Subgroups — #36562628

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	#26030597	C₂	1	1	yes	maximal, characteristic
2	#26029189	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028970	C₂	1	1	yes	maximal, characteristic
4	#26021229	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020960	C₂	1	2	yes	maximal, complemented
6	#26020953	C₂	1	2	yes	maximal, complemented
7	#26020392	C₂	1	1	yes	maximal, characteristic, complemented
8	#26020097	C₂	1	2	yes	maximal, complemented
9	#26020090	C₂	1	2	yes	maximal, complemented
10	C₄×#15667923	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1962	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668019	C₂²	1	2	yes	—
13	#19152155	C₂²	1	1	yes	characteristic
14	#19151966	C₄	1	2	yes	complemented
15	#19151962	C₄	1	2	yes	complemented
16	#19151957	C₄	1	4	yes	complemented
17	#19151727	—	2	4	no	—
18	#19151670	—	2	4	no	—
19	#19151626	C₂²	1	2	yes	—
20	#19151622	C₂²	1	1	yes	characteristic
21	#19151594	C₂²	1	2	yes	—
22	#19151594	C₂²	1	2	yes	—
23	#19151565	C₂²	1	2	yes	—
24	#19151556	C₂²	1	2	yes	—
25	C₂×#15667923	C₂²	1	1	yes	characteristic
26	C₂×#15667923	C₂²	1	1	yes	characteristic
27	C₂×#15667923	C₂²	1	1	yes	characteristic
28	C₂×#15667923	C₂²	1	1	yes	characteristic
29	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
30	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
31	#19151362	C₂²	1	2	yes	—
32	#19151362	C₂²	1	2	yes	—
33	#19151358	C₂²	1	2	yes	—
34	#19151358	C₂²	1	2	yes	—
35	#532×C₈₉	C₂²	1	1	yes	characteristic
36	#423×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#104×C₈₉	C₂²	1	2	yes	—
38	C₂×#104×C₈₉	C₂²	1	2	yes	—
39	C₄³×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	C₂²×#10623494	—	2	2	no	—
45	C₂²×#10623494	C₂³	1	1	yes	characteristic
46	C₂²×#10623494	C₂³	1	1	yes	characteristic
47	C₂²×#10623494	C₂³	1	1	yes	characteristic
48	C₂²×#10623494	C₂³	1	1	yes	characteristic
49	#15668012	C₂×C₄	1	2	yes	—
50	#15668004	C₂×C₄	1	2	yes	—
51	#15667991	—	2	2	no	—
52	C₄×Dic₁₇₈	—	2	2	no	—
53	#15667971	C₂³	1	2	yes	—
54	C₄⋊#10623494	—	2	2	no	—
55	#15667941	—	2	4	no	—
56	#15667936	C₂×C₄	1	4	yes	—
57	#15667936	C₂×C₄	1	4	yes	—
58	#15667933	C₂×C₄	1	4	yes	—
59	#15667933	C₂×C₄	1	4	yes	—
60	C₂²⋊#10623494	—	2	4	no	—
61	#15667923	—	2	2	no	—
62	#15667923	—	2	2	no	—
63	#15667923	C₂×C₄	1	2	yes	—
64	#15667923	C₂×C₄	1	2	yes	—
65	#15667923	D₄	1	4	yes	—
66	#15667922	—	2	2	no	—
67	C₄×#10623494	—	2	2	no	—
68	C₄×#10623494	—	2	2	no	—
69	C₈×Dic₈₉	—	2	2	no	—
70	C₂×#41×C₈₉	C₂³	1	2	yes	—
71	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
74	#124×C₈₉	C₂×C₄	1	2	yes	—
75	#121×C₈₉	—	2	2	no	—
76	C₄×D₄×C₈₉	—	2	2	no	—
77	C₂×C₄²×C₈₉	—	2	2	no	—
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₈₉	—	2	2	no	—
82	#104×C₈₉	—	2	4	no	—
83	#104×C₈₉	—	2	4	no	—
84	C₂×Dic₁₇₈	—	2	2	no	—
85	#13083558	—	2	2	no	—
86	#13083558	—	2	2	no	—
87	#13083558	—	4	4	no	—
88	C₂×#10623494	—	2	1	no	—
89	C₂×#10623494	—	2	1	no	—
90	C₂×#10623494	—	2	1	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	—	2	1	no	—
93	C₂×#10623494	—	2	1	no	—
94	C₂×#10623494	—	2	2	no	—
95	C₂×#10623494	—	4	4	no	—
96	C₂×#10623494	C₂²×C₄	1	2	yes	—
97	C₂×#10623494	C₂²×C₄	1	2	yes	—
98	C₂×#10623494	C₂×D₄	1	2	yes	—
99	C₂×#10623494	C₂×D₄	1	2	yes	—
