Subgroups — #36711233

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	#26033402	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033390	C₂	1	1	yes	maximal, characteristic, complemented
3	#26032040	C₂	1	2	yes	maximal, complemented
4	#26022513	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022378	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022373	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022321	C₂	1	2	yes	maximal, complemented
8	#26022224	C₂	1	2	yes	maximal, complemented
9	#26022217	C₂	1	2	yes	maximal, complemented
10	#26022090	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152412	C₂²	1	2	yes	complemented
13	#19152410	—	2	2	no	—
14	#19152400	C₂²	1	2	yes	—
15	#19152038	C₂²	1	2	yes	complemented
16	#19152037	—	2	2	no	—
17	#19152035	—	2	2	no	—
18	#19152033	C₂²	1	2	yes	—
19	#19152031	—	2	1	no	—
20	#19152027	—	2	1	no	—
21	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic, complemented
22	Q₃₂×D₈₉	—	2	1	no	—
23	C₂×#15667957	C₂²	1	2	yes	complemented
24	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
25	D₁₆×D₈₉	—	2	1	no	—
26	#19152014	C₂²	1	1	yes	characteristic
27	#19152014	C₂²	1	1	yes	characteristic, complemented
28	#19151982	C₂²	1	2	yes	complemented
29	#19151874	C₂²	1	2	yes	—
30	#19151869	C₂²	1	2	yes	complemented
31	#19151861	—	2	2	no	—
32	#19151777	—	2	2	no	—
33	#19151772	—	2	2	no	—
34	#19151716	C₂²	1	1	yes	characteristic
35	#19151419	C₂²	1	1	yes	characteristic, complemented
36	#19151413	C₂²	1	2	yes	complemented
37	#19151410	C₂²	1	2	yes	complemented
38	#19151381	C₂²	1	1	yes	characteristic
39	#19151381	C₂²	1	1	yes	characteristic, complemented
40	#508×C₈₉	C₂²	1	1	yes	characteristic
41	#443×C₈₉	C₂²	1	2	yes	complemented
42	#368×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	#359×C₈₉	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
45	#15668059	—	2	2	no	—
46	#15668058	—	2	2	no	—
47	#15668045	—	2	2	no	—
48	#15668045	—	4	2	no	—
49	#15668042	—	2	2	no	—
50	#15668041	—	4	2	no	—
51	#15668040	—	2	2	no	—
52	#15668040	—	4	2	no	—
53	#15668039	—	4	2	no	—
54	#15668037	—	2	2	no	—
55	#15668035	—	2	2	no	—
56	#15668034	—	4	2	no	—
57	#15668034	D₄	1	1	yes	characteristic
58	#15668031	—	2	2	no	—
59	C₂×Dic₃₅₆	C₂³	1	2	yes	—
60	Q₁₆×D₈₉	—	2	2	no	—
61	C₂×#13083563	—	2	2	no	—
62	C₂×#13083563	—	4	2	no	—
63	C₂×D₇₁₂	C₂³	1	2	yes	—
64	D₈×D₈₉	—	2	2	no	—
65	#15668018	C₂³	1	1	yes	characteristic
66	#15668018	C₂³	1	1	yes	characteristic
67	#15668000	—	2	2	no	—
68	Dic₇₁₂	—	2	1	no	—
69	Dic₇₁₂	—	2	1	no	—
70	#15667958	—	2	2	no	—
71	#15667957	—	4	2	no	—
72	#15667956	—	2	2	no	—
73	#15667955	—	2	2	no	—
74	D₁₄₂₄	—	2	1	no	—
75	D₁₄₂₄	—	2	1	no	—
76	#15667953	—	2	2	no	—
77	#15667952	C₂³	1	1	yes	characteristic
78	#15667952	C₂³	1	1	yes	characteristic
79	#15667951	—	2	1	no	—
80	#15667951	—	2	1	no	—
81	C₁₆×D₈₉	—	2	1	no	—
82	C₁₆×D₈₉	—	2	1	no	—
83	#15667948	D₄	1	1	yes	characteristic
84	#15667940	D₄	1	1	yes	characteristic
85	#15667940	D₄	1	1	yes	characteristic
86	#15667930	—	2	2	no	—
87	#15667928	—	2	2	no	—
88	#135×C₈₉	C₂³	1	2	yes	—
