Subgroups — #36700732

Order 22784.

672 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26025622	C₂	1	2	yes	maximal
2	#26021789	C₂	1	2	yes	maximal
3	#26021725	C₂	1	2	yes	maximal
4	#26021646	C₂	1	2	yes	maximal
5	#26021113	C₂	1	1	yes	maximal, characteristic
6	#26020985	C₂	1	1	yes	maximal, characteristic
7	#26020592	C₂	1	2	yes	maximal
8	#26020509	C₂	1	2	yes	maximal
9	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
12	C₂×#15667946	C₂²	1	1	yes	characteristic
13	C₂×#15667946	C₂²	1	2	yes	—
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667946	C₂²	1	1	yes	characteristic
16	C₂×#15667937	C₂²	1	2	yes	—
17	C₂×#15667937	C₂²	1	2	yes	—
18	C₂×#15667937	C₂²	1	2	yes	—
19	#19151531	—	2	2	no	—
20	#19151531	—	2	2	no	—
21	#19151505	—	2	2	no	—
22	#19151504	C₂²	1	2	yes	—
23	#19151504	C₂²	1	2	yes	—
24	#19151496	C₂²	1	2	yes	—
25	#19151496	C₂²	1	2	yes	—
26	#19151494	—	2	2	no	—
27	#19151468	C₂²	1	2	yes	—
28	#19151361	C₂²	1	2	yes	—
29	#19151361	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	#336×C₈₉	C₂²	1	2	yes	—
34	#336×C₈₉	C₂²	1	2	yes	—
35	C₂²×Dic₁₇₈	C₂³	1	2	yes	—
36	C₂×Dic₃₅₆	—	4	1	no	—
37	C₂×Dic₃₅₆	—	4	1	no	—
38	C₂×Dic₃₅₆	Q₈	1	4	yes	complemented
39	C₄×Dic₁₇₈	—	4	2	no	—
40	C₄×Dic₁₇₈	—	4	2	no	—
41	C₂×C₄⋊Dic₈₉	—	2	2	no	—
42	C₂×C₄⋊Dic₈₉	—	2	2	no	—
43	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
46	C₂×#13083553	—	2	2	no	—
47	C₂×#13083553	—	2	2	no	—
48	C₂×#13083553	—	2	2	no	—
49	C₂×C₄×Dic₈₉	—	2	2	no	—
50	#15667946	—	2	2	no	—
51	#15667946	—	4	1	no	—
52	#15667946	—	4	1	no	—
53	#15667946	—	4	2	no	—
54	#15667937	—	2	4	no	—
55	#15667937	—	4	2	no	—
56	#15667937	—	4	2	no	—
57	#15667919	—	2	2	no	—
58	#15667919	—	2	2	no	—
59	#15667919	—	2	2	no	—
60	#15667919	—	2	2	no	—
61	#15667919	—	2	2	no	—
62	#15667919	—	2	2	no	—
63	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×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	—	4	1	no	—
68	C₄×Q₈×C₈₉	—	4	1	no	—
69	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
73	C₄⋊C₈×C₈₉	—	4	1	no	—
74	C₄×C₈×C₈₉	—	4	1	no	—
75	C₂×Dic₁₇₈	—	2	2	no	—
76	C₂×Dic₁₇₈	—	2	2	no	—
77	C₂×Dic₁₇₈	—	2	2	no	—
78	C₂×Dic₁₇₈	—	2	2	no	—
79	C₂×Dic₁₇₈	—	2	2	no	—
80	C₂×Dic₁₇₈	C₂×Q₈	1	4	yes	—
81	C₂²×Dic₈₉	—	2	2	no	—
82	C₂²×Dic₈₉	—	2	2	no	—
83	C₂²×Dic₈₉	—	2	2	no	—
84	C₂²×Dic₈₉	—	2	2	no	—
85	Dic₃₅₆	—	4	4	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	4	4	no	—
93	C₄⋊Dic₈₉	—	8	2	no	—
94	#13083553	—	4	4	no	—
95	#13083553	—	8	2	no	—
96	#13083553	—	8	2	no	—
97	C₄×Dic₈₉	—	4	4	no	—
98	C₂×Q₈×C₈₉	—	4	1	no	—
99	C₂×Q₈×C₈₉	—	4	1	no	—
100	C₂×Q₈×C₈₉	Q₁₆	1	4	yes	complemented
101	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₈₉	—	2	1	no	—
107	C₂×C₈×C₈₉	—	2	1	no	—
108	C₂×C₈×C₈₉	—	2	1	no	—
109	C₂×C₈×C₈₉	—	2	1	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
114	C₂×C₈×C₈₉	#41	1	2	yes	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	2	no	—
117	C₄²×C₈₉	—	2	2	no	—
118	C₄²×C₈₉	—	2	2	no	—
119	C₄²×C₈₉	—	2	1	no	—
120	C₄²×C₈₉	—	2	1	no	—
121	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₈₉	—	4	4	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
135	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×Dic₈₉	—	2	4	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	4	2	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	C₂×Dic₈₉	—	4	2	no	—
