Subgroups — #36711670

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	D₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
3	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022444	C₂	1	2	yes	maximal, complemented
5	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022365	C₂	1	2	yes	maximal, complemented
7	#26022293	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022183	C₂	1	2	yes	maximal, complemented
9	C₄×#15667957	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021969	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₁₂	C₂²	1	2	yes	—
13	#19152318	—	2	1	no	—
14	D₄×D₃₅₆	—	2	1	no	—
15	C₂×#15667957	C₂²	1	1	yes	characteristic
16	C₂×#15667957	C₂²	1	2	yes	complemented
17	#19152003	C₂²	1	1	yes	characteristic, complemented
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151940	—	2	2	no	—
20	#19151884	—	2	1	no	—
21	#19151839	C₂²	1	2	yes	complemented
22	#19151834	C₂²	1	2	yes	—
23	#19151794	—	2	1	no	—
24	#19151734	—	2	2	no	—
25	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic, complemented
26	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
27	#19151636	C₂²	1	2	yes	complemented
28	#19151426	C₂²	1	1	yes	characteristic, complemented
29	#19151424	C₂²	1	2	yes	complemented
30	#19151403	C₂²	1	1	yes	characteristic, complemented
31	#19151400	C₂²	1	2	yes	—
32	#19151400	C₂²	1	1	yes	characteristic
33	#19151400	C₂²	1	2	yes	complemented
34	#19151400	C₂²	1	2	yes	complemented
35	#19151400	C₂²	1	2	yes	complemented
36	C₂×M₃₂×C₈₉	C₂²	1	2	yes	complemented
37	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₅₆	—	2	2	no	—
42	C₂×D₄×D₈₉	—	4	1	no	—
43	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
44	C₂×D₇₁₂	—	2	4	no	—
45	C₂×D₇₁₂	—	2	2	no	—
46	C₂×D₇₁₂	—	2	1	no	—
47	C₂×D₇₁₂	—	4	2	no	—
48	C₂×D₇₁₂	C₂³	1	2	yes	—
49	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
50	#15668012	—	2	1	no	—
51	#15668009	—	2	1	no	—
52	#15668005	—	4	2	no	—
53	#15667999	—	4	1	no	—
54	#15667991	—	2	2	no	—
55	#15667987	—	4	1	no	—
56	#15667986	—	2	2	no	—
57	#15667982	—	4	2	no	—
58	C₄×Dic₁₇₈	—	2	1	no	—
59	#15667977	—	4	1	no	—
60	C₄×D₃₅₆	—	2	1	no	—
61	#15667971	—	2	2	no	—
62	#15667957	—	2	2	no	—
63	#15667957	—	4	2	no	—
64	#15667946	—	2	1	no	—
65	#15667946	C₂³	1	1	yes	characteristic
66	#15667946	C₂³	1	2	yes	—
67	#15667946	C₂³	1	1	yes	characteristic
68	#15667944	—	2	2	no	—
69	#15667937	—	2	2	no	—
70	#15667937	—	2	1	no	—
71	#15667934	—	2	1	no	—
72	#15667934	—	2	2	no	—
73	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
74	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
75	M₃₂×C₈₉	—	4	2	no	—
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
78	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
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×Dic₁₇₈	—	2	1	no	—
84	C₂×Dic₁₇₈	—	4	1	no	—
85	C₂×D₃₅₆	—	2	1	no	—
86	C₂×D₃₅₆	—	2	2	no	—
87	C₂×D₃₅₆	—	4	4	no	—
88	C₂×D₃₅₆	—	4	2	no	—
89	C₂×D₃₅₆	—	4	1	no	—
90	C₂×D₃₅₆	—	4	1	no	—
91	C₂×D₃₅₆	—	8	2	no	—
92	C₂×C₂²⋊D₈₉	—	4	2	no	—
93	D₄×D₈₉	—	8	4	no	—
94	C₂×C₄×D₈₉	—	4	1	no	—
95	Dic₃₅₆	—	2	1	no	—
96	Dic₃₅₆	—	2	1	no	—
97	D₇₁₂	—	2	2	no	—
98	D₇₁₂	—	2	1	no	—
99	D₇₁₂	—	2	2	no	—
