Subgroups — #36562626

Order 22784.

738 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	#26029193	C₂	1	1	yes	maximal, characteristic, complemented
2	#26028970	C₂	1	1	yes	maximal, characteristic
3	#26021230	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021081	C₂	1	1	yes	maximal, characteristic
5	#26020965	C₂	1	2	yes	maximal, complemented
6	#26020958	C₂	1	2	yes	maximal, complemented
7	#26020393	C₂	1	1	yes	maximal, characteristic, complemented
8	#26020099	C₂	1	2	yes	maximal, complemented
9	#26020092	C₂	1	2	yes	maximal, complemented
10	#97×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3135	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668019	C₂²	1	2	yes	—
13	#19151729	—	2	4	no	—
14	#19151670	—	2	4	no	—
15	#19151625	C₂²	1	2	yes	—
16	#19151622	C₂²	1	1	yes	characteristic
17	#19151622	C₂²	1	1	yes	characteristic
18	C₂×#15667939	C₂²	1	2	yes	—
19	C₂×#15667939	C₂²	1	2	yes	—
20	#19151565	C₂²	1	2	yes	—
21	#19151554	C₂²	1	2	yes	—
22	C₂×#15667923	C₂²	1	1	yes	characteristic
23	C₂×#15667922	C₂²	1	1	yes	characteristic
24	C₂×#15667922	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
28	#19151363	C₂²	1	2	yes	—
29	#19151363	C₂²	1	2	yes	—
30	#19151357	C₂²	1	2	yes	—
31	#19151357	C₂²	1	2	yes	—
32	#533×C₈₉	C₂²	1	1	yes	characteristic
33	#532×C₈₉	C₂²	1	1	yes	characteristic
34	#491×C₈₉	C₄	1	2	yes	complemented
35	#490×C₈₉	C₄	1	2	yes	complemented
36	#489×C₈₉	C₄	1	4	yes	complemented
37	#416×C₈₉	C₂²	1	2	yes	—
38	#416×C₈₉	C₂²	1	2	yes	—
39	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
40	#15668019	—	2	4	no	—
41	#15668019	—	2	4	no	—
42	C₂×#13083558	—	2	2	no	—
43	C₂×#13083558	C₂³	1	2	yes	—
44	C₂×#13083558	C₂³	1	2	yes	—
45	C₂²×#10623494	—	2	2	no	—
46	C₂²×#10623494	C₂³	1	1	yes	characteristic
47	C₂²×#10623494	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₈₉	—	2	2	no	—
49	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊#10623494	—	2	4	no	—
52	C₄⋊#10623494	—	2	2	no	—
53	C₄⋊#10623494	—	2	2	no	—
54	#15667939	—	2	4	no	—
55	#15667939	—	2	4	no	—
56	C₂²⋊#10623494	—	2	4	no	—
57	#15667923	—	2	2	no	—
58	#15667923	D₄	1	4	yes	—
59	#15667922	—	2	2	no	—
60	#15667922	—	2	2	no	—
61	C₄×#10623494	—	2	2	no	—
62	C₈×Dic₈₉	—	2	2	no	—
63	C₈×Dic₈₉	—	2	2	no	—
64	C₂×#41×C₈₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
69	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
70	#125×C₈₉	C₂×C₄	1	2	yes	—
71	#124×C₈₉	C₂×C₄	1	2	yes	—
72	C₂²⋊Q₈×C₈₉	—	2	2	no	—
73	#121×C₈₉	—	2	2	no	—
74	C₄×Q₈×C₈₉	—	2	2	no	—
75	C₄×D₄×C₈₉	—	2	2	no	—
76	#117×C₈₉	C₂³	1	2	yes	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	#103×C₈₉	C₂×C₄	1	4	yes	—
79	#103×C₈₉	C₂×C₄	1	4	yes	—
80	#102×C₈₉	C₂×C₄	1	4	yes	—
81	#102×C₈₉	C₂×C₄	1	4	yes	—
82	#97×C₈₉	C₂×C₄	1	2	yes	—
83	#97×C₈₉	C₂×C₄	1	2	yes	—
84	C₂²×Dic₈₉	—	2	1	no	—
85	C₂²×Dic₈₉	—	2	1	no	—
86	#13083558	—	2	2	no	—
87	#13083558	—	2	2	no	—
88	#13083558	—	2	2	no	—
89	#13083558	—	2	2	no	—
90	#13083558	—	4	4	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	—	2	1	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	2	2	no	—
95	C₂×#10623494	—	4	4	no	—
96	C₂×#10623494	C₂×D₄	1	2	yes	—
97	C₂×#10623494	C₂×D₄	1	2	yes	—
98	C₂×#10623494	#41	1	2	yes	—
99	C₂×#10623494	#41	1	2	yes	—
100	C₄×Dic₈₉	—	2	1	no	—
101	C₄×Dic₈₉	—	2	1	no	—
102	C₄×Dic₈₉	—	4	4	no	—
103	C₄×Dic₈₉	#41	1	2	yes	—
104	C₄×Dic₈₉	#41	1	2	yes	—
105	#41×C₈₉	—	2	2	no	—
106	#41×C₈₉	—	2	2	no	—
107	#41×C₈₉	—	2	2	no	—
108	#41×C₈₉	—	2	2	no	—
109	C₂×Q₈×C₈₉	—	2	2	no	—
110	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₈₉	—	2	2	no	—
112	C₂×D₄×C₈₉	—	2	2	no	—