100	C₂×#10623494	#41	1	2	yes	—
101	C₂×#10623494	#41	1	2	yes	—
102	C₄⋊Dic₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
104	#13083553	—	2	4	no	—
105	C₂²⋊Dic₈₉	—	2	2	no	—
106	C₄×Dic₈₉	—	2	2	no	—
107	#41×C₈₉	—	2	2	no	—
108	#41×C₈₉	—	2	2	no	—
109	#41×C₈₉	—	2	2	no	—
110	#41×C₈₉	—	2	2	no	—
111	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₈₉	—	2	2	no	—
113	C₂×D₄×C₈₉	—	2	2	no	—
114	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₈₉	—	2	1	no	—
116	C₂²×C₄×C₈₉	—	2	1	no	—
117	C₂²×C₄×C₈₉	—	2	2	no	—
118	C₂²×C₄×C₈₉	#41	1	2	yes	—
119	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
120	M₁₆×C₈₉	—	2	2	no	—
121	M₁₆×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₂²⋊C₄×C₈₉	—	2	4	no	—
125	C₄²×C₈₉	—	2	1	no	—
126	C₄²×C₈₉	—	2	1	no	—
127	C₄²×C₈₉	—	4	4	no	—
128	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₈₉	C₂×D₄	1	2	yes	—
131	C₄²×C₈₉	#41	1	2	yes	—
132	C₄²×C₈₉	#41	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	D₄×C₈₉	—	2	2	no	—
138	D₄×C₈₉	—	2	2	no	—
139	D₄×C₈₉	—	4	2	no	—
140	D₄×C₈₉	—	4	2	no	—
141	D₄×C₈₉	—	8	2	no	—
142	C₂×C₄×C₈₉	—	2	4	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	4	2	no	—
150	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂²×D₄	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×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₈₉	—	2	4	no	—
159	Dic₁₇₈	—	8	2	no	—
160	C₈×C₈₉	—	4	2	no	—
161	#10623494	—	2	2	no	—
162	#10623494	—	2	2	no	—
163	#10623494	—	2	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	4	no	—
173	C₄×C₈₉	—	2	2	no	—
174	C₄×C₈₉	—	2	2	no	—
175	C₄×C₈₉	—	4	2	no	—
176	C₄×C₈₉	—	4	2	no	—
177	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
178	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
179	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
180	C₄×C₈₉	#532	1	1	yes	characteristic
181	Dic₈₉	—	4	4	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	—	89	1	no	—
189	#2961	—	89	1	no	—
190	#2932	—	89	1	no	—
191	#1994	—	89	1	no	—
192	#1962	D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
202	#532	C₂×D₈₉	1	1	yes	characteristic
203	#491	—	89	2	no	—
204	#490	—	89	2	no	—
205	#489	—	89	4	no	—
206	#446	—	178	4	no	—
207	#435	—	178	4	no	—
208	#424	—	89	2	no	—
209	#423	—	89	1	no	—
210	#423	C₂×D₈₉	1	1	yes	characteristic
211	C₂×#104	C₂×D₈₉	1	2	yes	—
212	C₂×#104	C₂×D₈₉	1	2	yes	—
213	#416	—	89	2	no	—
214	#416	—	89	2	no	—
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	—	89	1	no	—
221	C₂×C₄×C₈	—	89	1	no	—
222	C₂×C₄×C₈	—	89	1	no	—
223	C₄³	C₂×D₈₉	1	1	yes	characteristic
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	—	178	2	no	—
233	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
234	C₂²×C₈	—	89	1	no	—
235	C₂²×C₈	—	89	1	no	—
236	C₂²×C₈	—	89	1	no	—
237	C₂²×C₈	—	89	1	no	—
238	C₂²×C₈	—	178	2	no	—
239	C₄⋊Q₈	—	89	2	no	—
240	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
241	#127	C₄×D₈₉	1	1	yes	characteristic
242	#125	—	89	2	no	—
243	#124	C₄×D₈₉	1	2	yes	—
244	C₂²⋊Q₈	—	178	2	no	—
245	#121	—	2	2	no	—
246	C₄×Q₈	—	178	2	no	—
247	C₄×D₄	—	2	2	no	—
248	#117	—	89	2	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	178	4	no	—
255	C₄⋊C₈	—	178	2	no	—
256	#104	—	2	4	no	—
257	#104	—	2	4	no	—
258	#103	—	89	4	no	—
