Subgroups — #36471433

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667948	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151414	C₂	1	2	yes	maximal, complemented
3	#26021863	C₂	1	1	yes	maximal, characteristic
4	#26018360	C₂	1	2	yes	maximal, complemented
5	C₂×#467×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	—	2	1	no	—
7	C₂×#15667952	C₂²	1	2	yes	—
8	C₂×#15667948	—	2	4	no	—
9	C₂×#15667948	—	2	1	no	—
10	C₂×#15667948	C₂²	1	2	yes	—
11	#19151414	—	2	4	no	—
12	C₂²⋊#13083549	—	2	1	no	—
13	C₂²⋊#13083549	—	2	1	no	—
14	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
17	#467×C₈₉	—	2	1	no	—
18	#467×C₈₉	—	2	1	no	—
19	#467×C₈₉	C₄	1	2	yes	—
20	#467×C₈₉	C₄	1	2	yes	—
21	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#103×C₈₉	—	2	1	no	—
23	C₂×#13083558	—	2	4	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	4	2	no	—
26	C₂×#13083558	—	4	1	no	—
27	C₂²×#10623494	—	4	1	no	—
28	#15667952	—	2	2	no	—
29	#15667952	—	2	2	no	—
30	C₂×#13083549	—	2	2	no	—
31	#15667948	—	2	4	no	—
32	#15667948	—	2	2	no	—
33	#15667948	D₄	1	4	yes	—
34	C₂²×Q₈×C₈₉	—	2	1	no	—
35	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
36	C₂×Q₁₆×C₈₉	—	2	2	no	—
37	C₂×Q₁₆×C₈₉	—	2	1	no	—
38	C₂×Q₁₆×C₈₉	—	2	2	no	—
39	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	—	2	2	no	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂²⋊Q₈×C₈₉	—	2	2	no	—
49	C₂²⋊Q₈×C₈₉	—	2	2	no	—
50	C₂²⋊Q₈×C₈₉	—	4	1	no	—
51	C₂²⋊Q₈×C₈₉	—	4	1	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
54	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
55	#107×C₈₉	—	2	1	no	—
56	#107×C₈₉	C₂×C₄	1	2	yes	—
57	#103×C₈₉	—	2	2	no	—
58	#103×C₈₉	—	2	2	no	—
59	#13083558	—	2	4	no	—
60	#13083558	—	4	4	no	—
61	#13083558	—	4	2	no	—
62	C₂×#10623494	—	4	4	no	—
63	C₂×#10623494	—	4	2	no	—
64	#13083549	—	4	2	no	—
65	C₂⁴×C₈₉	SD₁₆	1	1	yes	characteristic
66	C₂×Q₈×C₈₉	—	2	2	no	—
67	C₂×Q₈×C₈₉	—	4	1	no	—
68	C₂×Q₈×C₈₉	—	4	1	no	—
69	C₂×Q₈×C₈₉	—	4	1	no	—
70	C₂×Q₈×C₈₉	—	4	1	no	—
71	C₂×Q₈×C₈₉	—	4	1	no	—
72	C₂²×C₄×C₈₉	—	2	1	no	—
73	C₂²×C₄×C₈₉	—	2	1	no	—
74	C₂²×C₄×C₈₉	—	2	1	no	—
75	C₂²×C₄×C₈₉	—	2	1	no	—
76	C₂²×C₄×C₈₉	—	4	1	no	—
77	C₂²×C₄×C₈₉	—	4	1	no	—
78	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₈₉	—	2	4	no	—
85	Q₁₆×C₈₉	—	4	2	no	—
86	C₂×C₈×C₈₉	—	2	1	no	—
87	C₂×C₈×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	#41	1	2	yes	—
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	4	2	no	—
100	C₄⋊C₄×C₈₉	—	4	2	no	—
101	C₄⋊C₄×C₈₉	—	4	1	no	—
102	C₂²⋊C₄×C₈₉	—	4	2	no	—
103	C₂²⋊C₄×C₈₉	—	4	2	no	—
104	C₂³×C₈₉	—	2	2	no	—
105	C₂³×C₈₉	—	2	1	no	—
106	C₂³×C₈₉	—	2	1	no	—
107	C₂³×C₈₉	—	4	1	no	—
108	C₂³×C₈₉	#102	1	1	yes	characteristic
109	C₂³×C₈₉	#102	1	1	yes	characteristic
110	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
111	Q₈×C₈₉	—	4	2	no	—
112	Q₈×C₈₉	—	4	2	no	—
113	Q₈×C₈₉	—	8	1	no	—
114	Q₈×C₈₉	—	8	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	8	2	no	—
128	C₂×C₄×C₈₉	—	8	1	no	—
129	C₂×C₄×C₈₉	—	8	1	no	—
130	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	C₄×D₄	1	2	yes	—
141	C₈×C₈₉	#121	1	2	yes	—
142	#10623494	—	4	4	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	4	2	no	—
151	C₂²×C₈₉	—	4	2	no	—
152	C₂²×C₈₉	#362	1	2	yes	complemented
153	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
154	C₂²×C₈₉	#448	1	1	yes	characteristic
155	C₂²×C₈₉	#481	1	1	yes	characteristic
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	#386	1	1	yes	characteristic
161	C₄×C₈₉	#417	1	1	yes	characteristic
162	C₄×C₈₉	#447	1	1	yes	characteristic
163	C₄×C₈₉	#480	1	1	yes	characteristic
164	#13003	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	2	no	—
166	C₂×C₈₉	—	2	1	no	—
167	C₂×C₈₉	—	2	1	no	—
168	C₂×C₈₉	—	4	1	no	—
169	C₂×C₈₉	#1874	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#362	1	1	yes	characteristic
171	C₂×C₈₉	#2148	1	1	yes	characteristic
172	C₂×#467	D₈₉	1	1	yes	characteristic
173	C₂²×#108	—	89	1	no	—
