Subgroups — #36709514

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022460	C₂	1	2	yes	maximal, complemented
3	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022302	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022297	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022192	C₂	1	2	yes	maximal
8	#26022121	C₂	1	2	yes	maximal, complemented
9	#26022116	C₂	1	2	yes	maximal, complemented
10	#26022055	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19152155	—	2	1	no	—
13	C₂×#15667958	C₂²	1	2	yes	—
14	C₂×#15667956	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151978	C₂²	1	1	yes	characteristic
17	#19151949	—	2	1	no	—
18	#19151839	—	2	1	no	—
19	#19151839	C₂²	1	2	yes	—
20	C₄×Dic₃₅₆	—	2	1	no	—
21	C₄×Dic₃₅₆	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151423	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	#19151402	C₂²	1	2	yes	—
27	#19151401	C₂²	1	2	yes	—
28	#19151401	C₂²	1	2	yes	—
29	#19151399	C₂²	1	2	yes	—
30	C₂²⋊#13083549	C₂²	1	2	yes	—
31	#19151374	C₂²	1	2	yes	—
32	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
33	C₄×#41×C₈₉	—	2	1	no	—
34	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₈₉	—	2	1	no	—
37	C₄×SD₁₆×C₈₉	—	2	1	no	—
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
40	#427×C₈₉	C₂²	1	1	yes	characteristic
41	#416×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×Dic₃₅₆	—	2	1	no	—
44	C₂×Dic₃₅₆	C₂³	1	2	yes	—
45	#15668012	—	2	1	no	—
46	#15668012	—	2	1	no	—
47	#15668004	—	4	1	no	—
48	#15667991	—	2	2	no	—
49	#15667991	—	4	1	no	—
50	C₄×Dic₁₇₈	—	2	2	no	—
51	C₄×Dic₁₇₈	—	4	1	no	—
52	#15667971	—	4	1	no	—
53	#15667958	—	2	4	no	—
54	#15667956	—	2	4	no	—
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	2	yes	—
57	#15667946	—	2	1	no	—
58	#15667946	C₂³	1	2	yes	—
59	#15667937	—	2	2	no	—
60	#15667937	—	2	2	no	—
61	C₂×#41×C₈₉	—	2	1	no	—
62	#135×C₈₉	—	2	2	no	—
63	#135×C₈₉	—	2	2	no	—
64	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×SD₁₆×C₈₉	—	2	1	no	—
66	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	—	2	1	no	—
71	C₄×D₄×C₈₉	—	2	1	no	—
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	4	1	no	—
74	C₄×D₄×C₈₉	—	4	1	no	—
75	#117×C₈₉	—	2	1	no	—
76	#117×C₈₉	—	4	1	no	—
77	C₂×C₄²×C₈₉	—	4	1	no	—
78	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#107×C₈₉	C₂³	1	1	yes	characteristic
81	#106×C₈₉	—	2	1	no	—
82	#103×C₈₉	—	2	1	no	—
83	#103×C₈₉	—	2	1	no	—
84	#102×C₈₉	—	2	1	no	—
85	#102×C₈₉	—	2	1	no	—
86	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₂×Dic₁₇₈	—	2	2	no	—
91	C₂×Dic₁₇₈	—	4	1	no	—
92	Dic₃₅₆	—	2	2	no	—
93	Dic₃₅₆	—	2	2	no	—
94	Dic₃₅₆	—	4	2	no	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	C₄⋊Dic₈₉	—	4	1	no	—
99	#13083553	—	4	2	no	—
100	#13083553	—	4	2	no	—
101	C₂²⋊Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	#13083549	—	2	2	no	—
104	#13083549	—	2	2	no	—
105	#41×C₈₉	—	2	4	no	—
106	#41×C₈₉	—	4	2	no	—
107	C₂×Q₈×C₈₉	—	2	1	no	—
108	C₂×D₄×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	—	4	1	no	—
110	C₂²×C₄×C₈₉	—	4	1	no	—
111	C₂²×C₄×C₈₉	—	4	1	no	—
112	C₂²×C₄×C₈₉	—	4	1	no	—
113	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	Q₁₆×C₈₉	—	2	1	no	—
117	Q₁₆×C₈₉	#41	1	2	yes	—
118	SD₁₆×C₈₉	—	2	4	no	—
119	D₈×C₈₉	—	2	1	no	—
120	D₈×C₈₉	#41	1	2	yes	—
121	C₂×C₈×C₈₉	—	2	1	no	—
122	C₂×C₈×C₈₉	—	2	1	no	—
123	C₂×C₈×C₈₉	—	4	1	no	—
124	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₈₉	—	2	1	no	—
129	C₄⋊C₄×C₈₉	—	2	1	no	—
130	C₄⋊C₄×C₈₉	—	4	1	no	—
131	C₄⋊C₄×C₈₉	—	4	1	no	—
132	C₂²⋊C₄×C₈₉	—	4	1	no	—
133	C₂²⋊C₄×C₈₉	—	4	1	no	—
134	C₂²⋊C₄×C₈₉	—	4	1	no	—
