Subgroups — #36709764

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030792	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030739	C₂	1	1	yes	maximal, characteristic
3	#26022492	C₂	1	2	yes	maximal, complemented
4	#26022490	C₂	1	2	yes	maximal, complemented
5	#26022485	C₂	1	2	yes	maximal, complemented
6	#26022474	C₂	1	2	yes	maximal, complemented
7	#26021967	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021964	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19151400	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021848	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668037	—	2	1	no	—
13	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
14	#19152226	—	2	1	no	—
15	#19152162	—	2	1	no	—
16	C₂×#15667952	C₂²	1	1	yes	characteristic
17	#19152005	—	2	2	no	—
18	#19152002	C₂²	1	2	yes	complemented
19	#19151999	C₂²	1	2	yes	complemented
20	#19151984	—	2	2	no	—
21	#19151983	C₂²	1	2	yes	—
22	#19151973	C₂²	1	2	yes	—
23	#19151641	—	2	1	no	—
24	C₂×#15667946	C₂²	1	1	yes	characteristic
25	#19151553	C₂²	1	1	yes	characteristic
26	#19151402	C₂²	1	2	yes	complemented
27	#19151400	C₂²	1	2	yes	—
28	#19151400	C₂²	1	2	yes	—
29	#19151399	C₂²	1	2	yes	complemented
30	#19151398	C₂²	1	2	yes	—
31	#19151398	C₂²	1	2	yes	—
32	#19151373	C₂²	1	2	yes	complemented
33	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
34	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	#427×C₈₉	C₂²	1	1	yes	characteristic
37	#358×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#358×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	#357×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	#357×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₅₆	—	2	1	no	—
42	#15668037	—	4	4	no	—
43	C₂×#13083561	—	2	2	no	—
44	C₂×D₇₁₂	—	2	1	no	—
45	C₂×D₇₁₂	—	2	2	no	—
46	C₂×D₇₁₂	—	2	2	no	—
47	C₂×D₇₁₂	C₂³	1	2	yes	—
48	C₂×#13083559	—	2	2	no	—
49	C₂×#13083558	D₄	1	1	yes	characteristic
50	#15668012	—	4	1	no	—
51	#15668011	—	2	2	no	—
52	#15668008	—	2	2	no	—
53	#15668004	—	4	1	no	—
54	#15668003	—	2	2	no	—
55	#15668001	—	2	2	no	—
56	#15667989	—	4	2	no	—
57	#15667986	—	4	1	no	—
58	#15667986	—	4	1	no	—
59	#15667980	—	4	2	no	—
60	#15667971	D₄	1	1	yes	characteristic
61	C₂×C₄⋊Dic₈₉	—	2	1	no	—
62	#15667952	—	2	2	no	—
63	C₂×#13083549	C₂³	1	2	yes	—
64	#15667946	—	2	1	no	—
65	#15667946	C₂³	1	2	yes	—
66	#15667945	—	2	2	no	—
67	#15667943	—	2	2	no	—
68	#15667922	D₄	1	2	yes	—
69	C₈×Dic₈₉	C₂³	1	2	yes	—
70	C₂×#41×C₈₉	—	2	1	no	—
71	#135×C₈₉	—	2	2	no	—
72	#135×C₈₉	—	2	2	no	—
73	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×SD₁₆×C₈₉	—	2	1	no	—
75	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	#117×C₈₉	—	2	1	no	—
78	C₂×C₁₆×C₈₉	—	2	2	no	—
79	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
81	#107×C₈₉	C₂³	1	1	yes	characteristic
82	#107×C₈₉	C₂³	1	1	yes	characteristic
83	#106×C₈₉	—	2	1	no	—
84	C₂³×D₈₉	—	4	1	no	—
85	C₂×Dic₁₇₈	—	4	2	no	—
86	C₂×D₃₅₆	—	2	2	no	—
87	C₂×D₃₅₆	—	4	2	no	—
88	C₂×D₃₅₆	—	4	2	no	—
