Subgroups — #36711226

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033390	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032038	C₂	1	2	yes	maximal, complemented
3	#26022514	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022374	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022372	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022323	C₂	1	2	yes	maximal, complemented
7	#26022220	C₂	1	2	yes	maximal, complemented
8	#26022216	C₂	1	2	yes	maximal, complemented
9	#26022133	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022087	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3417	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152406	—	2	2	no	—
13	#19152400	C₂²	1	2	yes	—
14	#19152035	—	2	2	no	—
15	#19152033	C₂²	1	2	yes	—
16	#19152031	—	2	1	no	—
17	C₂×#15667955	C₂²	1	2	yes	complemented
18	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
19	D₁₆×D₈₉	—	2	1	no	—
20	#19152014	C₂²	1	1	yes	characteristic
21	#19151983	C₂²	1	2	yes	complemented
22	#19151870	C₂²	1	2	yes	—
23	#19151868	C₂²	1	2	yes	complemented
24	#19151863	—	2	2	no	—
25	#19151774	—	2	2	no	—
26	#19151771	—	2	2	no	—
27	#19151715	C₂²	1	2	yes	complemented
28	#19151713	C₂²	1	1	yes	characteristic
29	#19151428	C₂²	1	1	yes	characteristic, complemented
30	#19151418	C₂²	1	1	yes	characteristic, complemented
31	#19151414	C₂²	1	2	yes	complemented
32	#19151412	C₂²	1	2	yes	complemented
33	#19151405	C₂²	1	2	yes	complemented
34	#19151383	C₂²	1	1	yes	characteristic
35	#19151382	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
37	#578×C₈₉	C₂²	1	2	yes	complemented
38	#510×C₈₉	—	2	2	no	—
39	#510×C₈₉	C₂²	1	2	yes	complemented
40	#508×C₈₉	—	2	1	no	—
41	#508×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×Q₃₂×C₈₉	—	2	1	no	—
43	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic, complemented
44	#504×C₈₉	C₂²	1	1	yes	characteristic, complemented
45	#15668058	—	2	2	no	—
46	#15668043	—	2	2	no	—
47	#15668043	—	4	2	no	—
48	#15668039	—	4	2	no	—
49	#15668037	—	2	2	no	—
50	#15668037	—	2	2	no	—
51	#15668037	—	4	2	no	—
52	#15668035	—	2	2	no	—
53	#15668032	—	4	2	no	—
54	#15668032	D₄	1	1	yes	characteristic
55	C₂×#13083561	—	2	2	no	—
56	C₂×#13083561	—	4	2	no	—
57	C₂×D₇₁₂	C₂³	1	2	yes	—
58	D₈×D₈₉	—	2	2	no	—
59	#15668019	—	2	2	no	—
60	#15668018	C₂³	1	1	yes	characteristic
61	#15668001	—	2	2	no	—
62	#15667955	—	2	2	no	—
63	#15667955	—	4	2	no	—
64	D₁₄₂₄	—	2	1	no	—
65	D₁₄₂₄	—	2	1	no	—
66	#15667953	—	2	2	no	—
67	#15667952	C₂³	1	1	yes	characteristic
68	#15667951	—	2	1	no	—
69	C₂×#13083549	C₂³	1	1	yes	characteristic
70	C₁₆×D₈₉	—	2	1	no	—
71	#15667948	C₂³	1	2	yes	—
72	#15667947	D₄	1	1	yes	characteristic
73	#15667939	—	2	2	no	—
74	#15667939	D₄	1	1	yes	characteristic
75	#15667938	D₄	1	1	yes	characteristic
76	#15667931	—	2	2	no	—
77	#15667928	—	2	2	no	—
78	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
79	2₋¹⁺⁴×C₈₉	—	2	2	no	—
80	#137×C₈₉	—	2	2	no	—
81	#137×C₈₉	—	4	2	no	—
82	#135×C₈₉	—	2	2	no	—
83	#135×C₈₉	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₈₉	—	2	2	no	—
85	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
86	#131×C₈₉	C₂³	1	1	yes	characteristic
87	Q₃₂×C₈₉	—	2	2	no	—
88	Q₃₂×C₈₉	—	2	1	no	—