89	#131×C₈₉	—	2	2	no	—
90	Q₃₂×C₈₉	D₄	1	1	yes	characteristic
91	SD₃₂×C₈₉	D₄	1	2	yes	—
92	D₁₆×C₈₉	D₄	1	1	yes	characteristic
93	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
95	#108×C₈₉	C₂³	1	2	yes	—
96	#104×C₈₉	—	2	2	no	—
97	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	#13083575	—	4	2	no	—
99	#13083574	—	4	2	no	—
100	#13083574	—	4	2	no	—
101	#13083574	C₂×D₄	1	1	yes	characteristic
102	#13083574	C₂×D₄	1	1	yes	characteristic
103	#13083573	—	4	2	no	—
104	#13083573	—	8	2	no	—
105	C₂×Dic₁₇₈	—	2	2	no	—
106	C₂×Dic₁₇₈	—	4	2	no	—
107	Q₈×D₈₉	—	4	2	no	—
108	C₂×D₃₅₆	—	2	2	no	—
109	C₂×D₃₅₆	—	4	2	no	—
110	D₄×D₈₉	—	4	2	no	—
111	D₄×D₈₉	—	8	2	no	—
112	Dic₃₅₆	—	2	2	no	—
113	Dic₃₅₆	—	2	2	no	—
114	Dic₃₅₆	—	4	2	no	—
115	Dic₃₅₆	D₈	1	1	yes	characteristic
116	#13083564	—	4	2	no	—
117	#13083563	—	4	2	no	—
118	#13083563	—	4	2	no	—
119	#13083563	—	4	2	no	—
120	#13083563	—	4	2	no	—
121	#13083563	D₈	1	1	yes	characteristic
122	#13083563	D₈	1	1	yes	characteristic
123	#13083562	—	4	2	no	—
124	#13083561	—	4	2	no	—
125	D₇₁₂	—	2	2	no	—
126	D₇₁₂	—	2	2	no	—
127	D₇₁₂	—	4	2	no	—
128	D₇₁₂	D₈	1	1	yes	characteristic
129	#13083559	—	4	2	no	—
130	#13083558	C₂×D₄	1	1	yes	characteristic
131	#13083558	C₂×D₄	1	1	yes	characteristic
132	#13083557	—	2	1	no	—
133	#13083557	—	2	1	no	—
134	C₈×D₈₉	—	2	1	no	—
135	C₈×D₈₉	—	2	1	no	—
136	#13083551	—	8	2	no	—
137	#13083549	—	2	1	no	—
138	#13083549	—	2	1	no	—
139	#41×C₈₉	—	2	2	no	—
140	Q₁₆×C₈₉	C₂×D₄	1	2	yes	—
141	SD₁₆×C₈₉	—	2	2	no	—
142	D₈×C₈₉	C₂×D₄	1	2	yes	—
143	M₁₆×C₈₉	—	2	2	no	—
144	M₁₆×C₈₉	—	4	2	no	—
145	C₂×C₈×C₈₉	—	4	2	no	—
146	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
148	C₄²×C₈₉	C₂×D₄	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	D₄×C₈₉	—	2	2	no	—
156	D₄×C₈₉	—	4	2	no	—
157	C₂×C₄×C₈₉	—	2	1	no	—
158	C₂×C₄×C₈₉	—	4	2	no	—
159	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₈₉	—	2	1	no	—
161	C₂²⋊D₈₉	—	2	1	no	—
162	C₂²⋊D₈₉	—	8	2	no	—
163	C₂²⋊D₈₉	—	8	2	no	—
164	C₂×Dic₈₉	—	4	2	no	—
165	C₂×Dic₈₉	—	8	2	no	—
166	D₃₅₆	—	4	2	no	—
167	D₃₅₆	—	4	2	no	—
168	D₃₅₆	—	8	2	no	—
169	D₃₅₆	C₂×D₈	1	1	yes	characteristic
170	D₃₅₆	C₂×D₈	1	1	yes	characteristic
171	C₄×D₈₉	—	2	1	no	—
172	C₄×D₈₉	—	2	1	no	—
173	C₄×D₈₉	—	8	2	no	—
174	C₄×D₈₉	—	8	2	no	—
175	Dic₁₇₈	—	4	2	no	—
176	Dic₁₇₈	—	4	2	no	—
177	Dic₁₇₈	—	8	2	no	—
178	Dic₁₇₈	C₂×D₈	1	1	yes	characteristic
179	Dic₁₇₈	C₂×D₈	1	1	yes	characteristic
180	C₈×C₈₉	—	4	2	no	—
181	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
182	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
183	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
184	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
185	#10623494	—	2	1	no	—
186	#10623494	—	2	1	no	—
187	C₂²×C₈₉	—	4	2	no	—
188	C₂²×C₈₉	#576	1	1	yes	characteristic
189	C₂×D₈₉	—	2	1	no	—
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	—	4	1	no	—
195	C₄×C₈₉	—	4	2	no	—
196	C₄×C₈₉	D₄²	1	1	yes	characteristic
197	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
198	Dic₈₉	—	2	1	no	—
199	Dic₈₉	—	2	1	no	—
200	Dic₈₉	—	8	2	no	—
201	#33859	—	89	1	no	maximal, Sylow
