Subgroups — #36709266

Order 22784.

588 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	#26028852	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022440	C₂	1	2	yes	maximal, complemented
3	#26022440	C₂	1	2	yes	maximal, complemented
4	#26022435	C₂	1	2	yes	maximal, complemented
5	#26022430	C₂	1	2	yes	maximal, complemented
6	#26022041	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022034	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021968	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19151399	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021866	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3058	C₂	1	1	yes	maximal, characteristic
12	#19152081	—	2	1	no	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	#19151640	—	2	1	no	—
16	#19151640	—	2	2	no	—
17	#19151636	C₂²	1	1	yes	characteristic
18	#19151636	C₂²	1	2	yes	complemented
19	C₂×#15667946	C₂²	1	1	yes	characteristic
20	C₂×#15667946	C₂²	1	2	yes	complemented
21	#19151425	C₂²	1	1	yes	characteristic, complemented
22	#19151425	C₂²	1	2	yes	complemented
23	#19151425	C₂²	1	1	yes	characteristic, complemented
24	#19151423	C₂²	1	1	yes	characteristic, complemented
25	#19151423	C₂²	1	2	yes	complemented
26	#19151423	C₂²	1	1	yes	characteristic, complemented
27	#19151399	C₂²	1	2	yes	—
28	#19151399	C₂²	1	2	yes	—
29	#19151399	C₂²	1	2	yes	—
30	#19151399	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	2	yes	complemented
32	C₂²⋊#13083549	C₂²	1	2	yes	complemented
33	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
35	#535×C₈₉	—	2	1	no	—
36	#469×C₈₉	—	2	2	no	—
37	#466×C₈₉	C₂²	1	2	yes	—
38	#466×C₈₉	C₂²	1	2	yes	—
39	#427×C₈₉	C₂²	1	1	yes	characteristic
40	#418×C₈₉	—	2	1	no	—
41	Q₈×Dic₈₉	—	4	1	no	—
42	D₄×Dic₈₉	—	4	2	no	—
43	D₄×Dic₈₉	—	4	1	no	—
44	#15667971	—	2	1	no	—
45	#15667971	—	2	2	no	—
46	C₂×C₄⋊Dic₈₉	—	2	1	no	—
47	C₂×C₄⋊Dic₈₉	—	2	2	no	—
48	#15667952	—	2	2	no	—
49	C₂×#13083549	—	2	2	no	—
50	C₂×#13083549	C₂³	1	1	yes	characteristic
51	C₂×#13083549	C₂³	1	2	yes	—
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	#15667946	—	2	1	no	—
54	#15667946	—	2	2	no	—
55	#15667946	C₂³	1	2	yes	—
56	#15667946	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667944	—	2	1	no	—
59	#15667944	—	2	2	no	—
60	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₈₉	—	2	1	no	—
62	C₂×D₈×C₈₉	—	2	1	no	—
63	C₂×D₈×C₈₉	—	2	2	no	—
64	C₂×D₈×C₈₉	—	2	2	no	—
65	C₂×D₈×C₈₉	C₂³	1	2	yes	—
66	#131×C₈₉	—	2	2	no	—
67	#131×C₈₉	—	2	2	no	—
68	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
69	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
70	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	#127×C₈₉	—	4	1	no	—
73	#124×C₈₉	—	4	1	no	—
74	#121×C₈₉	—	2	2	no	—
75	#121×C₈₉	—	2	2	no	—
76	#120×C₈₉	—	4	2	no	—
77	#117×C₈₉	—	2	1	no	—
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#106×C₈₉	—	2	1	no	—
81	#102×C₈₉	—	2	2	no	—
82	#102×C₈₉	—	2	2	no	—
83	C₂²×Dic₈₉	—	4	2	no	—
84	C₂²×Dic₈₉	—	4	1	no	—
85	C₄⋊Dic₈₉	—	2	1	no	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	4	2	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₂²⋊Dic₈₉	—	4	2	no	—
91	C₂²⋊Dic₈₉	—	4	1	no	—
92	C₄×Dic₈₉	—	4	2	no	—
93	C₄×Dic₈₉	—	4	1	no	—
94	#13083549	—	2	1	no	—
95	#13083549	—	2	1	no	—
96	#13083549	—	2	2	no	—
97	C₂⁴×C₈₉	—	4	1	no	—
98	#41×C₈₉	—	2	2	no	—
99	#41×C₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
101	C₂×D₄×C₈₉	—	2	2	no	—
102	C₂×D₄×C₈₉	—	4	2	no	—
103	C₂×D₄×C₈₉	—	4	2	no	—
104	C₂×D₄×C₈₉	—	4	2	no	—
105	C₂×D₄×C₈₉	—	4	1	no	—
106	C₂×D₄×C₈₉	—	4	2	no	—
107	C₂×D₄×C₈₉	D₈	1	2	yes	—
108	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	D₈×C₈₉	—	2	2	no	—
112	D₈×C₈₉	—	2	2	no	—
113	D₈×C₈₉	—	4	2	no	—
114	M₁₆×C₈₉	—	2	2	no	—
115	M₁₆×C₈₉	—	2	4	no	—
