Subgroups — #36538227

Order 22784.

690 conjugacy classes of subgroups.

The subgroup lattice has 370 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	#26030051	C₂	1	2	yes	maximal
3	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
4	#26019704	C₂	1	4	yes	maximal, complemented
5	#26019652	C₂	1	2	yes	maximal, complemented
6	#26019618	C₂	1	2	yes	maximal, complemented
7	#26018943	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₈₉	—	2	1	no	—
10	#19152155	C₂²	1	2	yes	—
11	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	#19152003	C₂²	1	1	yes	characteristic, complemented
13	#19151978	C₂²	1	1	yes	characteristic, complemented
14	#19151949	C₂²	1	2	yes	complemented
15	#19151899	—	2	4	no	—
16	#19151889	C₂²	1	4	yes	—
17	#19151879	C₂²	1	4	yes	—
18	#19151839	—	2	1	no	—
19	#19151839	—	2	1	no	—
20	C₄×Dic₃₅₆	—	2	1	no	—
21	C₄×Dic₃₅₆	—	2	1	no	—
22	C₈×D₃₅₆	—	2	1	no	—
23	C₈×D₃₅₆	—	2	1	no	—
24	C₈×D₃₅₆	C₂²	1	2	yes	—
25	C₈×D₃₅₆	C₂²	1	2	yes	—
26	#19151640	—	2	2	no	—
27	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
28	#19151604	—	2	2	no	—
29	C₂×#15667924	—	2	1	no	—
30	#19151346	C₂²	1	4	yes	complemented
31	#19151343	C₂²	1	4	yes	complemented
32	#423×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#335×C₈₉	C₂²	1	2	yes	complemented
35	#335×C₈₉	C₂²	1	2	yes	complemented
36	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₈₉	—	2	1	no	—
38	C₂×Dic₃₅₆	—	2	1	no	—
39	C₂×Dic₃₅₆	—	2	1	no	—
40	C₂²×#10623494	—	2	1	no	—
41	C₂×C₈×D₈₉	—	2	2	no	—
42	C₂×C₈×D₈₉	—	2	2	no	—
43	C₂×C₈×D₈₉	—	4	4	no	—
44	#15668012	C₂³	1	2	yes	—
45	#15668012	C₂³	1	2	yes	—
46	#15668004	—	2	2	no	—
47	#15667991	—	2	2	no	—
48	#15667991	—	2	2	no	—
49	C₄×Dic₁₇₈	—	2	2	no	—
50	C₄×Dic₁₇₈	—	2	2	no	—
51	C₄×D₃₅₆	—	2	1	no	—
52	C₄×D₃₅₆	—	2	1	no	—
53	C₄×D₃₅₆	D₄	1	2	yes	—
54	C₄×D₃₅₆	C₂³	1	2	yes	—
55	#15667971	—	2	2	no	—
56	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
57	C₂×#13083551	—	2	1	no	—
58	#15667946	—	2	4	no	—
59	#15667946	—	2	1	no	—
60	#15667946	—	2	1	no	—
61	#15667944	—	2	4	no	—
62	C₄⋊#10623494	—	2	1	no	—
63	C₄⋊#10623494	C₂³	1	2	yes	—
64	#15667937	—	2	2	no	—
65	#15667937	—	2	2	no	—
66	#15667934	—	2	4	no	—
67	#15667934	—	2	4	no	—
68	#15667924	—	2	2	no	—
69	#15667924	—	2	2	no	—
70	C₂×M₁₆×C₈₉	—	2	2	no	—
71	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	—	2	2	no	—
75	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂³×D₈₉	—	2	1	no	—
81	C₂×Dic₁₇₈	—	2	2	no	—
82	C₂×Dic₁₇₈	—	2	2	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×D₃₅₆	—	2	2	no	—
85	C₂×D₃₅₆	Q₁₆	1	4	yes	complemented
86	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
87	C₂²×Dic₈₉	—	2	1	no	—
88	C₂×C₄×D₈₉	—	2	2	no	—
89	C₂×C₄×D₈₉	—	2	2	no	—
90	C₂×C₄×D₈₉	—	4	2	no	—
91	C₂×C₄×D₈₉	—	4	2	no	—
92	Dic₃₅₆	—	4	2	no	—
93	Dic₃₅₆	—	4	2	no	—
94	C₂×#10623494	—	2	2	no	—
95	C₂×#10623494	—	4	2	no	—
96	C₈×D₈₉	—	4	4	no	—
97	C₈×D₈₉	—	4	4	no	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	2	2	no	—
101	C₄⋊Dic₈₉	—	2	2	no	—
102	C₄⋊Dic₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
104	#13083553	—	4	2	no	—
105	#13083553	—	4	2	no	—
106	C₂²⋊Dic₈₉	—	4	2	no	—
107	#13083551	—	2	2	no	—
108	#13083551	—	2	2	no	—
109	C₄×Dic₈₉	—	4	2	no	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
113	M₁₆×C₈₉	—	4	4	no	—
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	2	4	no	—
116	C₂×C₈×C₈₉	—	4	1	no	—
117	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	#41	1	1	yes	characteristic
124	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂³×C₈₉	#121	1	1	yes	characteristic
126	C₂²×D₈₉	—	2	2	no	—