202	C₂×C₈₉	—	2	1	no	—
203	C₂×C₈₉	—	8	2	no	—
204	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
205	D₈₉	—	4	1	no	—
206	D₈₉	—	4	1	no	—
207	D₈₉	—	16	2	no	—
208	#3417	—	89	1	no	—
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	D₈₉	1	1	yes	characteristic, complemented
218	#2170	—	89	1	no	—
219	C₈₉	#33859	1	1	yes	characteristic, complemented, Sylow
220	#578	—	89	2	no	—
221	#577	—	178	2	no	—
222	#576	—	89	2	no	—
223	#510	—	89	2	no	—
224	#510	—	178	2	no	—
225	#509	—	89	2	no	—
226	#509	—	178	2	no	—
227	#508	—	178	1	no	—
228	#508	—	178	1	no	—
229	#508	C₂×D₈₉	1	1	yes	characteristic
230	C₂×Q₃₂	—	89	1	no	—
231	C₂×Q₃₂	—	178	1	no	—
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	—	89	1	no	—
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	C₂×D₈₉	1	2	yes	complemented
245	#368	C₂×D₈₉	1	1	yes	characteristic, complemented
246	#364	—	89	1	no	—
247	#362	—	89	2	no	—
248	#361	—	89	2	no	—
249	#359	C₂×D₈₉	1	2	yes	complemented
250	#350	—	89	1	no	—
251	#350	—	89	1	no	—
252	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
253	2₋¹⁺⁴	—	178	2	no	—
254	H(5,𝔽₂)	—	178	2	no	—
255	#137	—	356	2	no	—
256	#137	—	356	2	no	—
257	#137	—	178	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	—	89	1	no	—
264	#135	—	356	2	no	—
265	#135	—	178	2	no	—
266	#135	—	178	2	no	—
267	#135	C₂²×D₈₉	1	2	yes	—
268	C₂×Q₁₆	—	89	2	no	—
269	C₂×Q₁₆	—	178	2	no	—
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	—	2	2	no	—
275	#131	—	89	1	no	—
276	#131	—	89	1	no	—
277	#124	—	178	2	no	—
278	Q₃₂	—	178	2	no	—
279	Q₃₂	—	178	1	no	—
280	Q₃₂	—	178	1	no	—
281	Q₃₂	D₃₅₆	1	1	yes	characteristic
282	SD₃₂	—	356	2	no	—
283	SD₃₂	—	178	2	no	—
284	SD₃₂	—	178	2	no	—
285	SD₃₂	D₃₅₆	1	2	yes	—
286	D₁₆	—	178	2	no	—
287	D₁₆	—	178	1	no	—
288	D₁₆	—	178	1	no	—
289	D₁₆	D₃₅₆	1	1	yes	characteristic
290	M₃₂	—	89	1	no	—
291	M₃₂	—	89	1	no	—
292	M₃₂	—	178	1	no	—
293	M₃₂	—	178	1	no	—
294	C₂×C₁₆	—	178	1	no	—
295	C₂×C₁₆	—	178	1	no	—
296	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
297	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
298	#108	—	89	1	no	—
299	#108	C₂²×D₈₉	1	2	yes	—
300	#104	—	2	2	no	—
301	#104	—	89	1	no	—
302	#104	—	89	1	no	—
303	#101	—	178	2	no	—
304	#100	—	178	2	no	—
305	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
306	#41	—	2	2	no	—
307	#41	—	89	1	no	—
308	#41	—	89	1	no	—
309	#41	—	356	2	no	—
310	#41	—	356	2	no	—
311	#41	—	356	2	no	—
312	#41	—	356	2	no	—
313	#41	—	712	2	no	—
314	C₂×Q₈	—	356	2	no	—
315	C₂×Q₈	—	356	2	no	—
316	C₂×Q₈	—	178	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	—	89	1	no	—
322	Q₁₆	—	356	2	no	—
323	Q₁₆	—	356	2	no	—
324	Q₁₆	—	178	2	no	—
325	Q₁₆	—	178	2	no	—
326	Q₁₆	C₂×D₃₅₆	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	89	1	no	—
329	SD₁₆	—	89	1	no	—
330	SD₁₆	—	356	2	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×D₃₅₆	1	2	yes	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	4	2	no	—
344	M₁₆	—	89	1	no	—
345	M₁₆	—	89	1	no	—
346	M₁₆	—	178	1	no	—
347	M₁₆	—	178	1	no	—
348	C₂×C₈	—	4	2	no	—
349	C₂×C₈	—	178	1	no	—