89	C₂×D₃₅₆	—	4	1	no	—
90	C₂×C₂²⋊D₈₉	—	4	2	no	—
91	C₂×C₂²⋊D₈₉	—	4	2	no	—
92	C₂²×Dic₈₉	—	4	1	no	—
93	#13083561	—	4	4	no	—
94	D₇₁₂	—	2	2	no	—
95	D₇₁₂	—	2	2	no	—
96	D₇₁₂	—	4	2	no	—
97	#13083559	—	4	4	no	—
98	#13083558	SD₁₆	1	4	yes	—
99	C₂×#10623494	C₂×D₄	1	2	yes	—
100	C₄⋊Dic₈₉	—	2	2	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	C₄⋊Dic₈₉	—	4	2	no	—
103	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
104	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
105	#13083553	—	4	2	no	—
106	#13083553	—	4	2	no	—
107	C₂²⋊Dic₈₉	D₈	1	2	yes	—
108	#13083551	—	4	2	no	—
109	#13083551	—	4	2	no	—
110	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
111	#13083549	—	2	2	no	—
112	#41×C₈₉	—	4	2	no	—
113	#41×C₈₉	—	4	2	no	—
114	C₂×Q₈×C₈₉	—	2	1	no	—
115	C₂×D₄×C₈₉	—	2	1	no	—
116	C₂×D₄×C₈₉	—	4	1	no	—
117	C₂²×C₄×C₈₉	—	4	1	no	—
118	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	Q₁₆×C₈₉	—	2	1	no	—
120	Q₁₆×C₈₉	—	2	1	no	—
121	SD₁₆×C₈₉	—	4	2	no	—
122	D₈×C₈₉	—	2	1	no	—
123	D₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₈₉	#41	1	2	yes	—
126	C₂×C₈×C₈₉	#41	1	2	yes	—
127	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₈₉	—	2	1	no	—
129	C₄⋊C₄×C₈₉	—	2	1	no	—
130	C₂²⋊C₄×C₈₉	—	4	1	no	—
131	C₄²×C₈₉	—	4	1	no	—
132	C₁₆×C₈₉	—	2	1	no	—
133	C₁₆×C₈₉	—	2	1	no	—
134	C₂³×C₈₉	—	4	1	no	—
135	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₈₉	—	4	2	no	—
137	C₂²×D₈₉	—	8	2	no	—
138	C₂²×D₈₉	—	16	1	no	—
139	C₂²×D₈₉	—	16	1	no	—
140	Q₈×C₈₉	—	4	1	no	—
141	Q₈×C₈₉	—	4	1	no	—
142	D₄×C₈₉	—	4	1	no	—
143	D₄×C₈₉	—	4	1	no	—
144	D₄×C₈₉	—	8	2	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	8	2	no	—
149	C₂×C₄×C₈₉	#124	1	2	yes	—
150	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₈₉	—	16	2	no	—
154	C₂²⋊D₈₉	—	16	2	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	C₂×Dic₈₉	—	16	1	no	—
157	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
158	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
159	D₃₅₆	—	4	2	no	—
160	D₃₅₆	—	4	2	no	—
161	D₃₅₆	—	8	2	no	—
162	Dic₁₇₈	—	16	2	no	—
163	C₈×C₈₉	#124	1	2	yes	—
164	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
165	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
166	#10623494	C₂×SD₁₆	1	4	yes	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	—	8	1	no	—
170	C₂²×C₈₉	—	8	1	no	—
171	C₂²×C₈₉	—	8	1	no	—
172	C₂²×C₈₉	#486	1	2	yes	—
173	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₈₉	—	8	2	no	—
175	C₂×D₈₉	—	8	2	no	—
176	C₂×D₈₉	—	8	2	no	—
177	C₂×D₈₉	—	16	4	no	—
178	C₄×C₈₉	—	8	1	no	—
179	C₄×C₈₉	—	8	1	no	—
180	C₄×C₈₉	—	8	1	no	—
181	C₄×C₈₉	#486	1	2	yes	—
182	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
183	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
184	Dic₈₉	—	2	2	no	—
185	Dic₈₉	—	2	2	no	—
186	Dic₈₉	—	8	2	no	—
187	#33702	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	2	2	no	—
189	C₂×C₈₉	—	16	1	no	—
190	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
191	C₂×C₈₉	#3415	1	1	yes	characteristic
192	C₂×C₈₉	#3416	1	1	yes	characteristic