127	C₂²×D₈₉	—	2	4	no	—
128	C₂²×D₈₉	—	4	2	no	—
129	C₂²×D₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
137	C₂×C₄×C₈₉	#135	1	2	yes	—
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	4	2	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	D₃₅₆	—	2	2	no	—
146	D₃₅₆	—	2	4	no	—
147	D₃₅₆	C₂×Q₁₆	1	4	yes	—
148	C₄×D₈₉	—	4	4	no	—
149	C₄×D₈₉	—	4	2	no	—
150	C₄×D₈₉	—	4	2	no	—
151	Dic₁₇₈	—	4	2	no	—
152	Dic₁₇₈	—	4	2	no	—
153	Dic₁₇₈	—	8	2	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	—	4	4	no	—
158	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
159	#10623494	—	4	2	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	#467	1	2	yes	—
163	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
164	C₂×D₈₉	—	2	4	no	—
165	C₂×D₈₉	—	4	8	no	—
166	C₂×D₈₉	—	4	2	no	—
167	C₂×D₈₉	—	4	2	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	#467	1	2	yes	—
172	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
173	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
174	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₄×C₈₉	#575	1	1	yes	characteristic
176	Dic₈₉	—	4	2	no	—
177	Dic₈₉	—	8	2	no	—
178	Dic₈₉	—	8	2	no	—
179	#19256	—	89	1	no	maximal, Sylow
180	C₂×C₈₉	—	2	2	no	—
181	C₂×C₈₉	#3036	1	1	yes	characteristic
182	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
183	C₂×C₈₉	#3067	1	1	yes	characteristic
184	D₈₉	—	4	4	no	—
185	#3170	—	89	1	no	—
186	#3077	—	89	2	no	—
187	C₂×C₈×D₄	—	89	1	no	—
188	#1683	—	89	4	no	—
189	#1675	—	89	2	no	—
190	#1669	—	89	2	no	—
191	#1567	D₈₉	1	1	yes	characteristic, complemented
192	#1550	—	89	2	no	—
193	C₈₉	#19256	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₈	—	178	1	no	—
195	#533	—	89	2	no	—
196	C₂×C₄×D₄	—	89	1	no	—
197	#500	—	89	1	no	—
198	#494	—	89	1	no	—
199	#487	—	89	2	no	—
200	#478	—	178	4	no	—
201	#476	—	89	4	no	—
202	#474	—	89	4	no	—
203	#467	—	178	1	no	—
204	#467	—	178	1	no	—
205	C₄×Q₁₆	—	178	1	no	—
206	C₄×Q₁₆	—	178	1	no	—
207	C₈×D₄	—	89	2	no	—
208	C₈×D₄	—	89	2	no	—
209	C₈×D₄	—	178	1	no	—
210	C₈×D₄	—	178	1	no	—
211	#428	—	178	2	no	—
212	#423	C₂×D₈₉	1	2	yes	—
213	C₂×C₄⋊C₈	—	89	1	no	—
214	#418	—	178	2	no	—
215	C₂×C₂²⋊C₈	—	178	1	no	—
216	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
217	#335	C₂×D₈₉	1	2	yes	complemented
218	#335	C₂×D₈₉	1	2	yes	complemented
219	#333	—	89	4	no	—
220	#332	—	89	4	no	—
221	C₂²×D₄	—	89	1	no	—
222	C₂³×C₄	—	178	1	no	—
223	C₂×Q₁₆	—	178	1	no	—
224	C₂×Q₁₆	—	178	1	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂²×C₈	—	356	4	no	—
227	C₂²×C₈	—	178	1	no	—
228	C₂²×C₈	—	178	2	no	—
229	C₂²×C₈	—	178	2	no	—
230	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
231	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
232	C₄⋊Q₈	—	89	2	no	—
233	C₄⋊Q₈	—	89	2	no	—
234	#125	—	178	2	no	—
235	C₂²⋊Q₈	—	178	2	no	—
236	C₂²⋊Q₈	—	178	2	no	—
237	C₄×Q₈	—	178	2	no	—
238	C₄×Q₈	—	178	2	no	—
239	C₄×D₄	—	89	2	no	—
240	C₄×D₄	—	89	2	no	—
241	C₄×D₄	—	178	1	no	—
242	C₄×D₄	—	178	1	no	—
243	#117	—	178	2	no	—
244	C₂×C₄⋊C₄	—	89	1	no	—
245	C₂×C₂²⋊C₄	—	178	1	no	—
246	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
247	#107	—	178	4	no	—
248	#107	—	178	1	no	—
249	#107	—	178	1	no	—
250	#106	—	178	4	no	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	89	2	no	—
253	C₄⋊C₈	—	178	1	no	—
254	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
255	#103	—	178	2	no	—
256	#103	—	178	2	no	—
257	#102	—	178	4	no	—
258	#102	—	178	4	no	—
259	C₂²⋊C₈	—	178	2	no	—
260	C₂²⋊C₈	—	178	2	no	—
261	C₄×C₈	D₃₅₆	1	1	yes	characteristic
