Subgroups — #36712156

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033241	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022491	C₂	1	2	yes	maximal, complemented
3	#26022481	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022401	C₂	1	2	yes	maximal, complemented
5	#26022386	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022307	C₂	1	2	yes	maximal
7	#26022297	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022116	C₂	1	2	yes	maximal, complemented
10	C₄×#15667953	C₂	1	1	yes	maximal, characteristic
11	Q₈×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152321	—	2	1	no	—
13	C₂×#15667956	C₂²	1	2	yes	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	#19151982	C₂²	1	2	yes	—
16	#19151978	C₂²	1	1	yes	characteristic
17	#19151821	—	2	2	no	—
18	#19151804	—	2	1	no	—
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
21	#19151423	C₂²	1	1	yes	characteristic
22	#19151423	C₂²	1	2	yes	—
23	C₄⋊#13083549	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151401	C₂²	1	2	yes	—
26	#19151401	C₂²	1	2	yes	—
27	#19151401	C₂²	1	2	yes	—
28	#19151399	C₂²	1	1	yes	characteristic
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	D₄×Q₈×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	2	yes	—
35	#500×C₈₉	C₂²	1	1	yes	characteristic
36	#474×C₈₉	—	2	1	no	—
37	#474×C₈₉	—	2	2	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	—	2	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	2	yes	—
43	C₂×#13083563	—	2	2	no	—
44	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
45	#15668012	—	2	1	no	—
46	#15668007	—	4	2	no	—
47	#15668000	—	2	2	no	—
48	#15667988	—	4	1	no	—
49	C₄×Dic₁₇₈	—	2	2	no	—
50	C₄×D₃₅₆	—	2	1	no	—
51	#15667970	—	4	1	no	—
52	#15667956	—	2	4	no	—
53	#15667953	—	2	2	no	—
54	C₂×#13083549	C₂³	1	1	yes	characteristic
55	C₂×#13083549	C₂³	1	1	yes	characteristic
56	C₂×#13083549	C₂³	1	2	yes	—
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667937	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂²×Q₈×C₈₉	—	4	1	no	—
61	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₈₉	—	2	1	no	—
63	C₄⋊Q₈×C₈₉	—	2	2	no	—
64	C₂²⋊Q₈×C₈₉	—	4	1	no	—
65	C₂²⋊Q₈×C₈₉	—	4	2	no	—
66	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
67	C₄×D₄×C₈₉	—	2	1	no	—
68	C₄×D₄×C₈₉	—	2	2	no	—
69	#107×C₈₉	—	2	2	no	—
70	#107×C₈₉	—	2	1	no	—
71	#107×C₈₉	C₂³	1	1	yes	characteristic
72	#107×C₈₉	C₂³	1	2	yes	—
73	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
75	#102×C₈₉	—	2	2	no	—
76	#102×C₈₉	—	2	1	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂×Dic₁₇₈	—	2	2	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×C₄×D₈₉	—	4	1	no	—
82	Dic₃₅₆	—	2	2	no	—
83	Dic₃₅₆	—	2	2	no	—
84	Dic₃₅₆	—	8	1	no	—
85	#13083563	—	8	2	no	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	#13083553	—	4	2	no	—
92	#13083551	—	4	2	no	—
93	#13083551	—	4	1	no	—
94	#13083551	—	4	2	no	—
95	C₄×Dic₈₉	—	4	2	no	—
96	#13083549	—	2	1	no	—
97	#13083549	—	2	2	no	—
98	#13083549	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	4	2	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	8	4	no	—
102	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
103	C₂×D₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	—	4	2	no	—
105	C₂²×C₄×C₈₉	—	4	1	no	—
106	D₈×C₈₉	—	2	1	no	—
107	D₈×C₈₉	#41	1	2	yes	—
108	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	2	1	no	—
113	C₄⋊C₄×C₈₉	—	4	2	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
117	C₂²⋊C₄×C₈₉	—	4	2	no	—
118	C₂²⋊C₄×C₈₉	—	4	1	no	—
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₉	C₂×D₄	1	2	yes	—
121	C₂³×C₈₉	—	4	1	no	—
122	C₂²×D₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	2	2	no	—
124	Q₈×C₈₉	—	8	4	no	—
125	Q₈×C₈₉	—	8	2	no	—
126	D₄×C₈₉	—	2	2	no	—
127	D₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	8	4	no	—
131	C₂×C₄×C₈₉	—	8	2	no	—
132	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	—	4	1	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	D₃₅₆	—	4	1	no	—
140	D₃₅₆	—	4	1	no	—
141	C₄×D₈₉	—	8	2	no	—
142	Dic₁₇₈	—	4	2	no	—
143	Dic₁₇₈	—	4	2	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₈₉	—	4	2	no	—
149	C₂²×C₈₉	—	8	1	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₈₉	—	8	1	no	—
153	C₂×D₈₉	—	8	1	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	—	8	1	no	—
160	C₄×C₈₉	#509	1	2	yes	—
161	C₄×C₈₉	#548	1	1	yes	characteristic
162	C₄×C₈₉	#577	1	1	yes	characteristic
163	Dic₈₉	—	8	2	no	—
164	Dic₈₉	—	8	1	no	—
165	#33934	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	8	2	no	—
167	C₂×C₈₉	#3358	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
169	C₂×C₈₉	#3417	1	1	yes	characteristic
170	D₈₉	—	16	1	no	—
171	#3393	—	89	1	no	—
172	Q₈×D₈	D₈₉	1	1	yes	characteristic, complemented
173	#2243	—	89	2	no	—
