Subgroups — #36711743

Order 22784.

580 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032600	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022452	C₂	1	2	yes	maximal, complemented
4	#26022389	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022367	C₂	1	2	yes	maximal, complemented
6	#26022294	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022175	C₂	1	2	yes	maximal, complemented
8	C₄×#15667953	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021964	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3283	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152252	—	2	1	no	—
13	C₂×#15667955	C₂²	1	2	yes	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	C₂×#15667952	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151912	—	2	2	no	—
18	#19151879	—	2	1	no	—
19	#19151834	C₂²	1	2	yes	—
20	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
21	C₂×#15667946	C₂²	1	2	yes	—
22	#19151423	C₂²	1	1	yes	characteristic
23	#19151423	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151402	C₂²	1	2	yes	—
26	#19151402	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151399	C₂²	1	2	yes	—
29	#19151399	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	1	yes	characteristic
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	C₂×#135×C₈₉	C₂²	1	2	yes	—
34	#547×C₈₉	—	2	1	no	—
35	#494×C₈₉	C₂²	1	1	yes	characteristic
36	#467×C₈₉	C₂²	1	2	yes	—
37	#461×C₈₉	—	2	1	no	—
38	#452×C₈₉	—	2	2	no	—
39	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083573	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	1	no	—
44	#15667996	—	4	2	no	—
45	#15667992	—	4	1	no	—
46	#15667986	—	2	2	no	—
47	C₄×D₃₅₆	—	2	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	2	no	—
49	#15667955	—	4	2	no	—
50	#15667953	—	2	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	2	no	—
56	#15667946	—	2	1	no	—
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667946	C₂³	1	2	yes	—
59	#15667934	—	2	2	no	—
60	#15667934	—	2	1	no	—
61	C₂×#41×C₈₉	—	2	2	no	—
62	#135×C₈₉	—	2	4	no	—
63	#135×C₈₉	—	4	2	no	—
64	C₂×Q₁₆×C₈₉	—	2	1	no	—
65	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
66	C₂×SD₁₆×C₈₉	—	2	2	no	—
67	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₉	—	2	1	no	—
70	#123×C₈₉	—	4	2	no	—
71	C₂²⋊Q₈×C₈₉	—	2	2	no	—
72	#121×C₈₉	—	4	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
78	#103×C₈₉	—	2	2	no	—
79	#102×C₈₉	—	2	1	no	—
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	#13083573	—	8	4	no	—
83	C₂×Dic₁₇₈	—	4	1	no	—
84	C₂×D₃₅₆	—	2	1	no	—
85	C₂×C₂²⋊D₈₉	—	4	2	no	—
86	C₂²×Dic₈₉	—	4	2	no	—
87	C₂×C₄×D₈₉	—	4	1	no	—
88	D₇₁₂	—	2	1	no	—
89	D₇₁₂	—	2	1	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	#13083551	—	4	1	no	—
95	#13083551	—	4	2	no	—
96	#13083549	—	2	1	no	—
97	#13083549	—	2	1	no	—
98	#13083549	—	2	2	no	—
99	#41×C₈₉	—	4	4	no	—
100	#41×C₈₉	—	8	2	no	—
101	C₂×Q₈×C₈₉	—	2	2	no	—
102	C₂×D₄×C₈₉	—	2	1	no	—
103	C₂×D₄×C₈₉	—	4	2	no	—
104	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	—	4	1	no	—
106	C₂²×C₄×C₈₉	—	4	2	no	—
107	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
108	Q₁₆×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	2	2	no	—
110	Q₁₆×C₈₉	—	8	1	no	—
111	SD₁₆×C₈₉	—	4	4	no	—
112	D₈×C₈₉	—	2	1	no	—
113	D₈×C₈₉	#41	1	2	yes	complemented
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	4	1	no	—
121	C₄⋊C₄×C₈₉	—	4	2	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
124	C₂²⋊C₄×C₈₉	—	4	2	no	—