116	C₂×C₈×C₈₉	—	2	2	no	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	#41	1	2	yes	—
121	C₂×C₈×C₈₉	#41	1	2	yes	—
122	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₄⋊C₄×C₈₉	—	2	1	no	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₂²⋊C₄×C₈₉	—	4	1	no	—
126	C₂²⋊C₄×C₈₉	—	4	2	no	—
127	C₂²⋊C₄×C₈₉	—	4	2	no	—
128	C₄²×C₈₉	—	4	1	no	—
129	C₂³×C₈₉	—	4	2	no	—
130	C₂³×C₈₉	—	8	2	no	—
131	C₂³×C₈₉	—	16	1	no	—
132	C₂³×C₈₉	—	16	1	no	—
133	C₂³×C₈₉	C₂×D₈	1	2	yes	—
134	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
135	Q₈×C₈₉	—	2	2	no	—
136	D₄×C₈₉	—	2	4	no	—
137	D₄×C₈₉	—	2	2	no	—
138	D₄×C₈₉	—	4	2	no	—
139	D₄×C₈₉	—	4	2	no	—
140	D₄×C₈₉	—	8	2	no	—
141	D₄×C₈₉	—	16	2	no	—
142	D₄×C₈₉	—	16	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	#123	1	2	yes	—
148	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#137	1	2	yes	—
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₈₉	—	4	1	no	—
153	C₂×Dic₈₉	—	4	2	no	—
154	C₂×Dic₈₉	—	4	1	no	—
155	C₂×Dic₈₉	—	16	1	no	—
156	C₂×Dic₈₉	—	16	2	no	—
157	C₈×C₈₉	—	2	2	no	—
158	C₈×C₈₉	#123	1	2	yes	—
159	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	—	2	2	no	—
164	C₂²×C₈₉	—	2	2	no	—
165	C₂²×C₈₉	—	2	2	no	—
166	C₂²×C₈₉	—	8	2	no	—
167	C₂²×C₈₉	—	8	2	no	—
168	C₂²×C₈₉	—	8	2	no	—
169	C₂²×C₈₉	—	16	4	no	—
170	C₂²×C₈₉	#480	1	2	yes	—
171	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₈₉	—	2	2	no	—
173	C₄×C₈₉	—	8	1	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	#480	1	2	yes	—
176	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
177	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
178	Dic₈₉	—	8	2	no	—
179	Dic₈₉	—	8	1	no	—
180	Dic₈₉	—	8	1	no	—
181	#33650	—	89	1	no	maximal, Sylow
182	C₂×C₈₉	—	2	2	no	—
183	C₂×C₈₉	—	4	2	no	—
184	C₂×C₈₉	—	16	2	no	—
185	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
186	C₂×C₈₉	#3415	1	1	yes	characteristic
187	C₂×C₈₉	#3416	1	1	yes	characteristic
188	#3058	D₈₉	1	1	yes	characteristic
189	#2913	—	89	1	no	—
190	#2234	—	89	2	no	—
191	#2234	—	89	2	no	—
192	#2233	—	89	2	no	—
193	#2232	—	89	2	no	—
194	#2159	—	89	1	no	—
195	#2158	—	89	1	no	—
196	#2140	—	89	1	no	—
197	C₂×#357	—	89	1	no	—
198	#2113	—	89	1	no	—
199	C₈₉	#33650	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
201	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
202	#535	—	2	1	no	—
203	#520	—	178	1	no	—
204	C₂×M₃₂	—	89	1	no	—
205	C₂²×C₁₆	—	89	1	no	—
206	#469	—	2	2	no	—
207	#466	C₂×D₈₉	1	2	yes	—
208	#466	C₂×D₈₉	1	2	yes	—
209	#428	—	178	1	no	—
210	#428	—	178	2	no	—
211	#427	—	89	1	no	—
212	#427	—	89	2	no	—
213	#427	C₂×D₈₉	1	1	yes	characteristic
214	C₂×#107	—	89	1	no	—
215	C₂×#107	—	89	2	no	—
216	#418	—	2	1	no	—
217	#367	—	89	1	no	—
218	#367	—	89	1	no	—
219	#367	—	89	2	no	—
220	#366	—	89	1	no	—
221	#366	—	89	1	no	—
222	#366	—	89	2	no	—
223	#357	—	89	2	no	—
224	#357	—	89	2	no	—
225	#357	—	89	2	no	—
226	#357	—	89	2	no	—
227	C₂²⋊C₁₆	—	89	2	no	—
228	C₂²⋊C₁₆	—	89	2	no	—
229	C₂×#41	C₂²⋊D₈₉	1	1	yes	characteristic
230	C₂²×D₄	—	2	1	no	—
231	C₂×D₈	—	2	1	no	—
232	C₂×D₈	—	2	2	no	—
233	C₂×D₈	—	2	2	no	—
234	C₂×D₈	C₂²×D₈₉	1	2	yes	—
235	#131	—	2	2	no	—
236	#131	—	2	2	no	—
237	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
238	C₂×M₁₆	C₂²⋊D₈₉	1	2	yes	—
239	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
240	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	#127	—	4	1	no	—
242	#124	—	4	1	no	—
243	#121	—	2	2	no	—
244	#121	—	2	2	no	—
245	#120	—	4	2	no	—
246	C₄×Q₈	—	356	1	no	—
247	C₄×D₄	—	356	2	no	—
248	C₄×D₄	—	356	1	no	—
249	#117	—	2	1	no	—
250	#117	—	178	2	no	—
251	#117	—	178	1	no	—
252	C₂×C₄⋊C₄	—	178	2	no	—
253	C₂×C₄⋊C₄	—	178	1	no	—
254	M₃₂	—	178	2	no	—
255	C₂×C₁₆	—	89	1	no	—
256	C₂×C₁₆	—	89	2	no	—
257	C₂×C₁₆	—	89	1	no	—