100	D₇₁₂	—	4	4	no	—
101	D₇₁₂	—	8	1	no	—
102	D₇₁₂	#41	1	2	yes	—
103	C₄⋊Dic₈₉	—	2	1	no	—
104	C₄⋊Dic₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	1	no	—
106	C₄⋊Dic₈₉	—	4	1	no	—
107	#13083553	—	4	2	no	—
108	#13083553	—	4	1	no	—
109	C₂²⋊Dic₈₉	—	4	2	no	—
110	C₂²⋊Dic₈₉	—	4	2	no	—
111	#13083551	—	4	1	no	—
112	#13083551	—	4	2	no	—
113	C₄×Dic₈₉	—	4	2	no	—
114	C₄×Dic₈₉	—	4	1	no	—
115	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	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	D₈	1	1	yes	characteristic
123	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₈₉	—	2	1	no	—
126	C₁₆×C₈₉	—	2	1	no	—
127	C₁₆×C₈₉	—	2	2	no	—
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	D₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	8	4	no	—
142	C₂×Dic₈₉	—	4	1	no	—
143	C₂×Dic₈₉	—	4	1	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	4	1	no	—
146	D₃₅₆	—	2	2	no	—
147	D₃₅₆	—	4	2	no	—
148	D₃₅₆	—	4	1	no	—
149	D₃₅₆	—	4	2	no	—
150	D₃₅₆	—	8	4	no	—
151	D₃₅₆	—	8	2	no	—
152	D₃₅₆	—	16	1	no	—
153	C₄×D₈₉	—	8	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	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	#509	1	2	yes	—
175	C₄×C₈₉	#546	1	1	yes	characteristic
176	C₄×C₈₉	#577	1	1	yes	characteristic
177	Dic₈₉	—	8	2	no	—
178	Dic₈₉	—	8	1	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	#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	—	89	1	no	—
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	D₈₉	1	1	yes	characteristic, complemented
198	#2140	—	89	2	no	—
199	C₈₉	#33891	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₈	—	89	2	no	—
201	#556	—	178	1	no	—
202	D₄²	—	178	1	no	—
203	C₂×SD₃₂	—	89	2	no	—
204	C₂×SD₃₂	—	89	1	no	—
205	C₂×M₃₂	C₂×D₈₉	1	2	yes	complemented
206	#500	—	89	1	no	—
207	#493	—	89	1	no	—
208	#485	—	178	2	no	—
209	#475	—	178	1	no	—
210	#467	—	89	2	no	—
211	#466	—	89	2	no	—
212	#459	—	178	1	no	—
213	#447	—	178	2	no	—
214	C₄×Q₁₆	—	89	1	no	—
215	C₄×D₈	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic, complemented
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	C₂×D₈₉	1	2	yes	complemented
228	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
229	C₂²×D₄	—	356	1	no	—
230	C₂²×D₄	—	178	2	no	—
231	C₂×Q₁₆	—	89	1	no	—
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	—	178	1	no	—
240	#127	—	178	1	no	—
241	#126	—	356	2	no	—
242	#124	—	356	1	no	—
243	C₂²⋊Q₈	—	178	2	no	—
244	#121	—	356	1	no	—
245	#121	—	178	2	no	—
246	#120	—	356	2	no	—
247	C₄×Q₈	—	356	1	no	—
248	C₄×Q₈	—	178	1	no	—
249	C₄×D₄	—	178	1	no	—
250	C₄×D₄	D₃₅₆	1	1	yes	characteristic
251	#117	—	178	2	no	—
252	SD₃₂	—	356	2	no	—
253	SD₃₂	—	178	2	no	—
254	M₃₂	—	4	2	no	—
255	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
258	#107	—	89	2	no	—
259	#107	—	89	1	no	—
260	#107	—	89	1	no	—
261	#107	—	178	1	no	—
262	#106	—	178	2	no	—
263	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
264	#103	—	178	1	no	—
265	#103	—	178	2	no	—
266	#102	—	178	1	no	—
267	#102	—	178	2	no	—
268	C₂²⋊C₈	D₃₅₆	1	2	yes	—
269	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂⁴	—	356	2	no	—
271	C₂×Q₈	—	356	1	no	—
272	C₂×Q₈	—	178	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	D₇₁₂	1	1	yes	characteristic