193	D₈₉	—	16	2	no	—
194	#3164	—	89	1	no	—
195	#3155	—	89	1	no	—
196	#2244	—	89	2	no	—
197	#2243	—	89	2	no	—
198	#2242	—	89	2	no	—
199	#2240	—	89	2	no	—
200	#2140	—	89	1	no	—
201	#2139	—	89	1	no	—
202	#2138	D₈₉	1	1	yes	characteristic, complemented
203	C₂×#357	—	89	1	no	—
204	#2108	—	89	1	no	—
205	C₈₉	#33702	1	1	yes	characteristic, complemented, Sylow
206	C₂×AGL(1,ℤ/8)	—	178	1	no	—
207	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
208	C₂²×D₈	—	89	1	no	—
209	#544	—	178	1	no	—
210	#535	—	178	1	no	—
211	C₂×M₃₂	—	89	1	no	—
212	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
213	#501	—	178	2	no	—
214	#500	—	89	2	no	—
215	#499	—	89	2	no	—
216	#496	—	178	2	no	—
217	#495	—	89	2	no	—
218	#493	—	89	2	no	—
219	#428	—	178	1	no	—
220	#427	C₂×D₈₉	1	1	yes	characteristic
221	C₂×#107	—	89	1	no	—
222	#405	—	89	1	no	—
223	#358	—	89	2	no	—
224	#358	C₂×D₈₉	1	1	yes	characteristic, complemented
225	#358	C₂×D₈₉	1	1	yes	characteristic, complemented
226	#357	—	89	2	no	—
227	#357	—	89	2	no	—
228	#357	—	89	2	no	—
229	#357	—	89	2	no	—
230	#357	—	89	2	no	—
231	#357	C₂×D₈₉	1	1	yes	characteristic, complemented
232	#357	C₂×D₈₉	1	1	yes	characteristic, complemented
233	#346	—	89	2	no	—
234	C₄×C₁₆	—	89	2	no	—
235	C₂×#41	—	2	1	no	—
236	C₂²×D₄	—	178	1	no	—
237	AGL(1,ℤ/8)	—	356	4	no	—
238	#135	—	2	2	no	—
239	#135	—	2	2	no	—
240	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×SD₁₆	—	178	2	no	—
243	C₂×D₈	—	89	2	no	—
244	C₂×D₈	—	178	2	no	—
245	C₂×D₈	—	178	1	no	—
246	C₂×D₈	—	178	2	no	—
247	C₂×D₈	—	178	2	no	—
248	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×M₁₆	—	89	1	no	—
250	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₄⋊Q₈	—	356	1	no	—
252	C₄⋊Q₈	—	178	2	no	—
253	#127	—	178	2	no	—
254	#125	—	356	1	no	—
255	#125	—	178	2	no	—
256	#124	—	178	2	no	—
257	C₂²⋊Q₈	—	356	2	no	—
258	#121	—	356	1	no	—
259	#121	—	356	1	no	—
260	#120	—	356	2	no	—
261	#117	—	2	1	no	—
262	#117	—	89	1	no	—
263	C₂×C₄⋊C₄	—	178	1	no	—
264	M₃₂	—	178	2	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	89	2	no	—
267	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
269	#107	—	89	2	no	—
270	#107	—	178	2	no	—
271	#107	—	178	1	no	—
272	#107	C₂²×D₈₉	1	1	yes	characteristic
273	#107	C₂²×D₈₉	1	1	yes	characteristic
274	#106	—	2	1	no	—
275	#106	—	178	2	no	—
276	#97	—	89	2	no	—
277	C₄×C₈	—	89	2	no	—
278	C₂⁴	—	356	1	no	—
279	#41	—	4	2	no	—
280	#41	—	4	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	356	2	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	356	2	no	—
286	C₂×D₄	—	356	2	no	—
287	C₂×D₄	—	356	2	no	—
288	C₂×D₄	—	356	1	no	—
289	C₂×D₄	—	356	2	no	—
290	C₂×D₄	—	178	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	356	1	no	—
293	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	2	1	no	—
296	SD₁₆	—	4	2	no	—
297	SD₁₆	—	356	4	no	—
298	D₈	—	2	1	no	—
299	D₈	—	2	1	no	—
300	D₈	—	356	4	no	—
301	D₈	—	356	2	no	—
302	D₈	—	178	2	no	—
303	D₈	—	178	2	no	—
304	M₁₆	—	89	4	no	—
305	C₂×C₈	—	89	2	no	—