125	C₂²⋊C₄×C₈₉	—	4	1	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	C₂×D₈	1	2	yes	—
130	C₂²×D₈₉	—	4	1	no	—
131	Q₈×C₈₉	—	4	2	no	—
132	Q₈×C₈₉	—	4	2	no	—
133	D₄×C₈₉	—	2	2	no	—
134	D₄×C₈₉	—	2	2	no	—
135	D₄×C₈₉	—	4	1	no	—
136	D₄×C₈₉	—	8	4	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	—	8	4	no	—
142	C₂×C₄×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
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	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	8	4	no	—
151	D₃₅₆	—	4	1	no	—
152	D₃₅₆	—	4	1	no	—
153	C₄×D₈₉	—	8	2	no	—
154	Dic₁₇₈	—	8	2	no	—
155	C₈×C₈₉	—	2	2	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
158	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²×C₈₉	—	2	2	no	—
160	C₂²×C₈₉	—	2	2	no	—
161	C₂²×C₈₉	—	2	2	no	—
162	C₂²×C₈₉	—	4	2	no	—
163	C₂²×C₈₉	—	8	1	no	—
164	C₂²×C₈₉	—	8	1	no	—
165	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
166	C₂×D₈₉	—	8	1	no	—
167	C₂×D₈₉	—	8	1	no	—
168	C₂×D₈₉	—	8	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	—	8	1	no	—
172	C₄×C₈₉	—	8	2	no	—
173	C₄×C₈₉	#509	1	2	yes	—
174	C₄×C₈₉	#546	1	1	yes	characteristic
175	C₄×C₈₉	#577	1	1	yes	characteristic
176	Dic₈₉	—	8	2	no	—
177	Dic₈₉	—	8	1	no	—
178	#33897	—	89	1	no	maximal, Sylow
179	C₂×C₈₉	—	4	2	no	—
180	C₂×C₈₉	—	8	2	no	—
181	C₂×C₈₉	#3294	1	1	yes	characteristic
182	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
183	C₂×C₈₉	#3417	1	1	yes	characteristic
184	D₈₉	—	16	1	no	—
185	#3334	—	89	1	no	—
186	#3283	D₈₉	1	1	yes	characteristic, complemented
187	#2241	—	89	1	no	—
188	#2236	—	89	2	no	—
189	#2224	—	89	1	no	—
190	#2220	—	89	2	no	—
191	#2208	—	89	1	no	—
192	#2186	—	89	2	no	—
193	C₄×D₁₆	—	89	1	no	—
194	#2169	—	89	1	no	—
195	#2139	—	89	2	no	—
196	C₈₉	#33897	1	1	yes	characteristic, complemented, Sylow
197	C₂×#135	C₂×D₈₉	1	2	yes	—
198	#547	—	2	1	no	—
199	#547	—	178	1	no	—
200	C₂×SD₃₂	—	89	2	no	—
201	C₂×D₁₆	—	89	1	no	—
202	C₂×M₃₂	—	89	2	no	—
203	#500	—	89	1	no	—
204	#494	C₂×D₈₉	1	1	yes	characteristic
205	#480	—	178	2	no	—
206	#474	—	178	1	no	—
207	#467	C₂×D₈₉	1	2	yes	—
208	#466	—	89	2	no	—
209	#461	—	2	1	no	—
210	#452	—	2	2	no	—
211	C₄×D₈	—	89	1	no	—
212	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
213	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
214	C₂×#107	—	89	2	no	—
215	#366	—	89	1	no	—
216	#366	—	89	2	no	—
217	C₄⋊C₁₆	—	89	1	no	—
218	#358	—	89	2	no	—
219	#358	—	89	2	no	—
220	#358	—	89	2	no	—
221	#357	—	89	1	no	—
222	#357	—	89	2	no	—
223	#357	—	89	1	no	—
224	C₂²⋊C₁₆	—	89	2	no	—
225	C₄×C₁₆	—	89	1	no	—
226	C₂×#41	—	2	2	no	—
227	C₂×#41	—	356	1	no	—
228	#135	—	2	4	no	—
229	#135	—	4	2	no	—
230	C₂×Q₁₆	—	2	1	no	—
231	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
232	C₂×SD₁₆	—	2	2	no	—
233	C₂×D₈	—	89	1	no	—
234	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	178	1	no	—
238	#123	—	4	2	no	—
239	#123	—	356	2	no	—
240	C₂²⋊Q₈	—	2	2	no	—
241	C₂²⋊Q₈	—	356	1	no	—
242	#121	—	4	1	no	—
243	#121	—	178	2	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	178	1	no	—
246	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
247	C₂×C₄⋊C₄	—	4	1	no	—
248	C₂×C₄⋊C₄	—	178	2	no	—
249	SD₃₂	—	356	2	no	—
250	D₁₆	—	178	2	no	—
251	M₃₂	—	356	2	no	—
252	C₂×C₁₆	—	89	1	no	—
253	C₂×C₁₆	—	89	1	no	—
254	C₂×C₁₆	—	89	2	no	—
255	#107	—	89	1	no	—
256	#107	—	89	2	no	—
257	#107	—	178	2	no	—
258	#107	—	178	1	no	—
259	#107	C₂²×D₈₉	1	1	yes	characteristic
260	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
261	#103	—	2	2	no	—
262	#102	—	2	1	no	—
