Subgroups — #36709762

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	#26030791	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030738	C₂	1	1	yes	maximal, characteristic
3	#26022494	C₂	1	2	yes	maximal, complemented
4	#26022489	C₂	1	2	yes	maximal, complemented
5	#26022484	C₂	1	2	yes	maximal, complemented
6	#26022476	C₂	1	2	yes	maximal, complemented
7	#26021969	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021965	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151400	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668040	—	2	1	no	—
13	C₂×#15668034	C₂²	1	1	yes	characteristic
14	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
15	#19152226	—	2	1	no	—
16	#19152167	—	2	1	no	—
17	#19152007	—	2	2	no	—
18	#19152003	C₂²	1	2	yes	complemented
19	#19152000	C₂²	1	2	yes	complemented
20	#19151987	—	2	2	no	—
21	#19151982	C₂²	1	2	yes	—
22	#19151975	C₂²	1	2	yes	—
23	#19151640	—	2	1	no	—
24	#19151636	C₂²	1	1	yes	characteristic
25	C₂×#15667946	C₂²	1	1	yes	characteristic
26	#19151403	C₂²	1	1	yes	characteristic, complemented
27	#19151403	C₂²	1	1	yes	characteristic, complemented
28	#19151403	C₂²	1	2	yes	complemented
29	#19151400	C₂²	1	2	yes	—
30	#19151400	C₂²	1	2	yes	—
31	#19151400	C₂²	1	2	yes	—
32	#19151400	C₂²	1	2	yes	—
33	#19151400	C₂²	1	1	yes	characteristic, complemented
34	#19151400	C₂²	1	1	yes	characteristic, complemented
35	#19151400	C₂²	1	2	yes	complemented
36	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	#346×C₈₉	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
41	C₂×#13083574	—	2	1	no	—
42	C₂²×D₃₅₆	—	2	1	no	—
43	#15668040	—	4	4	no	—
44	#15668034	—	2	2	no	—
45	#15668034	—	2	2	no	—
46	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
47	C₂×#13083563	—	2	2	no	—
48	C₂×#13083563	—	2	1	no	—
49	C₂×D₇₁₂	—	2	2	no	—
50	C₂×D₇₁₂	—	2	1	no	—
51	C₂×D₇₁₂	—	2	2	no	—
52	C₂×D₇₁₂	—	2	2	no	—
53	C₂×D₇₁₂	C₂³	1	2	yes	—
54	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
55	#15668012	—	2	2	no	—
56	#15668011	—	4	1	no	—
57	#15668009	—	2	2	no	—
58	#15668004	—	2	2	no	—
59	#15668003	—	4	1	no	—
60	#15668000	—	2	2	no	—
61	#15667989	—	4	2	no	—
62	#15667987	—	4	1	no	—
63	#15667987	—	4	1	no	—
64	#15667982	—	4	2	no	—
65	#15667971	—	2	1	no	—
66	C₂×C₄⋊Dic₈₉	—	2	1	no	—
67	#15667946	—	2	1	no	—
68	#15667946	—	2	2	no	—
69	#15667946	C₂³	1	2	yes	—
70	#15667946	C₂³	1	1	yes	characteristic
71	#15667946	C₂³	1	1	yes	characteristic
72	#15667944	—	2	1	no	—
73	#15667944	—	2	2	no	—
74	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	#117×C₈₉	D₄	1	1	yes	characteristic
77	M₃₂×C₈₉	—	2	2	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	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
82	#97×C₈₉	D₄	1	2	yes	—
83	C₄×C₈×C₈₉	C₂³	1	2	yes	—
84	C₂³×D₈₉	—	4	1	no	—
85	#13083574	—	4	2	no	—
86	#13083574	—	4	2	no	—