135	C₄²×C₈₉	—	4	1	no	—
136	C₄²×C₈₉	—	4	1	no	—
137	C₄²×C₈₉	—	4	1	no	—
138	C₄²×C₈₉	D₈	1	1	yes	characteristic
139	C₄²×C₈₉	D₈	1	1	yes	characteristic
140	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂³×C₈₉	—	4	1	no	—
143	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
144	Q₈×C₈₉	—	2	2	no	—
145	Q₈×C₈₉	—	4	1	no	—
146	D₄×C₈₉	—	2	2	no	—
147	D₄×C₈₉	—	4	4	no	—
148	D₄×C₈₉	—	4	1	no	—
149	C₂×C₄×C₈₉	—	2	1	no	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	4	1	no	—
153	C₂×C₄×C₈₉	—	4	4	no	—
154	C₂×C₄×C₈₉	—	4	1	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	—	4	1	no	—
157	C₂×C₄×C₈₉	—	8	2	no	—
158	C₂×C₄×C₈₉	—	8	2	no	—
159	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×Dic₈₉	—	4	2	no	—
166	C₂×Dic₈₉	—	4	2	no	—
167	Dic₁₇₈	—	4	2	no	—
168	Dic₁₇₈	—	4	2	no	—
169	Dic₁₇₈	—	8	2	no	—
170	C₈×C₈₉	—	2	1	no	—
171	C₈×C₈₉	—	2	1	no	—
172	C₈×C₈₉	—	2	1	no	—
173	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
174	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
175	C₂²×C₈₉	—	2	1	no	—
176	C₂²×C₈₉	—	2	1	no	—
177	C₂²×C₈₉	—	2	1	no	—
178	C₂²×C₈₉	—	4	2	no	—
179	C₂²×C₈₉	—	8	1	no	—
180	C₂²×C₈₉	—	8	1	no	—
181	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	—	2	1	no	—
184	C₄×C₈₉	—	2	1	no	—
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	4	2	no	—
187	C₄×C₈₉	—	8	1	no	—
188	C₄×C₈₉	—	8	1	no	—
189	C₄×C₈₉	#510	1	2	yes	—
190	C₄×C₈₉	#525	1	1	yes	characteristic
191	C₄×C₈₉	#575	1	1	yes	characteristic
192	Dic₈₉	—	8	2	no	—
193	Dic₈₉	—	8	2	no	—
194	#33676	—	89	1	no	maximal, Sylow
195	C₂×C₈₉	—	4	1	no	—
196	C₂×C₈₉	—	8	2	no	—
197	C₂×C₈₉	#3100	1	1	yes	characteristic
198	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
199	C₂×C₈₉	#3414	1	1	yes	characteristic
200	#3170	—	89	1	no	—
201	C₄×#135	D₈₉	1	1	yes	characteristic
202	#2237	—	89	2	no	—
203	#2227	—	89	1	no	—
204	#2226	—	89	1	no	—
205	#2209	—	89	1	no	—
206	#2208	—	89	1	no	—
207	#2189	—	89	2	no	—
208	#2176	—	89	2	no	—
209	#2175	—	89	2	no	—
210	#2163	—	89	1	no	—
211	C₈₉	#33676	1	1	yes	characteristic, complemented, Sylow
212	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
213	#533	—	178	1	no	—
214	C₄×#41	—	2	1	no	—
215	C₂×Q₃₂	—	89	2	no	—
216	C₂×SD₃₂	—	89	2	no	—
217	#500	—	89	1	no	—
218	#494	—	89	1	no	—
219	#487	—	178	1	no	—
220	#467	—	89	2	no	—
221	#467	—	178	1	no	—
222	C₄×Q₁₆	—	89	2	no	—
223	C₄×Q₁₆	—	178	1	no	—
224	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
225	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
226	C₄×SD₁₆	—	2	1	no	—
227	C₄×SD₁₆	—	2	1	no	—
228	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
229	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
230	#427	C₂×D₈₉	1	1	yes	characteristic
231	#416	—	2	1	no	—
232	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
233	#367	—	89	2	no	—
234	#366	—	89	2	no	—
235	C₄⋊C₁₆	—	89	1	no	—
236	C₄⋊C₁₆	—	89	1	no	—
237	#358	—	89	2	no	—
238	#358	—	89	2	no	—
239	#358	—	89	2	no	—
240	#357	—	89	2	no	—
241	C₂²⋊C₁₆	—	89	2	no	—
242	#346	—	89	2	no	—
243	C₂×#41	—	2	1	no	—
244	#135	—	2	2	no	—
245	#135	—	2	2	no	—
246	C₂×Q₁₆	—	89	2	no	—
247	C₂×Q₁₆	—	178	1	no	—
248	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×SD₁₆	—	2	1	no	—
250	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₄⋊Q₈	—	178	1	no	—
254	C₄⋊Q₈	—	178	1	no	—
255	#125	—	356	1	no	—
256	C₂²⋊Q₈	—	356	1	no	—
257	C₂²⋊Q₈	—	178	2	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₄×Q₈	—	356	1	no	—
261	C₄×Q₈	—	178	2	no	—
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	4	1	no	—
265	C₄×D₄	—	4	1	no	—
266	#117	—	2	1	no	—
267	#117	—	4	1	no	—
