Subgroups — #36711767

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 374 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	#26032011	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022439	C₂	1	2	yes	maximal, complemented
5	#26022394	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022354	C₂	1	2	yes	maximal, complemented
7	#26022287	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022174	C₂	1	2	yes	maximal, complemented
9	#26022085	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021963	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668035	C₂²	1	2	yes	—
13	#19152270	—	2	1	no	—
14	#19152253	—	2	1	no	—
15	C₂×#15667953	C₂²	1	2	yes	—
16	C₂×#15667951	C₂²	1	2	yes	—
17	#19152003	C₂²	1	1	yes	characteristic
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	—
23	#19151798	—	2	1	no	—
24	#19151746	—	2	2	no	—
25	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
26	#107×D₈₉	C₂²	1	2	yes	—
27	#19151425	C₂²	1	2	yes	—
28	C₄⋊#13083549	C₂²	1	1	yes	characteristic
29	#19151402	C₂²	1	2	yes	—
30	#19151400	C₂²	1	2	yes	—
31	#19151400	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
37	#367×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083575	—	2	2	no	—
42	C₂×#13083574	—	4	1	no	—
43	#15668035	—	2	4	no	—
44	#15668035	—	4	2	no	—
45	C₂×#13083561	—	2	2	no	—
46	C₂×D₇₁₂	—	2	1	no	—
47	C₂×D₇₁₂	C₂³	1	2	yes	—
48	C₂×C₈×D₈₉	C₂³	1	2	yes	—
49	#15668012	—	2	1	no	—
50	#15668009	—	2	1	no	—
51	#15668009	—	4	1	no	—
52	#15667994	—	4	2	no	—
53	#15667991	—	4	1	no	—
54	#15667987	—	2	2	no	—
55	#15667987	—	4	2	no	—
56	#15667983	—	2	2	no	—
57	C₄×D₃₅₆	—	4	1	no	—
58	C₄×D₃₅₆	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₈₉	—	2	2	no	—
60	#15667953	—	4	2	no	—
61	#15667951	—	4	2	no	—
62	C₂×#13083549	C₂³	1	2	yes	—
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×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	SD₃₂×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	#103×C₈₉	—	2	1	no	—
80	#102×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	#13083575	—	4	4	no	—
83	#13083575	—	8	2	no	—
84	#13083574	—	8	4	no	—
85	C₂×Dic₁₇₈	—	4	1	no	—
86	C₂×D₃₅₆	—	2	2	no	—
87	C₂×D₃₅₆	—	4	2	no	—
88	C₂×D₃₅₆	—	4	1	no	—
89	C₂×D₃₅₆	—	4	2	no	—
90	C₂×D₃₅₆	—	4	1	no	—
91	C₂×D₃₅₆	D₈	1	1	yes	characteristic
92	C₂×C₂²⋊D₈₉	—	4	2	no	—
93	C₂×C₄×D₈₉	—	4	2	no	—
94	C₂×C₄×D₈₉	—	4	2	no	—
95	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
96	#13083561	—	4	4	no	—
97	D₇₁₂	—	2	2	no	—
98	D₇₁₂	—	2	2	no	—
99	D₇₁₂	—	8	1	no	—
100	C₂×#10623494	C₂×D₄	1	2	yes	—
101	C₈×D₈₉	—	2	2	no	—
102	C₈×D₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	—	2	2	no	—
104	C₄⋊Dic₈₉	—	4	1	no	—
105	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
106	#13083553	—	4	2	no	—
107	C₂²⋊Dic₈₉	—	4	2	no	—
108	#13083551	—	4	2	no	—
109	#13083551	D₈	1	2	yes	—
110	#13083549	—	2	2	no	—
111	C₂×Q₈×C₈₉	—	2	1	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂²×C₄×C₈₉	—	4	1	no	—
114	Q₁₆×C₈₉	—	2	1	no	—
115	Q₁₆×C₈₉	#41	1	2	yes	complemented
116	D₈×C₈₉	—	2	1	no	—
117	D₈×C₈₉	—	2	1	no	—
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₂²⋊C₄×C₈₉	—	4	1	no	—
124	C₄²×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₈₉	—	2	1	no	—
127	C₁₆×C₈₉	—	2	1	no	—
128	C₂³×C₈₉	—	4	1	no	—
129	C₂²×D₈₉	—	4	2	no	—
130	C₂²×D₈₉	C₂×D₈	1	2	yes	—
131	Q₈×C₈₉	—	2	2	no	—
132	Q₈×C₈₉	—	4	1	no	—
133	D₄×C₈₉	—	4	1	no	—