283	C₂²×C₄	—	356	1	no	—
284	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
285	Q₁₆	—	178	1	no	—
286	Q₁₆	—	178	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×D₃₅₆	1	1	yes	characteristic
296	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
297	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
298	C₄⋊C₄	—	356	2	no	—
299	C₄⋊C₄	—	356	1	no	—
300	C₄⋊C₄	—	356	1	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
305	C₂²⋊C₄	—	356	2	no	—
306	C₂²⋊C₄	—	356	2	no	—
307	C₂²⋊C₄	—	356	1	no	—
308	C₂²⋊C₄	—	356	2	no	—
309	C₂²⋊C₄	D₇₁₂	1	2	yes	—
310	C₄²	—	356	2	no	—
311	C₄²	—	356	1	no	—
312	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	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×D₇₁₂	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	—	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	—	356	1	no	—
337	C₂×C₄	—	356	1	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	356	1	no	—
340	C₂×C₄	—	712	2	no	—
341	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
342	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
343	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	#41×D₈₉	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	—	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×D₇₁₂	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	1	no	—
362	C₄	—	712	1	no	—
363	C₄	—	712	1	no	—
364	C₄	—	712	2	no	—
365	C₄	—	712	1	no	—
366	C₄	#19152036	1	2	yes	—
367	C₄	#19152243	1	1	yes	characteristic
368	C₄	#19152410	1	1	yes	characteristic
369	C₂	—	4	2	no	—
370	C₂	—	1424	2	no	—
371	C₂	—	712	2	no	—
372	C₂	#26032066	1	1	yes	characteristic
373	C₂	C₂×#19152036	1	1	yes	characteristic
374	C₂	#26033400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2169	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	D₄×D₇₁₂	C₂
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#33891
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

Complements

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

N	H ≅ G/N	type
1. #26032683	C₂	semidirect
3. #26022476	C₂	semidirect
4. #26022444	C₂	semidirect
5. #26022390	C₂	semidirect
6. #26022365	C₂	semidirect
7. #26022293	C₂	semidirect
8. #26022183	C₂	semidirect
9. C₄×#15667957	C₂	semidirect
10. #26021969	C₂	semidirect
11. C₈₉×#2169	C₂	semidirect
16. C₂×#15667957	C₂²	semidirect
17. #19152003	C₂²	semidirect
21. #19151839	C₂²	semidirect
25. C₄×Dic₃₅₆	C₂²	semidirect
27. #19151636	C₂²	semidirect
28. #19151426	C₂²	semidirect
29. #19151424	C₂²	semidirect
30. #19151403	C₂²	semidirect
33. #19151400	C₂²	semidirect
34. #19151400	C₂²	semidirect
35. #19151400	C₂²	semidirect
36. C₂×M₃₂×C₈₉	C₂²	semidirect
38. C₄⋊C₁₆×C₈₉	C₂²	semidirect
39. C₂²⋊C₁₆×C₈₉	C₂²	semidirect
40. C₄×C₁₆×C₈₉	C₂²	semidirect
197. #2169	D₈₉	semidirect
199. C₈₉	#33891	semidirect
205. C₂×M₃₂	C₂×D₈₉	semidirect
220. C₄⋊C₁₆	C₂×D₈₉	semidirect
227. C₂²⋊C₁₆	C₂×D₈₉	semidirect
228. C₄×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26032683	2
D₄×D₇₁₂	2
#26022476	2
#26022444	2
#26022390	2
#26022365	2
#26022293	2
#26022183	2
C₄×#15667957	2
#26021969	2
C₈₉×#2169	2
#33891	89