87	C₂×Dic₁₇₈	—	2	1	no	—
88	C₂×Dic₁₇₈	—	4	2	no	—
89	C₂×D₃₅₆	—	2	2	no	—
90	C₂×D₃₅₆	—	2	1	no	—
91	C₂×D₃₅₆	—	4	2	no	—
92	C₂×D₃₅₆	—	4	2	no	—
93	C₂×D₃₅₆	—	4	1	no	—
94	C₂×D₃₅₆	—	4	2	no	—
95	C₂×D₃₅₆	—	4	2	no	—
96	C₂×C₂²⋊D₈₉	—	4	1	no	—
97	C₂²×Dic₈₉	—	4	1	no	—
98	C₂×C₄×D₈₉	—	4	1	no	—
99	Dic₃₅₆	—	2	1	no	—
100	Dic₃₅₆	—	2	1	no	—
101	#13083563	—	4	2	no	—
102	#13083563	—	4	4	no	—
103	D₇₁₂	—	2	2	no	—
104	D₇₁₂	—	2	2	no	—
105	D₇₁₂	—	2	1	no	—
106	D₇₁₂	—	2	1	no	—
107	D₇₁₂	—	4	2	no	—
108	D₇₁₂	—	4	4	no	—
109	C₄⋊Dic₈₉	—	2	1	no	—
110	C₄⋊Dic₈₉	—	2	1	no	—
111	C₄⋊Dic₈₉	—	2	2	no	—
112	C₄⋊Dic₈₉	—	4	2	no	—
113	C₄⋊Dic₈₉	—	4	1	no	—
114	#13083553	—	4	2	no	—
115	#13083553	—	4	2	no	—
116	C₂²⋊Dic₈₉	—	4	1	no	—
117	#13083551	—	4	2	no	—
118	#13083551	—	4	2	no	—
119	C₄×Dic₈₉	—	4	1	no	—
120	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	M₁₆×C₈₉	SD₁₆	1	4	yes	—
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₈₉	#41	1	2	yes	—
125	C₂×C₈×C₈₉	#41	1	2	yes	—
126	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
128	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
129	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
130	C₄²×C₈₉	C₂×D₄	1	2	yes	—
131	C₁₆×C₈₉	—	2	1	no	—
132	C₁₆×C₈₉	—	2	1	no	—
133	C₁₆×C₈₉	—	2	2	no	—
134	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₈₉	—	4	2	no	—
136	C₂²×D₈₉	—	4	1	no	—
137	C₂²×D₈₉	—	8	2	no	—
138	C₂²×D₈₉	—	16	1	no	—
139	C₂²×D₈₉	—	16	1	no	—
140	C₂×C₄×C₈₉	#124	1	2	yes	—
141	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₈₉	—	8	2	no	—
147	C₂×Dic₈₉	—	4	1	no	—
148	C₂×Dic₈₉	—	4	1	no	—
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	16	1	no	—
152	D₃₅₆	—	4	2	no	—
153	D₃₅₆	—	4	1	no	—
154	D₃₅₆	—	4	1	no	—
155	D₃₅₆	—	4	2	no	—
156	D₃₅₆	—	8	2	no	—
157	D₃₅₆	—	16	2	no	—
158	D₃₅₆	—	16	2	no	—
159	C₄×D₈₉	—	8	2	no	—
160	Dic₁₇₈	—	4	1	no	—
161	Dic₁₇₈	—	4	1	no	—
162	Dic₁₇₈	—	16	2	no	—
163	C₈×C₈₉	#124	1	2	yes	—
164	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
165	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
166	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	#486	1	2	yes	—
170	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₈₉	—	8	1	no	—
172	C₂×D₈₉	—	8	2	no	—
173	C₂×D₈₉	—	8	1	no	—
174	C₂×D₈₉	—	8	2	no	—
175	C₂×D₈₉	—	8	1	no	—
176	C₂×D₈₉	—	8	2	no	—
177	C₂×D₈₉	—	16	4	no	—
178	C₄×C₈₉	—	2	2	no	—
179	C₄×C₈₉	—	2	2	no	—
180	C₄×C₈₉	#486	1	2	yes	—
181	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
182	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
183	Dic₈₉	—	8	1	no	—
184	Dic₈₉	—	8	1	no	—
185	Dic₈₉	—	8	2	no	—
186	Dic₈₉	—	8	1	no	—
187	#33702	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	2	2	no	—
189	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
190	C₂×C₈₉	#3415	1	1	yes	characteristic
191	C₂×C₈₉	#3416	1	1	yes	characteristic
192	D₈₉	—	16	1	no	—