134	D₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	8	2	no	—
139	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₈₉	—	8	4	no	—
143	C₂×Dic₈₉	—	4	2	no	—
144	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
145	D₃₅₆	—	2	2	no	—
146	D₃₅₆	—	4	2	no	—
147	D₃₅₆	—	4	2	no	—
148	D₃₅₆	—	8	4	no	—
149	D₃₅₆	—	8	2	no	—
150	D₃₅₆	—	16	1	no	—
151	D₃₅₆	—	16	2	no	—
152	C₄×D₈₉	—	2	2	no	—
153	C₄×D₈₉	—	2	2	no	—
154	C₄×D₈₉	—	8	4	no	—
155	C₄×D₈₉	—	8	4	no	—
156	Dic₁₇₈	—	8	2	no	—
157	C₈×C₈₉	—	2	1	no	—
158	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
160	#10623494	—	2	2	no	—
161	C₂²×C₈₉	—	8	1	no	—
162	C₂²×C₈₉	—	8	1	no	—
163	C₂²×C₈₉	—	8	1	no	—
164	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₈₉	—	2	2	no	—
166	C₂×D₈₉	—	2	2	no	—
167	C₂×D₈₉	—	2	2	no	—
168	C₂×D₈₉	—	8	2	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	8	2	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	4	2	no	—
173	C₄×C₈₉	—	8	1	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	—	2	2	no	—
179	Dic₈₉	—	8	2	no	—
180	#33899	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	—	16	1	no	—
182	C₂×C₈₉	#3294	1	1	yes	characteristic
183	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
184	C₂×C₈₉	#3416	1	1	yes	characteristic
185	D₈₉	—	4	2	no	—
186	D₈₉	—	16	2	no	—
187	#3334	—	89	1	no	—
188	#3287	—	89	1	no	—
189	#2240	—	89	1	no	—
190	#2233	—	89	2	no	—
191	#2225	—	89	1	no	—
192	#2218	—	89	2	no	—
193	#2206	—	89	1	no	—
194	#2186	—	89	2	no	—
195	C₄×SD₃₂	D₈₉	1	1	yes	characteristic, complemented
196	#2169	—	89	1	no	—
197	#2139	—	89	2	no	—
198	C₈₉	#33899	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	89	2	no	—
200	#550	—	178	1	no	—
201	#547	—	178	1	no	—
202	C₂×SD₃₂	C₂×D₈₉	1	1	yes	characteristic
203	C₂×D₁₆	—	89	2	no	—
204	C₂×M₃₂	—	89	2	no	—
205	#500	—	89	1	no	—
206	#493	—	89	1	no	—
207	#480	—	178	2	no	—
208	#474	—	178	1	no	—
209	#466	—	89	2	no	—
210	#466	—	89	2	no	—
211	#460	—	178	1	no	—
212	#449	—	178	2	no	—
213	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
214	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
215	C₈×D₄	—	89	1	no	—
216	C₂×#107	—	89	2	no	—
217	#367	—	89	2	no	—
218	#367	C₂×D₈₉	1	1	yes	characteristic
219	C₄⋊C₁₆	—	89	1	no	—
220	#358	—	89	2	no	—
221	#358	C₂×D₈₉	1	1	yes	characteristic
222	#357	—	89	2	no	—
223	#357	—	89	2	no	—
224	#357	—	89	2	no	—
225	#357	C₂×D₈₉	1	1	yes	characteristic
226	C₂²⋊C₁₆	—	89	2	no	—
227	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
228	C₂×#41	—	356	1	no	—
229	C₂×#41	—	178	2	no	—
230	#135	—	356	2	no	—
231	#135	—	178	4	no	—
232	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×SD₁₆	—	178	2	no	—
234	C₂×D₈	—	89	2	no	—
235	C₂×D₈	—	178	1	no	—
236	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂²×C₈	—	89	2	no	—
238	C₄⋊Q₈	—	178	1	no	—
239	#127	—	356	1	no	—
240	#127	—	178	1	no	—
241	#123	—	356	2	no	—
242	C₂²⋊Q₈	—	356	1	no	—
243	#121	—	356	2	no	—
244	#121	—	178	2	no	—
245	#121	—	178	2	no	—
246	C₄×Q₈	—	2	1	no	—
247	C₄×D₄	—	2	1	no	—
248	C₄×D₄	—	89	1	no	—
249	C₄×D₄	—	356	1	no	—
250	C₂×C₄⋊C₄	—	178	2	no	—
251	SD₃₂	—	2	2	no	—
252	D₁₆	—	356	2	no	—
253	M₃₂	—	356	2	no	—
254	C₂×C₁₆	—	89	2	no	—
255	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
257	#107	—	89	2	no	—
258	#107	—	178	2	no	—
259	#107	—	178	1	no	—
260	#107	C₂²×D₈₉	1	1	yes	characteristic
261	#107	C₂²×D₈₉	1	1	yes	characteristic
262	C₄⋊C₈	—	89	1	no	—