174	C₂×#362	—	89	2	no	—
175	#2112	—	89	1	no	—
176	#1387	—	89	2	no	—
177	C₈₉	#13003	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
179	C₂²×M₁₆	—	178	1	no	—
180	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
181	C₂×C₂²⋊Q₈	—	2	1	no	—
182	C₂×M₃₂	—	89	2	no	—
183	#467	—	2	1	no	—
184	#467	—	2	1	no	—
185	#467	Dic₈₉	1	2	yes	—
186	#467	Dic₈₉	1	2	yes	—
187	C₂×#108	—	89	2	no	—
188	C₂×#108	—	178	4	no	—
189	C₂×#108	—	178	1	no	—
190	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
191	C₂×#103	—	2	1	no	—
192	#362	—	178	4	no	—
193	C₂²⋊C₁₆	—	178	1	no	—
194	C₂²⋊C₁₆	—	178	1	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	2	no	—
200	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
201	C₂×M₁₆	—	356	2	no	—
202	C₂×M₁₆	—	356	1	no	—
203	C₂×M₁₆	—	178	4	no	—
204	C₂×M₁₆	—	178	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	356	1	no	—
208	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
209	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
211	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
212	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	1	no	—
217	C₂²⋊Q₈	—	4	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	4	1	no	—
220	C₂×C₂²⋊C₄	—	4	1	no	—
221	M₃₂	—	178	2	no	—
222	M₃₂	—	178	2	no	—
223	C₂×C₁₆	—	178	2	no	—
224	#108	—	89	4	no	—
225	#108	—	178	4	no	—
226	#108	—	178	2	no	—
227	#107	—	2	1	no	—
228	#107	C₂×Dic₈₉	1	2	yes	—
229	#103	—	2	2	no	—
230	#103	—	2	2	no	—
231	C₂⁴	#13083562	1	1	yes	characteristic
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
246	C₂²×C₄	#13083559	1	1	yes	characteristic
247	C₂²×C₄	#13083559	1	1	yes	characteristic
248	C₂²×C₄	#13083562	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	4	2	no	—
252	M₁₆	—	356	4	no	—
253	M₁₆	—	356	2	no	—
254	M₁₆	—	178	4	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	356	4	no	—
262	C₂×C₈	—	356	2	no	—
263	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
266	C₂×C₈	D₄×D₈₉	1	2	yes	—
267	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
268	C₂×C₈	#13083573	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₁₆	—	356	2	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	#15667933	1	1	yes	characteristic
281	C₂³	#15667933	1	1	yes	characteristic
282	C₂³	C₂×#13083562	1	1	yes	characteristic
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	#15667933	1	1	yes	characteristic
303	C₂×C₄	#15667933	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
305	C₂×C₄	#15667980	1	1	yes	characteristic
306	C₂×C₄	#15667993	1	1	yes	characteristic
307	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
308	C₂×C₄	#15668037	1	1	yes	characteristic
309	C₂×C₄	#15668042	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	356	4	no	—
313	C₈	D₄×Dic₈₉	1	2	yes	—
314	C₈	#15667984	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	#19151414	1	2	yes	complemented
325	C₂²	C₂×#15667933	1	1	yes	characteristic
326	C₂²	#19151736	1	1	yes	characteristic
327	C₂²	#19151914	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	#19151478	1	1	yes	characteristic
333	C₄	#19151599	1	1	yes	characteristic
334	C₄	#19151731	1	1	yes	characteristic
335	C₄	#19151909	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	4	1	no	—
340	C₂	#26020570	1	1	yes	characteristic
341	C₂	C₂×#19151414	1	1	yes	characteristic
342	C₂	#26022001	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667933
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#467×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂⁴×C₈₉	SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#467	D₈₉
hypocenter	C₈₉	#13003
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#19151414	C₂	semidirect
4. #26018360	C₂	semidirect
152. C₂²×C₈₉	#362	semidirect
177. C₈₉	#13003	semidirect
324. C₂²	#19151414	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667948	2
C₂×#19151414	2
#26021863	2
#26018360	2
C₂×#467×C₈₉	2
#13003	89