268	#117	—	356	1	no	—
269	C₂×C₄²	—	4	1	no	—
270	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
271	Q₃₂	—	178	4	no	—
272	SD₃₂	—	178	4	no	—
273	C₂×C₁₆	—	89	2	no	—
274	C₂×C₁₆	—	89	2	no	—
275	#107	—	89	2	no	—
276	#107	—	178	1	no	—
277	#107	C₂²×D₈₉	1	1	yes	characteristic
278	#107	C₂²×D₈₉	1	1	yes	characteristic
279	#106	—	2	1	no	—
280	#103	—	2	1	no	—
281	#103	—	2	1	no	—
282	#103	—	178	2	no	—
283	#103	—	178	2	no	—
284	#102	—	2	1	no	—
285	#102	—	2	1	no	—
286	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
287	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
290	#41	—	2	4	no	—
291	#41	—	4	2	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	356	1	no	—
294	C₂×Q₈	—	178	2	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	#13083559	1	1	yes	characteristic
301	C₂²×C₄	#13083559	1	1	yes	characteristic
302	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
303	Q₁₆	—	2	1	no	—
304	Q₁₆	—	356	2	no	—
305	Q₁₆	—	178	2	no	—
306	Q₁₆	—	178	2	no	—
307	Q₁₆	#13083574	1	2	yes	—
308	SD₁₆	—	2	4	no	—
309	D₈	—	2	1	no	—
310	D₈	#13083574	1	2	yes	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
317	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	356	1	no	—
323	C₄⋊C₄	—	356	1	no	—
324	C₄⋊C₄	—	356	1	no	—
325	C₄⋊C₄	—	356	2	no	—
326	C₄⋊C₄	—	356	2	no	—
327	C₄⋊C₄	—	178	2	no	—
328	C₂²⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₂²⋊C₄	—	4	1	no	—
331	C₂²⋊C₄	—	356	2	no	—
332	C₄²	—	4	1	no	—
333	C₄²	—	4	1	no	—
334	C₄²	—	4	1	no	—
335	C₄²	—	356	2	no	—
336	C₄²	#13083559	1	1	yes	characteristic
337	C₄²	#13083559	1	1	yes	characteristic
338	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
339	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
340	C₁₆	—	178	2	no	—
341	C₁₆	—	178	2	no	—
342	C₂³	—	4	1	no	—
343	C₂³	C₂×#13083559	1	1	yes	characteristic
344	Q₈	—	2	2	no	—
345	Q₈	—	4	1	no	—
346	Q₈	—	356	2	no	—
347	Q₈	—	356	2	no	—
348	Q₈	—	712	2	no	—
349	D₄	—	2	2	no	—
350	D₄	—	4	4	no	—
351	D₄	—	4	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	—	8	2	no	—
362	C₂×C₄	—	356	2	no	—
363	C₂×C₄	—	356	2	no	—
364	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
365	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
366	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
367	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
368	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
369	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	C₂×#13083574	1	1	yes	characteristic
374	C₈	C₂×#13083574	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	1	no	—
380	C₂²	—	8	1	no	—
381	C₂²	C₂²×#13083559	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	8	1	no	—
389	C₄	—	712	2	no	—
390	C₄	—	712	2	no	—
391	C₄	#19152038	1	2	yes	—
392	C₄	#19152109	1	1	yes	characteristic
393	C₄	#19152396	1	1	yes	characteristic
394	C₂	—	4	1	no	—
395	C₂	—	8	2	no	—
396	C₂	#26030252	1	1	yes	characteristic
397	C₂	C₂×#19152038	1	1	yes	characteristic
398	C₂	#26033386	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#135×C₈₉	C₂²
hypercenter	C₄×#135	D₈₉
hypocenter	C₈₉	#33676
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #26030838	C₂	semidirect
2. #26022460	C₂	semidirect
3. #26022404	C₂	semidirect
4. #26022399	C₂	semidirect
5. #26022302	C₂	semidirect
6. #26022297	C₂	semidirect
8. #26022121	C₂	semidirect
9. #26022116	C₂	semidirect
10. #26022055	C₂	semidirect
211. C₈₉	#33676	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26022460	2
#26022404	2
#26022399	2
#26022302	2
#26022297	2
#26022192	2
#26022121	2
#26022116	2
#26022055	2
C₄×#135×C₈₉	2
#33676	89