262	C₄×C₈	D₃₅₆	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₂⁴	—	178	1	no	—
266	C₂×Q₈	—	178	2	no	—
267	C₂×Q₈	—	178	2	no	—
268	C₂×D₄	—	89	4	no	—
269	C₂×D₄	—	89	2	no	—
270	C₂×D₄	—	178	1	no	—
271	C₂×D₄	—	178	2	no	—
272	C₂²×C₄	—	356	2	no	—
273	C₂²×C₄	—	356	2	no	—
274	C₂²×C₄	—	178	1	no	—
275	C₂²×C₄	—	178	2	no	—
276	C₂²×C₄	—	178	2	no	—
277	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
279	C₂²×C₄	#13083575	1	1	yes	characteristic
280	Q₁₆	—	356	2	no	—
281	Q₁₆	—	356	2	no	—
282	M₁₆	—	4	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	356	4	no	—
287	C₂×C₈	—	356	2	no	—
288	C₂×C₈	—	356	4	no	—
289	C₂×C₈	—	178	2	no	—
290	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
291	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
292	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
293	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
294	C₄⋊C₄	—	89	2	no	—
295	C₄⋊C₄	—	356	2	no	—
296	C₄⋊C₄	—	356	2	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₂²⋊C₄	—	356	2	no	—
303	C₂²⋊C₄	—	178	2	no	—
304	C₂²⋊C₄	—	178	2	no	—
305	C₄²	—	356	2	no	—
306	C₄²	D₄×D₈₉	1	1	yes	characteristic
307	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
308	C₄²	#13083575	1	1	yes	characteristic
309	C₄²	C₂³×D₈₉	1	1	yes	characteristic
310	C₂³	—	356	2	no	—
311	C₂³	—	356	2	no	—
312	C₂³	—	178	2	no	—
313	C₂³	—	178	4	no	—
314	C₂³	#15667987	1	1	yes	characteristic
315	Q₈	—	356	2	no	—
316	Q₈	—	356	2	no	—
317	Q₈	—	712	2	no	—
318	D₄	—	89	4	no	—
319	D₄	—	178	4	no	—
320	D₄	—	178	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	356	4	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	—	356	2	no	—
329	C₂×C₄	—	356	2	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	#15667987	1	1	yes	characteristic
332	C₂×C₄	#15667987	1	1	yes	characteristic
333	C₂×C₄	#15667987	1	1	yes	characteristic
334	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
335	C₂×C₄	#15668031	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
338	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	4	4	no	—
343	C₈	—	356	2	no	—
344	C₈	#15668040	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	356	8	no	—
348	C₂²	—	356	2	no	—
349	C₂²	—	356	2	no	—
350	C₂²	—	178	4	no	—
351	C₂²	#19151840	1	2	yes	—
352	C₂²	C₂×#15667987	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	356	2	no	—
357	C₄	—	712	2	no	—
358	C₄	—	712	2	no	—
359	C₄	#19151840	1	2	yes	—
360	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
361	C₄	C₂×#15668031	1	1	yes	characteristic
362	C₄	C₂×#15668040	1	1	yes	characteristic
363	C₄	#19152398	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	356	4	no	—
366	C₂	#26029718	1	1	yes	characteristic
367	C₂	C₂×#19151840	1	1	yes	characteristic
368	C₂	#26029973	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1567	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	#1567	D₈₉
hypocenter	C₈₉	#19256
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. #26030838	C₂	semidirect
4. #26019704	C₂	semidirect
5. #26019652	C₂	semidirect
6. #26019618	C₂	semidirect
7. #26018943	C₂	semidirect
8. C₈₉×#1567	C₂	semidirect
12. #19152003	C₂²	semidirect
13. #19151978	C₂²	semidirect
14. #19151949	C₂²	semidirect
30. #19151346	C₂²	semidirect
31. #19151343	C₂²	semidirect
34. #335×C₈₉	C₂²	semidirect
35. #335×C₈₉	C₂²	semidirect
85. C₂×D₃₅₆	Q₁₆	semidirect
191. #1567	D₈₉	semidirect
193. C₈₉	#19256	semidirect
217. #335	C₂×D₈₉	semidirect
218. #335	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26030051	2
C₂×C₈×D₃₅₆	2
#26019704	2
#26019652	2
#26019618	2
#26018943	2
C₈₉×#1567	2
#19256	89