113	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₈₉	—	2	2	no	—
115	C₂²×C₄×C₈₉	#41	1	2	yes	—
116	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂×C₈×C₈₉	—	2	1	no	—
118	C₂×C₈×C₈₉	—	2	1	no	—
119	C₂×C₈×C₈₉	—	2	1	no	—
120	C₂×C₈×C₈₉	—	2	1	no	—
121	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄⋊C₄×C₈₉	—	2	4	no	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
127	C₂²⋊C₄×C₈₉	—	2	2	no	—
128	C₂²⋊C₄×C₈₉	—	2	4	no	—
129	C₄²×C₈₉	—	2	2	no	—
130	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₈₉	C₂×D₄	1	2	yes	—
133	C₂³×C₈₉	—	2	2	no	—
134	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
135	Q₈×C₈₉	—	2	2	no	—
136	Q₈×C₈₉	—	2	2	no	—
137	Q₈×C₈₉	—	8	2	no	—
138	D₄×C₈₉	—	2	2	no	—
139	D₄×C₈₉	—	2	2	no	—
140	D₄×C₈₉	—	4	2	no	—
141	D₄×C₈₉	—	4	2	no	—
142	D₄×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	—	2	4	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₈₉	—	2	4	no	—
157	C₂×Dic₈₉	—	2	2	no	—
158	C₂×Dic₈₉	—	2	2	no	—
159	C₂×Dic₈₉	—	4	2	no	—
160	C₈×C₈₉	—	2	2	no	—
161	C₈×C₈₉	—	2	2	no	—
162	#10623494	—	2	2	no	—
163	#10623494	—	4	2	no	—
164	#10623494	—	4	2	no	—
165	#10623494	#124	1	4	yes	—
166	C₂²×C₈₉	—	2	2	no	—
167	C₂²×C₈₉	—	4	4	no	—
168	C₂²×C₈₉	—	4	2	no	—
169	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₈₉	#529	1	1	yes	characteristic
171	C₂²×C₈₉	#529	1	1	yes	characteristic
172	C₄×C₈₉	—	2	2	no	—
173	C₄×C₈₉	—	2	2	no	—
174	C₄×C₈₉	—	4	4	no	—
175	C₄×C₈₉	—	4	2	no	—
176	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
177	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
178	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
179	C₄×C₈₉	#532	1	1	yes	characteristic
180	Dic₈₉	—	2	4	no	—
181	Dic₈₉	—	4	2	no	—
182	#21662	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	1	no	—
184	C₂×C₈₉	—	2	1	no	—
185	C₂×C₈₉	—	8	2	no	—
186	C₂×C₈₉	C₄×#124	1	1	yes	characteristic
187	C₂×C₈₉	#2956	1	2	yes	—
188	#3135	D₈₉	1	1	yes	characteristic
189	#2961	—	89	1	no	—
190	#2932	—	89	1	no	—
191	#1994	—	89	1	no	—
192	#1962	—	89	1	no	—
193	#1938	—	89	2	no	—
194	#1937	—	89	2	no	—
195	#1838	—	89	1	no	—
196	#1765	—	89	2	no	—
197	#1764	—	89	2	no	—
198	C₄×#97	—	89	1	no	—
199	C₈₉	#21662	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	89	2	no	—
201	#533	C₂×D₈₉	1	1	yes	characteristic
202	#532	C₂×D₈₉	1	1	yes	characteristic
203	#491	Dic₈₉	1	2	yes	complemented
204	#490	Dic₈₉	1	2	yes	complemented
205	#489	Dic₈₉	1	4	yes	complemented
206	#446	—	178	4	no	—
207	#435	—	178	4	no	—
208	#424	—	89	2	no	—
209	#423	—	89	1	no	—
210	#423	—	89	1	no	—
211	C₂×#104	—	89	2	no	—
212	C₂×#104	—	89	2	no	—
213	#416	C₂×D₈₉	1	2	yes	—
214	#416	C₂×D₈₉	1	2	yes	—
215	#408	—	89	2	no	—
216	#405	—	89	2	no	—
217	C₂×#97	—	89	1	no	—
218	C₂×#97	—	89	1	no	—
219	C₂×#97	—	89	1	no	—
220	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
221	C₂×C₄×C₈	—	89	1	no	—
222	C₂×C₄×C₈	—	89	1	no	—
223	C₄³	—	89	1	no	—
224	#341	—	89	2	no	—
225	#341	—	89	2	no	—
226	#339	—	89	2	no	—
227	#339	—	89	2	no	—
228	C₂×#41	C₂²×D₈₉	1	2	yes	—
229	#131	—	178	4	no	—
230	#131	—	178	4	no	—
231	C₂×M₁₆	—	89	2	no	—
232	C₂×M₁₆	—	89	2	no	—
233	C₂×M₁₆	—	178	2	no	—
234	C₂²×C₈	—	89	1	no	—
235	C₂²×C₈	—	89	1	no	—
236	C₂²×C₈	—	178	2	no	—
237	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
240	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
241	#127	C₂×Dic₈₉	1	1	yes	characteristic
242	#125	C₂×Dic₈₉	1	2	yes	—
243	#124	C₂×Dic₈₉	1	2	yes	—
244	C₂²⋊Q₈	—	2	2	no	—
245	#121	—	2	2	no	—
246	C₄×Q₈	—	2	2	no	—
247	C₄×D₄	—	2	2	no	—
248	#117	C₂²×D₈₉	1	2	yes	—
249	C₂×C₄²	—	89	1	no	—
250	C₂×C₄²	—	89	1	no	—