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	—	89	1	no	—
203	C₂×#357	—	89	1	no	—
204	#2108	D₈₉	1	1	yes	characteristic, complemented
205	C₈₉	#33702	1	1	yes	characteristic, complemented, Sylow
206	C₂×AGL(1,ℤ/8)	—	178	1	no	—
207	C₂×#135	—	89	1	no	—
208	C₂²×D₈	—	89	1	no	—
209	#544	—	178	1	no	—
210	#535	—	178	1	no	—
211	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
221	C₂×#107	—	89	1	no	—
222	#405	C₂×D₈₉	1	1	yes	characteristic
223	#358	—	89	1	no	—
224	#358	—	89	1	no	—
225	#358	—	89	2	no	—
226	#357	—	89	2	no	—
227	#357	—	89	2	no	—
228	#357	—	89	2	no	—
229	#357	—	89	2	no	—
230	#357	—	89	1	no	—
231	#357	—	89	1	no	—
232	#357	—	89	2	no	—
233	#346	C₂×D₈₉	1	2	yes	complemented
234	C₄×C₁₆	C₂×D₈₉	1	2	yes	complemented
235	C₂×#41	—	178	1	no	—
236	C₂²×D₄	—	178	1	no	—
237	AGL(1,ℤ/8)	—	356	4	no	—
238	#135	—	178	2	no	—
239	#135	—	178	2	no	—
240	C₂×Q₁₆	—	89	1	no	—
241	C₂×SD₁₆	—	178	1	no	—
242	C₂×SD₁₆	—	178	2	no	—
243	C₂×D₈	—	89	2	no	—
244	C₂×D₈	—	89	1	no	—
245	C₂×D₈	—	178	1	no	—
246	C₂×D₈	—	178	2	no	—
247	C₂×D₈	—	178	2	no	—
248	C₂×D₈	—	178	2	no	—
249	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
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	—	178	1	no	—
262	#117	D₃₅₆	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	178	1	no	—
264	M₃₂	—	2	2	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
268	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
269	#107	—	89	2	no	—
270	#107	—	89	1	no	—
271	#107	—	89	1	no	—
272	#107	—	178	1	no	—
273	#107	—	178	2	no	—
274	#106	—	178	1	no	—
275	#106	—	178	2	no	—
276	#97	D₃₅₆	1	2	yes	—
277	C₄×C₈	C₂²×D₈₉	1	2	yes	—
278	C₂⁴	—	356	1	no	—
279	#41	—	356	2	no	—
280	#41	—	356	2	no	—
281	C₂×Q₈	—	356	2	no	—
282	C₂×Q₈	—	178	1	no	—
283	C₂×D₄	—	356	1	no	—
284	C₂×D₄	—	356	2	no	—
285	C₂×D₄	—	356	2	no	—
286	C₂×D₄	—	356	2	no	—
287	C₂×D₄	—	356	1	no	—
288	C₂×D₄	—	356	2	no	—
289	C₂×D₄	—	178	2	no	—
290	C₂×D₄	—	178	1	no	—
291	C₂²×C₄	—	356	1	no	—
292	C₂²×C₄	—	356	1	no	—
293	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
294	Q₁₆	—	178	1	no	—
295	Q₁₆	—	178	1	no	—
296	SD₁₆	—	356	2	no	—
297	SD₁₆	—	356	4	no	—
298	D₈	—	356	2	no	—
299	D₈	—	356	4	no	—
300	D₈	—	178	2	no	—
301	D₈	—	178	2	no	—
302	D₈	—	178	1	no	—
303	D₈	—	178	1	no	—
304	M₁₆	#13083563	1	4	yes	—
305	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
306	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
307	C₂×C₈	#13083574	1	2	yes	—
308	C₂×C₈	#13083574	1	2	yes	—
309	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
310	C₄⋊C₄	—	356	1	no	—
311	C₄⋊C₄	—	356	2	no	—
312	C₄⋊C₄	—	356	2	no	—
313	C₄⋊C₄	—	356	2	no	—
314	C₄⋊C₄	—	178	1	no	—
315	C₄⋊C₄	—	178	2	no	—
316	C₄⋊C₄	—	178	1	no	—
317	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
318	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
319	C₂²⋊C₄	—	356	1	no	—
320	C₂²⋊C₄	—	356	2	no	—