350	C₂×C₈	—	178	1	no	—
351	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
352	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
353	C₂²⋊C₄	—	712	2	no	—
354	C₄²	D₄×D₈₉	1	1	yes	characteristic
355	C₁₆	—	2	1	no	—
356	C₁₆	—	178	1	no	—
357	C₁₆	—	178	1	no	—
358	C₁₆	C₂×D₃₅₆	1	1	yes	characteristic
359	C₁₆	C₂×D₃₅₆	1	1	yes	characteristic
360	C₂³	—	356	2	no	—
361	C₂³	—	712	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	89	1	no	—
364	Q₈	—	89	1	no	—
365	Q₈	—	356	2	no	—
366	Q₈	—	356	2	no	—
367	Q₈	—	712	2	no	—
368	D₄	—	2	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	89	1	no	—
371	D₄	—	89	1	no	—
372	D₄	—	356	2	no	—
373	D₄	—	356	2	no	—
374	D₄	—	178	1	no	—
375	D₄	—	178	1	no	—
376	D₄	—	712	2	no	—
377	D₄	—	712	2	no	—
378	D₄	—	712	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	356	2	no	—
382	C₂×C₄	—	178	1	no	—
383	C₂×C₄	—	178	1	no	—
384	C₂×C₄	—	712	2	no	—
385	C₂×C₄	—	712	2	no	—
386	C₂×C₄	—	712	2	no	—
387	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
388	C₈	—	4	2	no	—
389	C₈	—	178	1	no	—
390	C₈	—	178	1	no	—
391	C₈	D₈×D₈₉	1	1	yes	characteristic
392	C₈	D₈×D₈₉	1	1	yes	characteristic
393	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
394	C₈	#15668058	1	1	yes	characteristic
395	C₂²	—	4	2	no	—
396	C₂²	—	2848	2	no	—
397	C₂²	—	1424	2	no	—
398	C₂²	—	178	1	no	—
399	C₂²	—	178	1	no	—
400	C₂²	—	712	2	no	—
401	C₂²	#19152400	1	1	yes	characteristic
402	C₄	—	4	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	178	1	no	—
405	C₄	—	178	1	no	—
406	C₄	—	712	2	no	—
407	C₄	D₄×D₃₅₆	1	1	yes	characteristic
408	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	8	2	no	—
411	C₂	—	356	1	no	—
412	C₂	—	356	1	no	—
413	C₂	—	1424	2	no	—
414	C₂	D₈×D₃₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₅₆
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2178	C₂
socle	C₂×C₈₉	D₄×D₈
hypercenter	#2178	D₈₉
hypocenter	C₈₉	#33859
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
1. #26033402	C₂	semidirect
2. #26033390	C₂	semidirect
3. #26032040	C₂	semidirect
4. #26022513	C₂	semidirect
5. #26022378	C₂	semidirect
6. #26022373	C₂	semidirect
7. #26022321	C₂	semidirect
8. #26022224	C₂	semidirect
9. #26022217	C₂	semidirect
10. #26022090	C₂	semidirect
11. C₈₉×#2178	C₂	semidirect
12. #19152412	C₂²	semidirect
15. #19152038	C₂²	semidirect
21. C₂×Dic₇₁₂	C₂²	semidirect
23. C₂×#15667957	C₂²	semidirect
27. #19152014	C₂²	semidirect
28. #19151982	C₂²	semidirect
30. #19151869	C₂²	semidirect
35. #19151419	C₂²	semidirect
36. #19151413	C₂²	semidirect
37. #19151410	C₂²	semidirect
39. #19151381	C₂²	semidirect
41. #443×C₈₉	C₂²	semidirect
42. #368×C₈₉	C₂²	semidirect
43. #359×C₈₉	C₂²	semidirect
44. C₄×C₁₆×C₈₉	C₂²	semidirect
217. #2178	D₈₉	semidirect
219. C₈₉	#33859	semidirect
244. #443	C₂×D₈₉	semidirect
245. #368	C₂×D₈₉	semidirect
249. #359	C₂×D₈₉	semidirect
252. 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
#26033402	2
#26033390	2
#26032040	2
#26022513	2
#26022378	2
#26022373	2
#26022321	2
#26022224	2
#26022217	2
#26022090	2
C₈₉×#2178	2
#33859	89