263	#102	—	178	1	no	—
264	#102	—	178	2	no	—
265	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
266	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	#41	—	4	4	no	—
268	#41	—	8	2	no	—
269	#41	—	712	4	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	356	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	356	2	no	—
275	C₂×D₄	—	178	1	no	—
276	C₂×D₄	#13083559	1	1	yes	characteristic
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	2	no	—
279	C₂²×C₄	—	356	2	no	—
280	C₂²×C₄	—	356	1	no	—
281	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
282	Q₁₆	—	2	2	no	—
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	8	1	no	—
285	SD₁₆	—	4	4	no	—
286	D₈	—	2	1	no	—
287	D₈	—	178	1	no	—
288	D₈	—	178	1	no	—
289	D₈	#13083573	1	2	yes	complemented
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
294	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	356	2	no	—
300	C₄⋊C₄	—	356	1	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	—	178	2	no	—
303	C₄⋊C₄	#13083559	1	1	yes	characteristic
304	C₂²⋊C₄	—	4	2	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	356	1	no	—
307	C₂²⋊C₄	—	356	2	no	—
308	C₂²⋊C₄	#13083559	1	2	yes	—
309	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₁₆	—	178	1	no	—
311	C₁₆	—	178	1	no	—
312	C₁₆	—	178	2	no	—
313	C₂³	—	4	1	no	—
314	C₂³	—	356	1	no	—
315	C₂³	C₂×#13083559	1	2	yes	—
316	Q₈	—	4	2	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	712	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	4	1	no	—
322	D₄	—	8	4	no	—
323	D₄	—	356	1	no	—
324	D₄	—	356	1	no	—
325	D₄	—	712	4	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	356	1	no	—
333	C₂×C₄	—	356	2	no	—
334	C₂×C₄	—	712	4	no	—
335	C₂×C₄	—	712	2	no	—
336	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
337	C₂×C₄	C₂×#13083559	1	2	yes	—
338	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
339	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	2	2	no	—
342	C₈	C₂×#13083573	1	1	yes	characteristic
343	C₈	#15668057	1	1	yes	characteristic
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	8	1	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	712	1	no	—
351	C₂²	—	712	1	no	—
352	C₂²	—	712	1	no	—
353	C₂²	C₂²×#13083559	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	1	no	—
358	C₄	—	712	1	no	—
359	C₄	—	712	2	no	—
360	C₄	#19152033	1	2	yes	—
361	C₄	#19152242	1	1	yes	characteristic
362	C₄	#19152409	1	1	yes	characteristic
363	C₂	—	4	2	no	—
364	C₂	—	8	2	no	—
365	C₂	—	1424	1	no	—
366	C₂	#26032065	1	1	yes	characteristic
367	C₂	C₂×#19152033	1	1	yes	characteristic
368	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3283	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3283	D₈₉
hypocenter	C₈₉	#33897
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3283

Complements

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

N	H ≅ G/N	type
1. #26032600	C₂	semidirect
2. #26022479	C₂	semidirect
3. #26022452	C₂	semidirect
4. #26022389	C₂	semidirect
5. #26022367	C₂	semidirect
6. #26022294	C₂	semidirect
7. #26022175	C₂	semidirect
8. C₄×#15667953	C₂	semidirect
9. #26022084	C₂	semidirect
10. #26021964	C₂	semidirect
11. C₈₉×#3283	C₂	semidirect
113. D₈×C₈₉	#41	semidirect
186. #3283	D₈₉	semidirect
196. C₈₉	#33897	semidirect
289. D₈	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#26032600	2
#26022479	2
#26022452	2
#26022389	2
#26022367	2
#26022294	2
#26022175	2
C₄×#15667953	2
#26022084	2
#26021964	2
C₈₉×#3283	2
#33897	89