259	#103	—	89	4	no	—
260	#102	—	89	4	no	—
261	#102	—	89	4	no	—
262	C₂²⋊C₈	—	178	4	no	—
263	#97	—	89	2	no	—
264	#97	—	89	2	no	—
265	#97	—	89	4	no	—
266	#97	—	178	2	no	—
267	#97	—	178	2	no	—
268	#97	—	178	2	no	—
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	—	178	2	no	—
277	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	C₂×C₄×D₈₉	1	2	yes	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	#13083573	1	2	yes	—
285	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	356	4	no	—
289	M₁₆	—	178	2	no	—
290	M₁₆	—	178	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	89	2	no	—
293	C₂×C₈	—	89	2	no	—
294	C₂×C₈	—	89	2	no	—
295	C₂×C₈	—	89	2	no	—
296	C₂×C₈	—	89	2	no	—
297	C₂×C₈	—	89	2	no	—
298	C₂×C₈	—	356	4	no	—
299	C₂×C₈	—	178	2	no	—
300	C₂×C₈	—	178	1	no	—
301	C₂×C₈	—	178	1	no	—
302	C₂×C₈	—	178	1	no	—
303	C₂×C₈	—	178	1	no	—
304	C₂×C₈	—	178	1	no	—
305	C₂×C₈	—	178	1	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	89	4	no	—
308	C₄⋊C₄	—	178	4	no	—
309	C₄⋊C₄	—	178	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₂²⋊C₄	—	178	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	4	4	no	—
315	C₄²	—	178	2	no	—
316	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
318	C₄²	D₄×D₈₉	1	2	yes	—
319	C₄²	#13083573	1	2	yes	—
320	C₄²	#13083573	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	—	712	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	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	178	4	no	—
340	C₂×C₄	#15667972	1	2	yes	—
341	C₂×C₄	#15667972	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₈₉	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	C₂×#13083573	1	1	yes	characteristic
347	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
348	C₈	—	4	2	no	—
349	C₈	—	89	4	no	—
350	C₈	—	356	2	no	—
351	C₈	—	178	2	no	—
352	C₈	—	178	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×#15667972	1	1	yes	characteristic
358	C₂²	#19152129	1	1	yes	characteristic
359	C₂²	#19152129	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	356	4	no	—
366	C₄	C₄×#13083573	1	1	yes	characteristic
367	C₄	C₄×#13083573	1	1	yes	characteristic
368	C₄	#124×D₈₉	1	1	yes	characteristic
369	C₄	#19152146	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	8	2	no	—
373	C₂	#26022799	1	1	yes	characteristic
374	C₂	#26029158	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×#1962	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#532×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1962	D₈₉
hypocenter	C₈₉	#21662
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

Complements

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

N	H ≅ G/N	type
2. #26029189	C₂	semidirect
4. #26021229	C₂	semidirect
5. #26020960	C₂	semidirect
6. #26020953	C₂	semidirect
7. #26020392	C₂	semidirect
8. #26020097	C₂	semidirect
9. #26020090	C₂	semidirect
10. C₄×#15667923	C₂	semidirect
14. #19151966	C₄	semidirect
15. #19151962	C₄	semidirect
16. #19151957	C₄	semidirect
199. C₈₉	#21662	semidirect

Maximal subgroups

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

subgroup	index
#26030597	2
#26029189	2
#26028970	2
#26021229	2
#26020960	2
#26020953	2
#26020392	2
#26020097	2
#26020090	2
C₄×#15667923	2
C₈₉×#1962	2
#21662	89