258	C₂×C₁₆	—	178	2	no	—
259	#107	—	89	2	no	—
260	#107	—	89	1	no	—
261	#107	—	89	1	no	—
262	#107	—	178	2	no	—
263	#107	—	178	1	no	—
264	#107	C₂²×D₈₉	1	1	yes	characteristic
265	#107	C₂²×D₈₉	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	178	2	no	—
268	#106	—	178	1	no	—
269	#102	—	2	2	no	—
270	#102	—	2	2	no	—
271	C₂⁴	—	4	1	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	C₂×Q₈	#13083559	1	1	yes	characteristic
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	#13083559	1	2	yes	—
282	C₂×D₄	#13083559	1	1	yes	characteristic
283	C₂²×C₄	—	356	2	no	—
284	C₂²×C₄	—	356	1	no	—
285	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
286	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	D₈	—	2	2	no	—
288	D₈	—	2	2	no	—
289	D₈	—	4	2	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
295	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₂×C₈	#13083573	1	2	yes	—
297	C₂×C₈	#13083573	1	2	yes	—
298	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	356	2	no	—
302	C₄⋊C₄	—	356	1	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₄⋊C₄	—	178	2	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	356	2	no	—
310	C₂²⋊C₄	—	356	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	356	2	no	—
313	C₄²	—	356	1	no	—
314	C₁₆	—	178	1	no	—
315	C₁₆	—	178	1	no	—
316	C₁₆	—	178	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	8	2	no	—
319	C₂³	—	16	1	no	—
320	C₂³	—	16	1	no	—
321	C₂³	C₂×#13083559	1	2	yes	—
322	C₂³	C₂×#13083559	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	D₄	—	2	4	no	—
325	D₄	—	2	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	16	2	no	—
330	D₄	—	16	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	356	1	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	356	1	no	—
338	C₂×C₄	—	1424	1	no	—
339	C₂×C₄	—	1424	2	no	—
340	C₂×C₄	#15667993	1	2	yes	—
341	C₂×C₄	C₂×#13083559	1	2	yes	—
342	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
343	C₂×C₄	#15668042	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	#15667993	1	2	yes	—
347	C₈	C₂×#13083573	1	1	yes	characteristic
348	C₈	C₂×#13083573	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	16	4	no	—
358	C₂²	#19151909	1	2	yes	—
359	C₂²	C₂²×#13083559	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	712	2	no	—
364	C₄	—	712	1	no	—
365	C₄	—	712	1	no	—
366	C₄	#19151909	1	2	yes	—
367	C₄	C₂×#15667993	1	1	yes	characteristic
368	C₄	C₂×#15668042	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	4	2	no	—
371	C₂	—	16	2	no	—
372	C₂	C₂×#19151909	1	1	yes	characteristic
373	C₂	#26033391	1	1	yes	characteristic
374	C₂	#26033399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3058	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3058	D₈₉
hypocenter	C₈₉	#33650
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3058

Complements

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

N	H ≅ G/N	type
1. #26028852	C₂	semidirect
2. #26022440	C₂	semidirect
3. #26022440	C₂	semidirect
4. #26022435	C₂	semidirect
5. #26022430	C₂	semidirect
6. #26022041	C₂	semidirect
7. #26022034	C₂	semidirect
8. #26021968	C₂	semidirect
9. C₂×#19151399	C₂	semidirect
10. #26021866	C₂	semidirect
18. #19151636	C₂²	semidirect
20. C₂×#15667946	C₂²	semidirect
21. #19151425	C₂²	semidirect
22. #19151425	C₂²	semidirect
23. #19151425	C₂²	semidirect
24. #19151423	C₂²	semidirect
25. #19151423	C₂²	semidirect
26. #19151423	C₂²	semidirect
31. C₂²⋊#13083549	C₂²	semidirect
32. C₂²⋊#13083549	C₂²	semidirect
199. C₈₉	#33650	semidirect

Maximal subgroups

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

subgroup	index
#26028852	2
#26022440	2
#26022435	2
#26022430	2
#26022041	2
#26022034	2
#26021968	2
C₂×#19151399	2
#26021866	2
C₈₉×#3058	2
#33650	89