934	—	2	1	no	—
62	#15667934	—	2	2	no	—
63	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	#126×C₈₉	—	4	2	no	—
67	#124×C₈₉	—	4	1	no	—
68	C₂²⋊Q₈×C₈₉	—	2	2	no	—
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	—	4	1	no	—
71	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
72	#117×C₈₉	—	2	2	no	—
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#106×C₈₉	—	2	2	no	—
77	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
78	#103×C₈₉	—	2	1	no	—
79	#103×C₈₉	—	2	2	no	—
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×D₈₉	—	4	2	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×D₃₅₆	—	2	2	no	—
85	C₂×D₃₅₆	—	4	4	no	—
86	C₂×D₃₅₆	—	4	2	no	—
87	C₂×D₃₅₆	—	4	1	no	—
88	C₂×D₃₅₆	—	4	1	no	—
89	C₂×D₃₅₆	—	8	2	no	—
90	C₂×C₂²⋊D₈₉	—	4	2	no	—
91	D₄×D₈₉	—	8	4	no	—
92	C₂×C₄×D₈₉	—	4	1	no	—
93	D₇₁₂	—	2	2	no	—
94	D₇₁₂	—	2	1	no	—
95	D₇₁₂	—	2	2	no	—
96	D₇₁₂	—	4	4	no	—
97	D₇₁₂	—	8	1	no	—
98	D₇₁₂	#41	1	2	yes	—
99	C₄⋊Dic₈₉	—	2	1	no	—
100	#13083551	—	4	1	no	—
101	#13083551	—	4	2	no	—
102	#13083549	—	2	1	no	—
103	#13083549	—	2	1	no	—
104	#13083549	—	2	2	no	—
105	C₂×Q₈×C₈₉	—	2	1	no	—
106	C₂×Q₈×C₈₉	—	4	1	no	—
107	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
109	Q₁₆×C₈₉	—	2	1	no	—
110	Q₁₆×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	4	1	no	—
119	C₄⋊C₄×C₈₉	—	4	2	no	—
120	C₄⋊C₄×C₈₉	—	4	1	no	—
121	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
122	C₂²⋊C₄×C₈₉	—	4	2	no	—
123	C₂²⋊C₄×C₈₉	—	4	2	no	—
124	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
125	C₄²×C₈₉	—	4	2	no	—
126	C₄²×C₈₉	—	4	1	no	—
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₈₉	C₂×D₈	1	2	yes	—
129	C₂²×D₈₉	—	4	1	no	—
130	C₂²×D₈₉	—	4	2	no	—
131	C₂²×D₈₉	—	8	4	no	—
132	C₂²×D₈₉	—	8	4	no	—
133	C₂²×D₈₉	—	16	2	no	—
134	Q₈×C₈₉	—	4	1	no	—
135	Q₈×C₈₉	—	4	1	no	—
136	Q₈×C₈₉	—	8	2	no	—
137	D₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	4	1	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
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	C₂×Dic₈₉	—	4	1	no	—
149	D₃₅₆	—	2	2	no	—
150	D₃₅₆	—	4	2	no	—
151	D₃₅₆	—	4	1	no	—
152	D₃₅₆	—	4	2	no	—
153	D₃₅₆	—	8	4	no	—
154	D₃₅₆	—	8	2	no	—
155	D₃₅₆	—	16	1	no	—
156	C₄×D₈₉	—	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	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₈₉	—	4	2	no	—
166	C₂×D₈₉	—	8	1	no	—
167	C₂×D₈₉	—	8	1	no	—
168	C₂×D₈₉	—	8	2	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	8	2	no	—
171	C₂×D₈₉	—	16	8	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	2	2	no	—
174	C₄×C₈₉	—	8	2	no	—
175	C₄×C₈₉	—	8	1	no	—
176	C₄×C₈₉	—	8	1	no	—
177	C₄×C₈₉	#509	1	2	yes	—
178	C₄×C₈₉	#546	1	1	yes	characteristic
179	C₄×C₈₉	#577	1	1	yes	characteristic
180	Dic₈₉	—	8	1	no	—
181	#33891	—	89	1	no	maximal, Sylow
182	C₂×C₈₉	—	4	2	no	—
183	C₂×C₈₉	#3294	1	1	yes	characteristic
184	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
185	C₂×C₈₉	#3416	1	1	yes	characteristic
186	D₈₉	—	8	2	no	—
187	D₈₉	—	16	2	no	—
188	#3342	D₈₉	1	1	yes	characteristic, complemented
189	D₄×D₈	—	89	1	no	—
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	—	89	2	no	—
201	#556	—	2	1	no	—
202	D₄²	—	178	1	no	—
203	C₂×SD₃₂	—	89	2	no	—
204	C₂×SD₃₂	—	89	1	no	—
205	C₂×M₃₂	—	89	2	no	—
206	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
207	#493	—	89	1	no	—
208	#485	—	2	2	no	—
209	#475	—	2	1	no	—
210	#467	C₂×D₈₉	1	2	yes	complemented
211	#466	—	89	2	no	—
212	#459	—	178	1	no	—
213	#447	—	178	2	no	—
214	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
215	C₄×D₈	—	89	1	no	—
216	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
217	#427	C₂×D₈₉	1	2	yes	complemented
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	—	356	1	no	—
230	C₂²×D₄	—	178	2	no	—
231	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×D₈	—	89	1	no	—
233	C₂×D₈	—	89	2	no	—
234	C₂×D₈	—	356	2	no	—
235	C₂×D₈	—	178	4	no	—
236	C₂×D₈	—	178	2	no	—
237	C₂×D₈	—	178	1	no	—
238	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
239	C₄⋊Q₈	—	2	1	no	—
240	#127	—	178	1	no	—
241	#126	—	4	2	no	—
242	#124	—	4	1	no	—
243	C₂²⋊Q₈	—	2	2	no	—
244	#121	—	356	1	no	—