251	C₂×C₄²	—	178	2	no	—
252	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
253	C₄⋊C₈	—	178	4	no	—
254	C₄⋊C₈	—	178	2	no	—
255	C₄⋊C₈	—	178	2	no	—
256	#104	—	178	4	no	—
257	#104	—	178	4	no	—
258	#103	C₂×Dic₈₉	1	4	yes	—
259	#103	C₂×Dic₈₉	1	4	yes	—
260	#102	C₂×Dic₈₉	1	4	yes	—
261	#102	C₂×Dic₈₉	1	4	yes	—
262	C₂²⋊C₈	—	178	4	no	—
263	#97	—	89	4	no	—
264	#97	—	178	2	no	—
265	#97	—	178	2	no	—
266	#97	—	178	2	no	—
267	#97	C₂×Dic₈₉	1	2	yes	—
268	#97	C₂×Dic₈₉	1	2	yes	—
269	C₄×C₈	—	178	2	no	—
270	C₄×C₈	—	178	2	no	—
271	C₄×C₈	—	178	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	#41	—	2	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	#13083573	1	2	yes	—
285	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
286	M₁₆	—	356	4	no	—
287	M₁₆	—	178	2	no	—
288	M₁₆	—	178	2	no	—
289	M₁₆	—	178	2	no	—
290	M₁₆	—	178	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	89	2	no	—
296	C₂×C₈	—	89	2	no	—
297	C₂×C₈	—	89	2	no	—
298	C₂×C₈	—	89	2	no	—
299	C₂×C₈	—	356	4	no	—
300	C₂×C₈	—	178	2	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	1	no	—
303	C₂×C₈	—	178	1	no	—
304	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
305	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	2	4	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	89	2	no	—
314	C₄²	—	89	2	no	—
315	C₄²	—	356	4	no	—
316	C₄²	—	178	1	no	—
317	C₄²	—	178	1	no	—
318	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
319	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
320	C₄²	D₄×D₈₉	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	C₂×#13083573	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	178	2	no	—
338	C₂×C₄	—	178	4	no	—
339	C₂×C₄	—	178	2	no	—
340	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
341	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
342	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
345	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
346	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
347	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	89	4	no	—
351	C₈	—	356	2	no	—
352	C₈	—	356	2	no	—
353	C₈	—	178	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
358	C₂²	#19152127	1	1	yes	characteristic
359	C₂²	#19152127	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	4	4	no	—
363	C₄	—	4	2	no	—
364	C₄	—	356	2	no	—
365	C₄	—	178	4	no	—
366	C₄	#41×Dic₈₉	1	1	yes	characteristic
367	C₄	#41×Dic₈₉	1	1	yes	characteristic
368	C₄	#124×D₈₉	1	1	yes	characteristic
369	C₄	#19152147	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	8	2	no	—
373	C₂	#124×Dic₈₉	1	1	yes	characteristic
374	C₂	#26029162	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#3135	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#532×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#3135	D₈₉
hypocenter	C₈₉	#21662
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

Complements

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

N	H ≅ G/N	type
1. #26029193	C₂	semidirect
3. #26021230	C₂	semidirect
5. #26020965	C₂	semidirect
6. #26020958	C₂	semidirect
7. #26020393	C₂	semidirect
8. #26020099	C₂	semidirect
9. #26020092	C₂	semidirect
10. #97×Dic₈₉	C₂	semidirect
34. #491×C₈₉	C₄	semidirect
35. #490×C₈₉	C₄	semidirect
36. #489×C₈₉	C₄	semidirect
199. C₈₉	#21662	semidirect
203. #491	Dic₈₉	semidirect
204. #490	Dic₈₉	semidirect
205. #489	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029193	2
#26028970	2
#26021230	2
#26021081	2
#26020965	2
#26020958	2
#26020393	2
#26020099	2
#26020092	2
#97×Dic₈₉	2
C₈₉×#3135	2
#21662	89