174	#2241	—	89	1	no	—
175	#2227	—	89	2	no	—
176	#2224	—	89	1	no	—
177	#2210	—	89	2	no	—
178	#2208	—	89	1	no	—
179	#2183	—	89	1	no	—
180	#2175	—	89	2	no	—
181	C₄×D₁₆	—	89	1	no	—
182	C₈₉	#33934	1	1	yes	characteristic, complemented, Sylow
183	#556	—	178	1	no	—
184	D₄×Q₈	—	2	1	no	—
185	C₂×SD₃₂	—	89	2	no	—
186	C₂×D₁₆	—	89	1	no	—
187	#500	C₂×D₈₉	1	1	yes	characteristic
188	#500	C₂×D₈₉	1	2	yes	—
189	#495	—	89	2	no	—
190	#494	—	89	1	no	—
191	#474	—	2	1	no	—
192	#474	—	2	2	no	—
193	#464	—	178	2	no	—
194	#461	—	178	1	no	—
195	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
196	C₄×Q₁₆	—	89	2	no	—
197	C₄×D₈	—	89	1	no	—
198	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
199	C₄×D₈	C₂×D₈₉	1	2	yes	—
200	#366	—	89	1	no	—
201	#366	—	89	2	no	—
202	C₄⋊C₁₆	—	89	1	no	—
203	C₄⋊C₁₆	—	89	2	no	—
204	#358	—	89	2	no	—
205	#358	—	89	2	no	—
206	#358	—	89	2	no	—
207	#357	—	89	1	no	—
208	#357	—	89	1	no	—
209	#357	—	89	2	no	—
210	#346	—	89	2	no	—
211	C₄×C₁₆	—	89	1	no	—
212	C₂²×Q₈	—	4	1	no	—
213	C₂×Q₁₆	—	89	2	no	—
214	C₂×Q₁₆	—	178	1	no	—
215	C₂×SD₁₆	—	178	2	no	—
216	C₂×D₈	—	89	1	no	—
217	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	—	178	1	no	—
221	#126	—	356	2	no	—
222	#124	—	178	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	4	2	no	—
225	C₂²⋊Q₈	—	356	1	no	—
226	C₄×Q₈	—	178	2	no	—
227	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	178	1	no	—
231	#117	—	356	1	no	—
232	SD₃₂	—	178	4	no	—
233	D₁₆	—	178	2	no	—
234	C₂×C₁₆	—	89	1	no	—
235	C₂×C₁₆	—	89	1	no	—
236	C₂×C₁₆	—	89	2	no	—
237	#107	—	2	1	no	—
238	#107	—	2	2	no	—
239	#107	—	89	1	no	—
240	#107	C₂²×D₈₉	1	1	yes	characteristic
241	#107	C₂²×D₈₉	1	2	yes	—
242	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
244	#103	—	178	2	no	—
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	#102	—	178	1	no	—
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₉	1	2	yes	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	8	4	no	—
253	C₂×Q₈	—	178	2	no	—
254	C₂×Q₈	#13083559	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	356	1	no	—
260	Q₁₆	—	178	2	no	—
261	Q₁₆	—	178	2	no	—
262	Q₁₆	—	712	1	no	—
263	SD₁₆	—	712	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	178	1	no	—
266	D₈	—	178	1	no	—
267	D₈	#13083575	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
270	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	356	1	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	356	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	#13083559	1	1	yes	characteristic
280	C₄⋊C₄	#13083559	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	356	2	no	—
284	C₂²⋊C₄	—	356	1	no	—
285	C₂²⋊C₄	—	356	2	no	—
286	C₄²	—	356	2	no	—
287	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
289	C₁₆	—	178	1	no	—
290	C₁₆	—	178	2	no	—
291	C₁₆	—	178	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	356	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	8	4	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	356	2	no	—
298	Q₈	—	356	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	356	1	no	—
302	D₄	—	356	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	356	1	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	712	2	no	—
310	C₂×C₄	C₂×#13083559	1	2	yes	—
311	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
312	C₂×C₄	C₂×#13083559	1	2	yes	—
313	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	C₂×#13083575	1	1	yes	characteristic
318	C₈	#15668060	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	712	1	no	—
323	C₂²	—	712	1	no	—
324	C₂²	—	712	1	no	—
325	C₂²	C₂²×#13083559	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	—	712	1	no	—
332	C₄	—	712	2	no	—
333	C₄	#19152034	1	2	yes	—
334	C₄	#19152259	1	1	yes	characteristic
335	C₄	#19152406	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	1424	1	no	—
338	C₂	#26032887	1	1	yes	characteristic
339	C₂	C₂×#19152034	1	1	yes	characteristic
340	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	Q₈×D₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15667953	C₂²
hypercenter	Q₈×D₈	D₈₉
hypocenter	C₈₉	#33934
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×D₈

Complements

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

N	H ≅ G/N	type
1. #26033241	C₂	semidirect
2. #26022491	C₂	semidirect
3. #26022481	C₂	semidirect
4. #26022401	C₂	semidirect
5. #26022386	C₂	semidirect
7. #26022297	C₂	semidirect
8. #26022161	C₂	semidirect
9. #26022116	C₂	semidirect
11. Q₈×D₈×C₈₉	C₂	semidirect
172. Q₈×D₈	D₈₉	semidirect
182. C₈₉	#33934	semidirect

Maximal subgroups

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

subgroup	index
#26033241	2
#26022491	2
#26022481	2
#26022401	2
#26022386	2
#26022307	2
#26022297	2
#26022161	2
#26022116	2
C₄×#15667953	2
Q₈×D₈×C₈₉	2
#33934	89