147	C₂×Dic₈₉	—	4	2	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	C₂×Dic₈₉	—	4	2	no	—
153	Dic₁₇₈	—	2	4	no	—
154	Dic₁₇₈	—	2	4	no	—
155	Dic₁₇₈	—	4	2	no	—
156	Dic₁₇₈	—	4	2	no	—
157	C₈×C₈₉	—	2	2	no	—
158	C₈×C₈₉	—	2	2	no	—
159	C₈×C₈₉	—	4	2	no	—
160	C₂²×C₈₉	#534	1	1	yes	characteristic
161	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
162	C₂²×C₈₉	#550	1	1	yes	characteristic
163	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
164	C₂²×C₈₉	#575	1	1	yes	characteristic
165	C₂²×C₈₉	#576	1	1	yes	characteristic
166	C₂²×C₈₉	#578	1	1	yes	characteristic
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	4	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	2	no	—
171	C₄×C₈₉	#539	1	1	yes	characteristic
172	C₄×C₈₉	#543	1	1	yes	characteristic
173	C₄×C₈₉	#554	1	1	yes	characteristic
174	C₄×C₈₉	#562	1	1	yes	characteristic
175	Dic₈₉	—	4	4	no	—
176	Dic₈₉	—	8	2	no	—
177	Dic₈₉	—	8	2	no	—
178	Dic₈₉	—	8	2	no	—
179	#32650	—	89	1	no	maximal, Sylow
180	C₂×C₈₉	#2732	1	1	yes	characteristic
181	C₂×C₈₉	#3189	1	1	yes	characteristic
182	C₂×C₈₉	#3203	1	1	yes	characteristic
183	C₂×C₈₉	Q₈×Q₁₆	1	1	yes	characteristic
184	C₂×C₈₉	#3390	1	1	yes	characteristic
185	C₂×C₈₉	#3395	1	1	yes	characteristic
186	C₂×C₈₉	#3403	1	1	yes	characteristic
187	#2582	—	89	2	no	—
188	#2095	—	89	2	no	—
189	#2083	—	89	2	no	—
190	#2068	—	89	2	no	—
191	#1984	D₈₉	1	1	yes	characteristic
192	#1969	—	89	1	no	—
193	#1942	—	89	1	no	—
194	#1878	—	89	2	no	—
195	#1863	—	89	2	no	—
196	C₈₉	#32650	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₁₆	—	89	1	no	—
198	C₂×C₄×Q₈	—	89	2	no	—
199	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
200	C₂×#107	—	89	2	no	—
201	C₂×#107	—	89	1	no	—
202	C₂×#107	—	89	1	no	—
203	C₂×#107	—	89	1	no	—
204	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
205	C₂×#103	—	89	2	no	—
206	C₂×#103	—	89	2	no	—
207	C₂×#103	—	89	2	no	—
208	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
209	#398	—	178	2	no	—
210	#398	—	178	2	no	—
211	#391	—	89	2	no	—
212	#391	—	89	2	no	—
213	#391	—	178	2	no	—
214	#389	—	89	2	no	—
215	#389	—	89	2	no	—
216	#389	—	178	2	no	—
217	#384	—	89	2	no	—
218	#340	—	89	2	no	—
219	#340	—	89	2	no	—
220	#336	C₂×D₈₉	1	2	yes	—
221	#336	C₂×D₈₉	1	2	yes	—
222	C₂²×Q₈	—	89	2	no	—
223	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
224	C₂×Q₁₆	—	89	4	no	—
225	C₂×Q₁₆	—	356	1	no	—
226	C₂×Q₁₆	—	356	1	no	—
227	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
228	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
230	C₄×Q₈	—	4	1	no	—
231	C₄×Q₈	—	4	1	no	—
232	C₄×Q₈	—	356	2	no	—
233	C₄×Q₈	—	356	2	no	—
234	C₂×C₄⋊C₄	—	89	2	no	—
235	C₂×C₄⋊C₄	—	89	2	no	—
236	C₂×C₄⋊C₄	—	89	2	no	—
237	C₂×C₄⋊C₄	—	178	2	no	—
238	C₂×C₄⋊C₄	—	178	2	no	—
239	C₂×C₄⋊C₄	—	178	2	no	—
240	C₂×C₄⋊C₄	—	178	2	no	—
241	C₂×C₄⋊C₄	—	178	2	no	—
242	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
243	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
244	C₂×C₄²	—	178	2	no	—
245	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
246	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
247	#107	—	356	1	no	—
248	#107	—	356	2	no	—
249	#107	—	356	1	no	—
250	#107	—	178	2	no	—
251	C₄⋊C₈	—	4	1	no	—
252	#103	—	356	2	no	—
253	#103	—	356	2	no	—
254	#103	—	178	4	no	—
255	C₄×C₈	—	4	1	no	—
256	#95	—	178	2	no	—
