Subgroups — #36448314

Order 22784.

466 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029361	C₂	1	1	yes	maximal, characteristic
2	#26020403	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#15667938	C₂	1	1	yes	maximal, characteristic
4	#26019004	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018640	C₂	1	1	yes	maximal, characteristic
6	#26018084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083574	C₂²	1	1	yes	characteristic
9	#19151722	—	2	1	no	—
10	#19151663	—	2	1	no	—
11	C₈×C₂²⋊D₈₉	—	2	2	no	—
12	#19151651	C₂²	1	1	yes	characteristic
13	#19151650	—	2	1	no	—
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×#15667941	—	2	2	no	—
16	C₄⋊C₈×D₈₉	—	2	1	no	—
17	C₂×#15667938	C₄	1	1	yes	characteristic
18	C₂×#15667938	C₄	1	1	yes	characteristic
19	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
20	#19151359	C₄	1	1	yes	characteristic
21	#19151359	C₄	1	1	yes	characteristic
22	#19151351	—	2	1	no	—
23	#19151349	—	2	1	no	—
24	#19151330	—	2	1	no	—
25	#19151327	—	2	1	no	—
26	#431×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10623494	—	2	1	no	—
31	C₂×C₈×D₈₉	—	2	2	no	—
32	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
33	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
34	C₄×D₃₅₆	D₄	1	1	yes	characteristic
35	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₉	—	2	1	no	—
37	C₄×C₂²⋊D₈₉	—	2	1	no	—
38	#15667971	C₂×C₄	1	1	yes	characteristic
39	#15667969	—	2	1	no	—
40	C₂×C₄×Dic₈₉	—	2	1	no	—
41	C₂×C₄×Dic₈₉	—	2	1	no	—
42	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₈₉	—	2	1	no	—
45	C₄⋊#10623494	—	2	1	no	—
46	C₄⋊#10623494	—	2	1	no	—
47	C₄⋊#10623494	D₄	1	1	yes	characteristic
48	C₄⋊#10623494	D₄	1	1	yes	characteristic
49	#15667941	—	2	2	no	—
50	#15667941	—	4	4	no	—
51	#15667938	—	2	2	no	—
52	#15667938	C₈	1	4	yes	complemented
53	C₂²⋊#10623494	—	2	1	no	—
54	#15667924	—	2	2	no	—
55	C₄×#10623494	—	2	1	no	—
56	C₈×Dic₈₉	—	2	2	no	—
57	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	D₄	1	2	yes	—
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₈₉	—	2	1	no	—
62	C₄⋊C₈×C₈₉	—	2	1	no	—
63	C₄⋊C₈×C₈₉	—	2	1	no	—
64	C₂²⋊C₈×C₈₉	—	2	1	no	—
65	#97×C₈₉	—	2	1	no	—
66	C₄×C₈×C₈₉	—	2	1	no	—
67	C₄×C₈×C₈₉	—	2	1	no	—
68	#13083574	—	2	1	no	—
69	#13083574	—	2	1	no	—
70	#13083574	C₂×C₈	1	2	yes	—
71	#13083574	M₁₆	1	2	yes	—
72	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₈₉	—	2	1	no	—
75	C₂²×Dic₈₉	—	2	1	no	—
76	C₂²×Dic₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	—	2	1	no	—
80	C₂×#10623494	—	2	1	no	—
81	C₂×#10623494	—	2	1	no	—
82	C₂×#10623494	—	4	2	no	—
83	C₂×#10623494	—	4	1	no	—
84	C₈×D₈₉	—	8	2	no	—
85	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
86	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	C₄×Dic₈₉	—	2	2	no	—
93	C₄×Dic₈₉	—	2	2	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	—	2	1	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₄×Dic₈₉	C₂×C₈	1	2	yes	complemented
102	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
105	M₁₆×C₈₉	—	2	1	no	—
106	M₁₆×C₈₉	—	2	1	no	—
107	M₁₆×C₈₉	—	2	1	no	—
108	M₁₆×C₈₉	C₂×C₈	1	2	yes	complemented
109	C₂×C₈×C₈₉	—	2	1	no	—
110	C₂×C₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	4	2	no	—
113	C₂×C₈×C₈₉	—	4	1	no	—
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₈₉	#41	1	1	yes	characteristic
118	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
119	C₂²×D₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	2	2	no	—
134	C₂²⋊D₈₉	—	2	2	no	—