89	Q₃₂×C₈₉	—	2	1	no	—
90	Q₃₂×C₈₉	D₄	1	1	yes	characteristic
91	SD₃₂×C₈₉	—	2	2	no	—
92	SD₃₂×C₈₉	D₄	1	2	yes	—
93	D₁₆×C₈₉	D₄	1	1	yes	characteristic
94	M₃₂×C₈₉	—	2	1	no	—
95	M₃₂×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₈₉	—	2	1	no	—
97	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
98	#13083575	—	4	2	no	—
99	#13083574	—	4	2	no	—
100	#13083574	C₂×D₄	1	1	yes	characteristic
101	C₂×D₃₅₆	—	2	2	no	—
102	C₂×D₃₅₆	—	4	2	no	—
103	D₄×D₈₉	—	4	2	no	—
104	D₄×D₈₉	—	8	2	no	—
105	#13083564	—	4	2	no	—
106	#13083564	D₈	1	1	yes	characteristic
107	#13083562	D₈	1	1	yes	characteristic
108	#13083561	—	4	2	no	—
109	#13083561	—	4	2	no	—
110	#13083561	—	4	2	no	—
111	#13083561	—	4	2	no	—
112	#13083561	—	4	2	no	—
113	#13083561	D₈	1	1	yes	characteristic
114	D₇₁₂	—	2	2	no	—
115	D₇₁₂	—	2	2	no	—
116	#13083559	—	4	2	no	—
117	#13083559	—	4	2	no	—
118	#13083559	D₈	1	1	yes	characteristic
119	#13083558	—	2	2	no	—
120	#13083558	—	4	2	no	—
121	#13083558	C₂×D₄	1	1	yes	characteristic
122	#13083557	—	2	1	no	—
123	C₂×#10623494	—	4	2	no	—
124	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
125	C₈×D₈₉	—	2	1	no	—
126	#13083551	—	8	2	no	—
127	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	#13083549	—	2	1	no	—
129	#13083549	—	2	1	no	—
130	#41×C₈₉	—	2	2	no	—
131	#41×C₈₉	—	4	2	no	—
132	#41×C₈₉	—	4	2	no	—
133	#41×C₈₉	—	8	2	no	—
134	#41×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₈₉	—	2	2	no	—
136	C₂×Q₈×C₈₉	—	4	2	no	—
137	C₂×Q₈×C₈₉	—	4	2	no	—
138	Q₁₆×C₈₉	—	2	2	no	—
139	Q₁₆×C₈₉	—	2	2	no	—
140	Q₁₆×C₈₉	—	4	2	no	—
141	Q₁₆×C₈₉	C₂×D₄	1	2	yes	—
142	SD₁₆×C₈₉	—	2	2	no	—
143	SD₁₆×C₈₉	—	4	2	no	—
144	D₈×C₈₉	C₂×D₄	1	2	yes	—
145	M₁₆×C₈₉	—	2	1	no	—
146	M₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₈₉	—	2	1	no	—
148	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
149	C₁₆×C₈₉	—	2	1	no	—
150	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
151	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
152	C₂²×D₈₉	—	4	2	no	—
153	C₂²×D₈₉	—	8	2	no	—
154	Q₈×C₈₉	—	2	2	no	—
155	Q₈×C₈₉	—	4	2	no	—
156	Q₈×C₈₉	—	4	2	no	—
157	Q₈×C₈₉	—	8	2	no	—
158	Q₈×C₈₉	C₂×D₈	1	1	yes	characteristic
159	D₄×C₈₉	—	2	1	no	—
160	D₄×C₈₉	—	2	2	no	—
161	D₄×C₈₉	—	4	2	no	—
162	D₄×C₈₉	—	8	2	no	—
163	D₄×C₈₉	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₈₉	—	2	1	no	—
165	C₂×C₄×C₈₉	—	4	2	no	—
166	C₂×C₄×C₈₉	—	4	2	no	—
167	C₂×C₄×C₈₉	—	8	2	no	—
168	C₂×C₄×C₈₉	—	8	2	no	—
169	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
170	C₂²⋊D₈₉	—	2	1	no	—
171	C₂²⋊D₈₉	—	8	2	no	—
172	C₂×Dic₈₉	—	2	1	no	—
173	D₃₅₆	—	4	2	no	—
174	D₃₅₆	—	4	2	no	—
175	D₃₅₆	—	8	2	no	—
176	D₃₅₆	C₂×D₈	1	1	yes	characteristic
177	C₄×D₈₉	—	2	1	no	—
178	C₄×D₈₉	—	8	2	no	—
179	Dic₁₇₈	C₂×D₈	1	1	yes	characteristic
180	C₈×C₈₉	—	2	1	no	—
181	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
183	#10623494	—	2	1	no	—
184	#10623494	—	4	2	no	—
185	#10623494	C₂×D₈	1	1	yes	characteristic
186	#10623494	C₂×D₈	1	1	yes	characteristic
187	C₂²×C₈₉	—	2	1	no	—
188	C₂²×C₈₉	—	4	2	no	—
189	C₂²×C₈₉	#576	1	1	yes	characteristic
190	C₂×D₈₉	—	2	1	no	—
191	C₂×D₈₉	—	8	2	no	—
192	C₂×D₈₉	—	16	2	no	—
193	C₂×D₈₉	—	32	2	no	—
194	C₄×C₈₉	—	2	1	no	—
195	C₄×C₈₉	—	4	2	no	—
196	C₄×C₈₉	—	8	2	no	—