321	C₂²⋊C₄	—	356	2	no	—
322	C₂²⋊C₄	D₇₁₂	1	2	yes	—
323	C₄²	—	356	1	no	—
324	C₄²	C₂×D₃₅₆	1	2	yes	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	—	2	2	no	—
328	C₂³	—	356	2	no	—
329	C₂³	—	356	1	no	—
330	C₂³	—	1424	1	no	—
331	C₂³	—	1424	1	no	—
332	C₂³	—	712	2	no	—
333	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
334	Q₈	—	356	1	no	—
335	Q₈	—	356	1	no	—
336	Q₈	—	1424	2	no	—
337	D₄	—	356	2	no	—
338	D₄	—	356	1	no	—
339	D₄	—	356	1	no	—
340	D₄	—	356	2	no	—
341	D₄	—	1424	2	no	—
342	D₄	—	1424	2	no	—
343	D₄	—	712	2	no	—
344	D₄	—	712	2	no	—
345	C₂×C₄	—	356	1	no	—
346	C₂×C₄	—	356	1	no	—
347	C₂×C₄	—	356	1	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	1424	1	no	—
350	C₂×C₄	—	712	2	no	—
351	C₂×C₄	#15668000	1	2	yes	—
352	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
353	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
354	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
355	C₂×C₄	C₂×#13083563	1	2	yes	—
356	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
357	C₈	#15668000	1	2	yes	—
358	C₈	C₂×#13083563	1	4	yes	—
359	C₈	C₂×#13083574	1	1	yes	characteristic
360	C₈	C₂×#13083574	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	1424	4	no	—
364	C₂²	—	712	1	no	—
365	C₂²	—	712	2	no	—
366	C₂²	—	712	1	no	—
367	C₂²	—	712	2	no	—
368	C₂²	—	712	1	no	—
369	C₂²	—	712	2	no	—
370	C₂²	#19151945	1	2	yes	—
371	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	712	1	no	—
375	C₄	—	712	2	no	—
376	C₄	—	712	1	no	—
377	C₄	—	712	1	no	—
378	C₄	#19151945	1	2	yes	—
379	C₄	C₂×#15668000	1	1	yes	characteristic
380	C₄	C₂²×#13083563	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	1424	2	no	—
383	C₂	—	1424	1	no	—
384	C₂	C₂×#19151945	1	1	yes	characteristic
385	C₂	#26033393	1	1	yes	characteristic
386	C₂	#26033400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26030738	C₂
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#33702
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #26030791	C₂	semidirect
3. #26022494	C₂	semidirect
4. #26022489	C₂	semidirect
5. #26022484	C₂	semidirect
6. #26022476	C₂	semidirect
7. #26021969	C₂	semidirect
8. #26021965	C₂	semidirect
9. #26021960	C₂	semidirect
10. C₂×#19151400	C₂	semidirect
11. C₈₉×#2108	C₂	semidirect
18. #19152003	C₂²	semidirect
19. #19152000	C₂²	semidirect
26. #19151403	C₂²	semidirect
27. #19151403	C₂²	semidirect
28. #19151403	C₂²	semidirect
33. #19151400	C₂²	semidirect
34. #19151400	C₂²	semidirect
35. #19151400	C₂²	semidirect
39. #346×C₈₉	C₂²	semidirect
40. C₄×C₁₆×C₈₉	C₂²	semidirect
204. #2108	D₈₉	semidirect
205. C₈₉	#33702	semidirect
233. #346	C₂×D₈₉	semidirect
234. 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
#26030791	2
#26030738	2
#26022494	2
#26022489	2
#26022484	2
#26022476	2
#26021969	2
#26021965	2
#26021960	2
C₂×#19151400	2
C₈₉×#2108	2
#33702	89