135	C₂×Dic₈₉	—	2	2	no	—
136	C₂×Dic₈₉	—	2	2	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	2	no	—
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	2	1	no	—
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	—	4	1	no	—
143	C₂×Dic₈₉	—	4	1	no	—
144	D₃₅₆	—	2	1	no	—
145	D₃₅₆	C₄⋊C₈	1	2	yes	complemented
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	4	1	no	—
149	C₄×D₈₉	—	4	1	no	—
150	C₄×D₈₉	—	8	1	no	—
151	Dic₁₇₈	—	2	1	no	—
152	Dic₁₇₈	C₄⋊C₈	1	2	yes	complemented
153	C₈×C₈₉	—	2	2	no	—
154	C₈×C₈₉	—	4	1	no	—
155	C₈×C₈₉	—	4	2	no	—
156	C₈×C₈₉	—	8	1	no	—
157	#10623494	—	4	1	no	—
158	#10623494	—	4	1	no	—
159	#10623494	—	8	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	#385	1	1	yes	characteristic
163	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₈₉	#435	1	1	yes	characteristic
165	C₂×D₈₉	—	2	2	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₄×C₈₉	#424	1	1	yes	characteristic
173	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
174	C₄×C₈₉	#435	1	1	yes	characteristic
175	C₄×C₈₉	#453	1	1	yes	characteristic
176	C₄×C₈₉	#456	1	1	yes	characteristic
177	Dic₈₉	—	2	4	no	—
178	Dic₈₉	—	2	2	no	—
179	Dic₈₉	—	4	2	no	—
180	Dic₈₉	—	4	1	no	—
181	Dic₈₉	—	4	1	no	—
182	#8839	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	1	no	—
184	C₂×C₈₉	—	2	1	no	—
185	C₂×C₈₉	#1850	1	1	yes	characteristic
186	C₂×C₈₉	#1859	1	1	yes	characteristic
187	C₂×C₈₉	#1861	1	1	yes	characteristic
188	D₈₉	—	4	2	no	—
189	#2988	—	89	1	no	—
190	#1842	—	89	1	no	—
191	C₄×#104	—	89	1	no	—
192	#1565	—	89	1	no	—
193	#1477	—	89	1	no	—
194	#1283	D₈₉	1	1	yes	characteristic, complemented
195	#1276	—	89	1	no	—
196	C₈₉	#8839	1	1	yes	characteristic, complemented, Sylow
197	C₄×#41	—	89	1	no	—
198	C₈×Q₈	—	178	1	no	—
199	C₈×D₄	—	178	1	no	—
200	C₈×D₄	—	178	2	no	—
201	#431	—	89	1	no	—
202	#431	—	178	1	no	—
203	#431	C₂×D₈₉	1	1	yes	characteristic
204	C₂×C₄⋊C₈	—	89	1	no	—
205	C₂×C₄⋊C₈	—	178	1	no	—
206	C₂×C₄⋊C₈	—	178	2	no	—
207	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
208	C₂×#104	—	89	1	no	—
209	C₂×#104	—	89	1	no	—
210	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
211	C₄³	—	89	1	no	—
212	#339	—	89	1	no	—
213	#339	—	89	1	no	—
214	#335	—	178	1	no	—
215	#334	—	178	1	no	—
216	Q₈⋊C₈	—	178	1	no	—
217	D₄⋊C₈	—	178	1	no	—
218	C₂×#41	—	89	1	no	—
219	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
220	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
221	C₂²×C₈	—	178	2	no	—
222	C₂²×C₈	—	178	1	no	—
223	C₂²×C₈	D₃₅₆	1	2	yes	—
224	C₄×Q₈	—	89	1	no	—
225	C₄×Q₈	—	89	1	no	—
226	C₄×D₄	—	89	1	no	—
227	C₄×D₄	—	89	1	no	—
228	C₄×D₄	—	178	1	no	—
229	C₄×D₄	—	178	1	no	—
230	#117	—	89	1	no	—
231	#117	—	178	1	no	—
232	C₂×C₄²	—	89	1	no	—
233	C₂×C₄²	—	89	1	no	—
234	C₂×C₄²	—	178	1	no	—
235	C₂×C₄²	—	178	1	no	—
236	C₂×C₄²	—	178	1	no	—
237	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	89	1	no	—
242	C₄⋊C₈	—	89	1	no	—
243	C₄⋊C₈	—	356	4	no	—
244	C₄⋊C₈	—	178	2	no	—
245	C₄⋊C₈	—	178	1	no	—
246	C₄⋊C₈	—	178	1	no	—
247	#104	—	89	4	no	—
248	#104	—	178	2	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	178	2	no	—
251	C₂²⋊C₈	—	178	1	no	—
252	#97	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	178	2	no	—
256	C₄×C₈	—	178	1	no	—
257	#41	—	89	2	no	—
258	#41	—	89	2	no	—
259	#41	—	178	1	no	—
260	#41	—	178	1	no	—
261	C₂×Q₈	—	89	1	no	—
262	C₂×D₄	—	89	1	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂²×C₄	—	178	1	no	—
265	C₂²×C₄	—	178	1	no	—