197	C₄×C₈₉	D₄²	1	1	yes	characteristic
198	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
199	Dic₈₉	—	2	1	no	—
200	Dic₈₉	—	4	1	no	—
201	#33859	—	89	1	no	maximal, Sylow
202	C₂×C₈₉	—	2	1	no	—
203	C₂×C₈₉	—	4	1	no	—
204	C₂×C₈₉	—	8	2	no	—
205	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
206	D₈₉	—	4	1	no	—
207	D₈₉	—	16	2	no	—
208	#3417	D₈₉	1	1	yes	characteristic, complemented
209	#3415	—	89	1	no	—
210	#3290	—	89	2	no	—
211	#2249	—	89	1	no	—
212	#2222	—	89	1	no	—
213	#2221	—	89	1	no	—
214	#2212	—	89	2	no	—
215	#2194	—	89	2	no	—
216	#2193	—	89	2	no	—
217	#2178	—	89	1	no	—
218	#2170	—	89	1	no	—
219	C₈₉	#33859	1	1	yes	characteristic, complemented, Sylow
220	#578	C₂×D₈₉	1	2	yes	complemented
221	#577	—	178	2	no	—
222	#576	—	89	2	no	—
223	#510	—	2	2	no	—
224	#510	C₂×D₈₉	1	2	yes	complemented
225	#509	—	89	2	no	—
226	#509	—	178	2	no	—
227	#508	—	2	1	no	—
228	#508	—	178	1	no	—
229	#508	C₂×D₈₉	1	1	yes	characteristic
230	C₂×Q₃₂	—	2	1	no	—
231	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic, complemented
232	C₂×SD₃₂	—	89	2	no	—
233	C₂×D₁₆	—	89	1	no	—
234	C₂×D₁₆	—	178	1	no	—
235	#504	—	89	1	no	—
236	#504	C₂×D₈₉	1	1	yes	characteristic, complemented
237	#495	—	89	2	no	—
238	#473	—	89	2	no	—
239	#472	—	89	2	no	—
240	#471	—	178	2	no	—
241	#455	—	178	2	no	—
242	#454	—	178	2	no	—
243	#443	—	89	1	no	—
244	#443	—	89	2	no	—
245	#368	—	89	1	no	—
246	#364	—	89	1	no	—
247	#362	—	89	2	no	—
248	#361	—	89	2	no	—
249	#359	—	89	2	no	—
250	#350	—	89	1	no	—
251	#350	—	89	1	no	—
252	C₄×C₁₆	—	89	1	no	—
253	2₋¹⁺⁴	—	2	2	no	—
254	H(5,𝔽₂)	—	178	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	356	2	no	—
258	#137	—	178	2	no	—
259	AGL(1,ℤ/8)	—	356	2	no	—
260	AGL(1,ℤ/8)	—	356	2	no	—
261	AGL(1,ℤ/8)	—	178	2	no	—
262	AGL(1,ℤ/8)	—	178	2	no	—
263	#135	—	2	2	no	—
264	#135	—	89	1	no	—
265	#135	—	356	2	no	—
266	#135	—	178	2	no	—
267	#135	C₂²×D₈₉	1	2	yes	—
268	C₂×Q₁₆	—	2	2	no	—
269	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
270	C₂×SD₁₆	—	356	2	no	—
271	C₂×SD₁₆	—	178	2	no	—
272	C₂×D₈	—	89	2	no	—
273	C₂×D₈	—	178	2	no	—
274	#131	—	89	1	no	—
275	#131	—	178	2	no	—
276	#131	C₂²×D₈₉	1	1	yes	characteristic
277	#124	—	178	2	no	—
278	Q₃₂	—	2	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	356	2	no	—
284	SD₃₂	—	178	2	no	—
285	SD₃₂	C₂²⋊D₈₉	1	2	yes	—
286	D₁₆	—	178	2	no	—
287	D₁₆	—	178	1	no	—
288	D₁₆	—	178	1	no	—
289	D₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	89	1	no	—
292	M₃₂	—	178	1	no	—
293	M₃₂	C₂²×D₈₉	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	89	1	no	—
296	C₂×C₁₆	—	178	1	no	—
297	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
298	#108	—	89	1	no	—
299	#108	—	89	2	no	—
300	#104	—	89	1	no	—
301	#104	—	89	1	no	—
302	#104	—	178	2	no	—
303	#101	—	178	2	no	—
304	#100	—	178	2	no	—
305	C₄×C₈	—	89	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	8	2	no	—
310	#41	—	89	1	no	—
311	#41	—	356	2	no	—
312	#41	—	356	2	no	—
313	#41	D₄×D₈₉	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	356	2	no	—
318	C₂×D₄	—	356	2	no	—
319	C₂×D₄	—	178	2	no	—
320	C₂×D₄	—	712	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	89	1	no	—
325	Q₁₆	—	356	2	no	—