245	#121	—	178	2	no	—
246	#120	—	356	2	no	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×Q₈	—	4	1	no	—
249	C₄×D₄	—	178	1	no	—
250	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
251	#117	—	2	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	—	2	1	no	—
259	#107	—	89	1	no	—
260	#107	C₂²×D₈₉	1	2	yes	—
261	#107	C₂²×D₈₉	1	1	yes	characteristic
262	#106	—	2	2	no	—
263	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
264	#103	—	2	1	no	—
265	#103	—	2	2	no	—
266	#102	—	178	1	no	—
267	#102	—	178	2	no	—
268	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
269	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂⁴	—	356	2	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×D₄	—	356	4	no	—
274	C₂×D₄	—	356	2	no	—
275	C₂×D₄	—	356	1	no	—
276	C₂×D₄	—	356	2	no	—
277	C₂×D₄	—	356	1	no	—
278	C₂×D₄	—	178	2	no	—
279	C₂×D₄	—	178	1	no	—
280	C₂×D₄	—	712	2	no	—
281	C₂×D₄	—	712	4	no	—
282	C₂×D₄	#13083559	1	1	yes	characteristic
283	C₂²×C₄	—	356	1	no	—
284	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
285	Q₁₆	—	2	1	no	—
286	Q₁₆	—	2	1	no	—
287	D₈	—	89	2	no	—
288	D₈	—	356	4	no	—
289	D₈	—	178	2	no	—
290	D₈	—	178	2	no	—
291	D₈	—	178	1	no	—
292	D₈	—	712	1	no	—
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	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₄⋊C₄	#13083559	1	1	yes	characteristic
305	C₂²⋊C₄	—	4	2	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	356	1	no	—
308	C₂²⋊C₄	—	356	2	no	—
309	C₂²⋊C₄	#13083559	1	2	yes	—
310	C₄²	—	4	2	no	—
311	C₄²	—	4	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	—	356	1	no	—
317	C₂³	—	356	2	no	—
318	C₂³	—	1424	2	no	—
319	C₂³	—	712	4	no	—
320	C₂³	—	712	4	no	—
321	C₂³	C₂×#13083559	1	2	yes	—
322	Q₈	—	4	1	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	8	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	356	1	no	—
327	D₄	—	356	2	no	—
328	D₄	—	356	2	no	—
329	D₄	—	1424	1	no	—
330	D₄	—	178	2	no	—
331	D₄	—	712	4	no	—
332	D₄	—	712	4	no	—
333	D₄	—	712	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	356	1	no	—
340	C₂×C₄	—	712	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	#41×D₈₉	1	1	yes	characteristic
348	C₈	#15668058	1	1	yes	characteristic
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	356	2	no	—
353	C₂²	—	1424	8	no	—
354	C₂²	—	712	2	no	—
355	C₂²	—	712	2	no	—
356	C₂²	—	712	2	no	—
357	C₂²	—	712	1	no	—
358	C₂²	—	712	1	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	—	8	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	712	1	no	—
366	C₄	#19152033	1	2	yes	—
367	C₄	#19152236	1	1	yes	characteristic
368	C₄	#19152406	1	1	yes	characteristic
369	C₂	—	4	2	no	—
370	C₂	—	1424	2	no	—
371	C₂	—	712	2	no	—
372	C₂	#26032059	1	1	yes	characteristic
373	C₂	C₂×#19152033	1	1	yes	characteristic
374	C₂	#26033396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3342	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	D₄×D₇₁₂	C₂
hypercenter	#3342	D₈₉
hypocenter	C₈₉	#33891
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #26022477	C₂	semidirect
3. #26022443	C₂	semidirect
4. #26022386	C₂	semidirect
5. #26022367	C₂	semidirect
6. #26022292	C₂	semidirect
7. #26022180	C₂	semidirect
8. C₄×#15667955	C₂	semidirect
9. #26022084	C₂	semidirect
10. #26021967	C₂	semidirect
11. C₈₉×#3342	C₂	semidirect
15. C₂×#15667955	C₂²	semidirect
16. C₂×#15667952	C₂²	semidirect
22. #19151425	C₂²	semidirect
23. #19151423	C₂²	semidirect
24. C₄⋊#13083549	C₂²	semidirect
25. #19151402	C₂²	semidirect
28. #19151398	C₂²	semidirect
29. #19151398	C₂²	semidirect
30. #19151398	C₂²	semidirect
31. C₂²⋊#13083549	C₂²	semidirect
32. C₄×#13083549	C₂²	semidirect
34. #500×C₈₉	C₂²	semidirect
37. #467×C₈₉	C₂²	semidirect
38. C₄×Q₁₆×C₈₉	C₂²	semidirect
40. #427×C₈₉	C₂²	semidirect
188. #3342	D₈₉	semidirect
199. C₈₉	#33891	semidirect
206. #500	C₂×D₈₉	semidirect
210. #467	C₂×D₈₉	semidirect
214. C₄×Q₁₆	C₂×D₈₉	semidirect
217. #427	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
D₄×D₇₁₂	2
#26022477	2
#26022443	2
#26022386	2
#26022367	2
#26022292	2
#26022180	2
C₄×#15667955	2
#26022084	2
#26021967	2
C₈₉×#3342	2
#33891	89