306	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
307	C₂×C₈	#13083573	1	2	yes	—
308	C₂×C₈	#13083574	1	2	yes	—
309	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	89	1	no	—
313	C₄⋊C₄	—	89	1	no	—
314	C₄⋊C₄	—	356	2	no	—
315	C₄⋊C₄	—	356	2	no	—
316	C₄⋊C₄	—	356	2	no	—
317	C₄⋊C₄	—	356	1	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	89	2	no	—
321	C₂²⋊C₄	—	356	2	no	—
322	C₂²⋊C₄	—	356	2	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	89	2	no	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	—	178	2	no	—
328	C₂³	—	4	1	no	—
329	C₂³	—	356	2	no	—
330	C₂³	—	1424	1	no	—
331	C₂³	—	1424	1	no	—
332	C₂³	—	712	2	no	—
333	C₂³	D₈×D₈₉	1	1	yes	characteristic
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	1424	2	no	—
337	D₄	—	4	1	no	—
338	D₄	—	4	1	no	—
339	D₄	—	8	2	no	—
340	D₄	—	356	2	no	—
341	D₄	—	356	2	no	—
342	D₄	—	1424	2	no	—
343	D₄	—	1424	2	no	—
344	D₄	—	712	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	89	2	no	—
350	C₂×C₄	—	89	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	1424	1	no	—
353	C₂×C₄	#15667998	1	2	yes	—
354	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
355	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
357	C₈	—	89	4	no	—
358	C₈	#15667998	1	2	yes	—
359	C₈	C₂×#13083573	1	1	yes	characteristic
360	C₈	C₂×#13083574	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	1424	4	no	—
367	C₂²	—	712	2	no	—
368	C₂²	—	712	2	no	—
369	C₂²	—	712	2	no	—
370	C₂²	#19151943	1	2	yes	—
371	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	178	2	no	—
376	C₄	—	178	2	no	—
377	C₄	—	712	2	no	—
378	C₄	#19151943	1	2	yes	—
379	C₄	C₂×#15667998	1	1	yes	characteristic
380	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	16	1	no	—
383	C₂	—	1424	2	no	—
384	C₂	C₂×#19151943	1	1	yes	characteristic
385	C₂	#26033390	1	1	yes	characteristic
386	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	C₈₉×#2138	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26030739	C₂
hypercenter	#2138	D₈₉
hypocenter	C₈₉	#33702
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
1. #26030792	C₂	semidirect
3. #26022492	C₂	semidirect
4. #26022490	C₂	semidirect
5. #26022485	C₂	semidirect
6. #26022474	C₂	semidirect
7. #26021967	C₂	semidirect
8. #26021964	C₂	semidirect
9. C₂×#19151400	C₂	semidirect
10. #26021848	C₂	semidirect
11. C₈₉×#2138	C₂	semidirect
18. #19152002	C₂²	semidirect
19. #19151999	C₂²	semidirect
26. #19151402	C₂²	semidirect
29. #19151399	C₂²	semidirect
32. #19151373	C₂²	semidirect
33. C₁₆×Dic₈₉	C₂²	semidirect
37. #358×C₈₉	C₂²	semidirect
38. #358×C₈₉	C₂²	semidirect
39. #357×C₈₉	C₂²	semidirect
40. #357×C₈₉	C₂²	semidirect
202. #2138	D₈₉	semidirect
205. C₈₉	#33702	semidirect
224. #358	C₂×D₈₉	semidirect
225. #358	C₂×D₈₉	semidirect
231. #357	C₂×D₈₉	semidirect
232. #357	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030792	2
#26030739	2
#26022492	2
#26022490	2
#26022485	2
#26022474	2
#26021967	2
#26021964	2
C₂×#19151400	2
#26021848	2
C₈₉×#2138	2
#33702	89