263	#103	—	2	1	no	—
264	#102	—	2	1	no	—
265	#102	—	178	2	no	—
266	#102	—	178	2	no	—
267	C₂²⋊C₈	—	89	2	no	—
268	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
269	#41	—	356	4	no	—
270	#41	—	712	4	no	—
271	#41	—	712	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	356	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	89	1	no	—
276	C₂×D₄	—	356	2	no	—
277	C₂×D₄	—	356	2	no	—
278	C₂×D₄	—	356	2	no	—
279	C₂×D₄	—	356	1	no	—
280	C₂×D₄	—	356	1	no	—
281	C₂×D₄	—	178	2	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	89	2	no	—
284	C₂²×C₄	—	356	2	no	—
285	C₂²×C₄	—	356	2	no	—
286	Q₁₆	—	2	1	no	—
287	Q₁₆	#13083574	1	2	yes	complemented
288	SD₁₆	—	356	4	no	—
289	D₈	—	2	1	no	—
290	D₈	—	2	1	no	—
291	D₈	—	178	2	no	—
292	D₈	—	178	2	no	—
293	D₈	—	712	1	no	—
294	C₂×C₈	—	89	2	no	—
295	C₂×C₈	—	178	2	no	—
296	C₂×C₈	—	178	2	no	—
297	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	89	1	no	—
303	C₄⋊C₄	—	356	2	no	—
304	C₄⋊C₄	—	356	1	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	89	2	no	—
308	C₂²⋊C₄	—	356	2	no	—
309	C₂²⋊C₄	—	356	2	no	—
310	C₄²	—	4	1	no	—
311	C₄²	D₄×D₈₉	1	1	yes	characteristic
312	C₁₆	—	2	1	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	178	2	no	—
315	C₂³	—	4	1	no	—
316	C₂³	—	89	2	no	—
317	C₂³	—	356	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	4	1	no	—
320	Q₈	—	712	2	no	—
321	D₄	—	4	1	no	—
322	D₄	—	4	1	no	—
323	D₄	—	356	2	no	—
324	D₄	—	356	2	no	—
325	D₄	—	1424	1	no	—
326	D₄	—	1424	2	no	—
327	D₄	—	178	2	no	—
328	D₄	—	712	4	no	—
329	D₄	—	712	4	no	—
330	D₄	—	712	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	89	2	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	178	2	no	—
338	C₂×C₄	—	178	2	no	—
339	C₂×C₄	—	712	4	no	—
340	C₂×C₄	—	712	4	no	—
341	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
342	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	178	2	no	—
346	C₈	C₂×#13083574	1	1	yes	characteristic
347	C₈	#15668058	1	1	yes	characteristic
348	C₂²	—	8	1	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	1	no	—
351	C₂²	—	178	2	no	—
352	C₂²	—	178	2	no	—
353	C₂²	—	178	2	no	—
354	C₂²	—	712	2	no	—
355	C₂²	—	712	2	no	—
356	C₂²	—	712	2	no	—
357	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	4	2	no	—
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	178	2	no	—
363	C₄	—	712	2	no	—
364	C₄	#19152035	1	2	yes	—
365	C₄	#19152244	1	1	yes	characteristic
366	C₄	#19152405	1	1	yes	characteristic
367	C₂	—	16	1	no	—
368	C₂	—	356	2	no	—
369	C₂	—	1424	2	no	—
370	C₂	#26032067	1	1	yes	characteristic
371	C₂	C₂×#19152035	1	1	yes	characteristic
372	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×SD₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₄×SD₃₂	D₈₉
hypocenter	C₈₉	#33899
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #26032601	C₂	semidirect
2. #26032011	C₂	semidirect
3. #26022476	C₂	semidirect
4. #26022439	C₂	semidirect
5. #26022394	C₂	semidirect
6. #26022354	C₂	semidirect
7. #26022287	C₂	semidirect
8. #26022174	C₂	semidirect
9. #26022085	C₂	semidirect
10. #26021963	C₂	semidirect
11. C₄×SD₃₂×C₈₉	C₂	semidirect
115. Q₁₆×C₈₉	#41	semidirect
195. C₄×SD₃₂	D₈₉	semidirect
198. C₈₉	#33899	semidirect
287. Q₁₆	#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
#26032011	2
#26022476	2
#26022439	2
#26022394	2
#26022354	2
#26022287	2
#26022174	2
#26022085	2
#26021963	2
C₄×SD₃₂×C₈₉	2
#33899	89