326	Q₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	89	1	no	—
330	SD₁₆	—	89	1	no	—
331	SD₁₆	—	356	2	no	—
332	SD₁₆	—	356	2	no	—
333	SD₁₆	—	356	2	no	—
334	SD₁₆	—	356	2	no	—
335	SD₁₆	—	356	2	no	—
336	D₈	—	89	1	no	—
337	D₈	—	356	2	no	—
338	D₈	—	356	2	no	—
339	D₈	—	178	2	no	—
340	D₈	—	178	2	no	—
341	D₈	C₂×C₂²⋊D₈₉	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	89	1	no	—
344	M₁₆	—	356	2	no	—
345	M₁₆	—	178	2	no	—
346	M₁₆	—	178	1	no	—
347	M₁₆	D₄×D₈₉	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	89	1	no	—
350	C₂×C₈	—	356	2	no	—
351	C₂×C₈	—	178	1	no	—
352	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
353	C₂²⋊C₄	—	712	2	no	—
354	C₄²	—	89	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	178	1	no	—
357	C₁₆	—	178	1	no	—
358	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
360	C₂³	—	356	2	no	—
361	C₂³	—	712	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	—	89	1	no	—
367	Q₈	D₈×D₈₉	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	89	1	no	—
373	D₄	—	356	2	no	—
374	D₄	—	356	2	no	—
375	D₄	—	178	1	no	—
376	D₄	—	712	2	no	—
377	D₄	—	712	2	no	—
378	D₄	D₈×D₈₉	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	178	1	no	—
385	C₂×C₄	—	178	1	no	—
386	C₂×C₄	—	712	2	no	—
387	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	89	1	no	—
390	C₈	—	89	1	no	—
391	C₈	—	356	2	no	—
392	C₈	—	178	1	no	—
393	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
394	C₈	#15668057	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	2848	2	no	—
398	C₂²	—	1424	2	no	—
399	C₂²	—	178	1	no	—
400	C₂²	—	712	2	no	—
401	C₂²	#19152399	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	—	356	1	no	—
406	C₄	—	178	1	no	—
407	C₄	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
408	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	356	1	no	—
413	C₂	—	1424	2	no	—
414	C₂	D₈×C₂²⋊D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3417	C₂
socle	C₂×C₈₉	D₄×D₈
hypercenter	#3417	D₈₉
hypocenter	C₈₉	#33859
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3417

Complements

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

N	H ≅ G/N	type
1. #26033390	C₂	semidirect
2. #26032038	C₂	semidirect
3. #26022514	C₂	semidirect
4. #26022374	C₂	semidirect
5. #26022372	C₂	semidirect
6. #26022323	C₂	semidirect
7. #26022220	C₂	semidirect
8. #26022216	C₂	semidirect
9. #26022133	C₂	semidirect
10. #26022087	C₂	semidirect
11. C₈₉×#3417	C₂	semidirect
17. C₂×#15667955	C₂²	semidirect
21. #19151983	C₂²	semidirect
23. #19151868	C₂²	semidirect
27. #19151715	C₂²	semidirect
29. #19151428	C₂²	semidirect
30. #19151418	C₂²	semidirect
31. #19151414	C₂²	semidirect
32. #19151412	C₂²	semidirect
33. #19151405	C₂²	semidirect
35. #19151382	C₂²	semidirect
36. C₁₆×Dic₈₉	C₂²	semidirect
37. #578×C₈₉	C₂²	semidirect
39. #510×C₈₉	C₂²	semidirect
43. C₂×Q₃₂×C₈₉	C₂²	semidirect
44. #504×C₈₉	C₂²	semidirect
208. #3417	D₈₉	semidirect
219. C₈₉	#33859	semidirect
220. #578	C₂×D₈₉	semidirect
224. #510	C₂×D₈₉	semidirect
231. C₂×Q₃₂	C₂×D₈₉	semidirect
236. #504	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033390	2
#26032038	2
#26022514	2
#26022374	2
#26022372	2
#26022323	2
#26022220	2
#26022216	2
#26022133	2
#26022087	2
C₈₉×#3417	2
#33859	89