266	C₂²×C₄	—	178	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	—	178	1	no	—
269	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
271	C₂²×C₄	#13083574	1	1	yes	characteristic
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	C₈×D₈₉	1	2	yes	complemented
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	356	2	no	—
282	C₂×C₈	—	356	1	no	—
283	C₂×C₈	—	178	1	no	—
284	C₂×C₈	—	178	1	no	—
285	C₂×C₈	—	712	2	no	—
286	C₄⋊C₄	—	89	1	no	—
287	C₄⋊C₄	—	89	1	no	—
288	C₄⋊C₄	—	178	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₄²	—	89	2	no	—
294	C₄²	—	356	2	no	—
295	C₄²	—	178	2	no	—
296	C₄²	—	178	2	no	—
297	C₄²	—	178	1	no	—
298	C₄²	—	178	1	no	—
299	C₄²	—	178	1	no	—
300	C₄²	—	178	1	no	—
301	C₄²	—	178	1	no	—
302	C₄²	—	178	1	no	—
303	C₄²	D₄×D₈₉	1	1	yes	characteristic
304	C₄²	D₄×D₈₉	1	1	yes	characteristic
305	C₄²	Q₈×D₈₉	1	1	yes	characteristic
306	C₄²	#13083573	1	1	yes	characteristic
307	C₂³	—	178	1	no	—
308	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
309	Q₈	—	89	2	no	—
310	Q₈	—	178	1	no	—
311	D₄	—	89	2	no	—
312	D₄	—	178	2	no	—
313	D₄	—	178	2	no	—
314	D₄	—	178	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	356	1	no	—
319	C₂×C₄	—	356	1	no	—
320	C₂×C₄	—	356	1	no	—
321	C₂×C₄	—	356	1	no	—
322	C₂×C₄	—	178	2	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	—	712	1	no	—
332	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
333	C₂×C₄	#15667972	1	1	yes	characteristic
334	C₂×C₄	#15667982	1	1	yes	characteristic
335	C₂×C₄	#15667990	1	1	yes	characteristic
336	C₂×C₄	#15667992	1	1	yes	characteristic
337	C₂×C₄	#15667994	1	1	yes	characteristic
338	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
339	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
340	C₂×C₄	#15668018	1	1	yes	characteristic
341	C₂×C₄	#15668020	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	4	2	no	—
345	C₈	—	8	1	no	—
346	C₈	—	356	1	no	—
347	C₈	—	356	1	no	—
348	C₈	—	712	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	356	1	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	178	2	no	—
354	C₂²	#19151472	1	1	yes	characteristic
355	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
356	C₂²	#19151671	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	356	2	no	—
361	C₄	—	356	1	no	—
362	C₄	—	356	1	no	—
363	C₄	—	178	4	no	—
364	C₄	—	178	2	no	—
365	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
366	C₄	#19151623	1	1	yes	characteristic
367	C₄	#19151661	1	1	yes	characteristic
368	C₄	#19151672	1	1	yes	characteristic
369	C₄	#19151768	1	1	yes	characteristic
370	C₄	#19151783	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	356	2	no	—
374	C₂	#26020431	1	1	yes	characteristic
375	C₂	#26020481	1	1	yes	characteristic
376	C₂	#26020495	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667982
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1283	C₂
socle	C₂²×C₈₉	#385
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1283	D₈₉
hypocenter	C₈₉	#8839
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
2. #26020403	C₂	semidirect
4. #26019004	C₂	semidirect
6. #26018084	C₂	semidirect
7. C₈₉×#1283	C₂	semidirect
52. #15667938	C₈	semidirect
101. C₄×Dic₈₉	C₂×C₈	semidirect
108. M₁₆×C₈₉	C₂×C₈	semidirect
145. D₃₅₆	C₄⋊C₈	semidirect
152. Dic₁₇₈	C₄⋊C₈	semidirect
194. #1283	D₈₉	semidirect
196. C₈₉	#8839	semidirect
275. M₁₆	C₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029361	2
#26020403	2
C₄×#15667938	2
#26019004	2
#26018640	2
#26018084	2
C₈₉×#1283	2
#8839	89