257	#95	—	178	2	no	—
258	#95	—	178	2	no	—
259	#95	—	178	2	no	—
260	#95	—	178	2	no	—
261	#95	—	178	2	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	89	4	no	—
265	C₂×Q₈	—	178	2	no	—
266	C₂×Q₈	—	178	2	no	—
267	C₂×Q₈	—	178	2	no	—
268	C₂×Q₈	—	178	2	no	—
269	C₂×Q₈	—	178	2	no	—
270	C₂×Q₈	Dic₃₅₆	1	4	yes	complemented
271	C₂²×C₄	—	178	2	no	—
272	C₂²×C₄	—	178	2	no	—
273	C₂²×C₄	—	178	2	no	—
274	C₂²×C₄	—	178	2	no	—
275	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
277	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
279	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
280	Q₁₆	—	356	4	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	Q₈×D₈₉	1	2	yes	—
289	C₂×C₈	#13083575	1	2	yes	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	356	4	no	—
293	C₄⋊C₄	—	356	4	no	—
294	C₄⋊C₄	—	178	2	no	—
295	C₄⋊C₄	—	178	2	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₄⋊C₄	—	712	2	no	—
301	C₄⋊C₄	—	712	2	no	—
302	C₄⋊C₄	—	712	2	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	356	4	no	—
308	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
309	Q₈	—	4	4	no	—
310	Q₈	—	356	2	no	—
311	Q₈	—	356	2	no	—
312	Q₈	—	178	4	no	—
313	Q₈	—	178	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	356	2	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	—	356	2	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	356	2	no	—
333	C₂×C₄	—	356	2	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	178	4	no	—
336	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
337	C₂×C₄	C₂×Dic₃₅₆	1	4	yes	—
338	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
339	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
340	C₂×C₄	#15668057	1	1	yes	characteristic
341	C₂×C₄	#15668057	1	1	yes	characteristic
342	C₂×C₄	#15668058	1	1	yes	characteristic
343	C₂×C₄	#15668059	1	1	yes	characteristic
344	C₈	—	2	2	no	—
345	C₈	—	2	2	no	—
346	C₈	—	4	2	no	—
347	C₂²	#19152160	1	1	yes	characteristic
348	C₂²	Q₈×D₃₅₆	1	1	yes	characteristic
349	C₂²	#19152270	1	1	yes	characteristic
350	C₂²	C₂²×Dic₃₅₆	1	1	yes	characteristic
351	C₂²	#19152398	1	1	yes	characteristic
352	C₂²	#19152403	1	1	yes	characteristic
353	C₂²	#19152415	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	356	4	no	—
359	C₄	—	712	2	no	—
360	C₄	—	712	2	no	—
361	C₄	—	712	2	no	—
362	C₄	#19152197	1	1	yes	characteristic
363	C₄	#19152223	1	1	yes	characteristic
364	C₄	#19152309	1	1	yes	characteristic
365	C₄	#19152342	1	1	yes	characteristic
366	C₂	#26027270	1	1	yes	characteristic
367	C₂	#26031027	1	1	yes	characteristic
368	C₂	#26031154	1	1	yes	characteristic
369	C₂	Q₈×Dic₃₅₆	1	1	yes	characteristic
370	C₂	#26033221	1	1	yes	characteristic
371	C₂	#26033258	1	1	yes	characteristic
372	C₂	#26033342	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1984	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
hypercenter	#1984	D₈₉
hypocenter	C₈₉	#32650
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ #1984

Complements

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

N	H ≅ G/N	type
38. C₂×Dic₃₅₆	Q₈	semidirect
100. C₂×Q₈×C₈₉	Q₁₆	semidirect
196. C₈₉	#32650	semidirect
270. C₂×Q₈	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26025622	2
#26021789	2
#26021725	2
#26021646	2
#26021113	2
#26020985	2
#26020592	2
#26020509	2
C₈₉×#1984	2
#32650	89