Subgroups — #36711795

Order 22784.

516 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032645	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022532	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022449	C₂	1	2	yes	maximal, complemented
4	#26022435	C₂	1	2	yes	maximal, complemented
5	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022391	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022346	C₂	1	2	yes	maximal, complemented
8	C₄×#15667956	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151425	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3334	C₂	1	1	yes	maximal, characteristic
12	#19152305	—	2	1	no	—
13	C₂×#15667956	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151934	—	2	2	no	—
17	#19151894	—	2	1	no	—
18	#19151839	C₂²	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	C₂×#15667946	C₂²	1	2	yes	—
21	#19151425	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151425	C₂²	1	2	yes	—
24	#19151425	C₂²	1	2	yes	—
25	#19151425	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	#19151401	C₂²	1	1	yes	characteristic
28	#19151401	C₂²	1	2	yes	—
29	#19151399	C₂²	1	1	yes	characteristic
30	#19151399	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#547×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#480×C₈₉	—	2	2	no	—
36	#474×C₈₉	—	2	1	no	—
37	#466×C₈₉	C₂²	1	2	yes	—
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#107×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668003	—	4	1	no	—
44	#15667991	—	2	2	no	—
45	#15667989	—	4	2	no	—
46	C₄×Dic₁₇₈	—	2	1	no	—
47	D₄×Dic₈₉	—	4	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	2	no	—
49	#15667956	—	2	2	no	—
50	C₂×#13083549	—	2	4	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	2	yes	—
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	#15667946	—	2	2	no	—
55	#15667946	—	2	1	no	—
56	#15667946	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	2	yes	—
58	#15667937	—	2	2	no	—
59	#15667937	—	2	1	no	—
60	C₂×#41×C₈₉	—	4	1	no	—
61	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₈₉	—	2	1	no	—
64	#123×C₈₉	—	4	2	no	—
65	C₂²⋊Q₈×C₈₉	—	4	1	no	—
66	#121×C₈₉	—	2	2	no	—
67	C₄×D₄×C₈₉	—	2	1	no	—
68	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
70	#107×C₈₉	—	2	2	no	—
71	#107×C₈₉	—	2	1	no	—
72	#107×C₈₉	C₂³	1	1	yes	characteristic
73	#107×C₈₉	C₂³	1	2	yes	—
74	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
75	#102×C₈₉	—	2	1	no	—
76	#102×C₈₉	—	2	2	no	—
77	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂²×Dic₈₉	—	4	2	no	—
82	Dic₃₅₆	—	2	1	no	—
83	Dic₃₅₆	—	2	1	no	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	—	4	1	no	—
87	C₄⋊Dic₈₉	—	4	2	no	—
88	#13083553	—	4	1	no	—
89	#13083553	—	4	2	no	—
90	C₂²⋊Dic₈₉	—	4	2	no	—
91	C₄×Dic₈₉	—	4	1	no	—
92	#13083549	—	2	2	no	—
93	#13083549	—	2	1	no	—
94	#13083549	#41	1	2	yes	complemented
95	#41×C₈₉	—	8	4	no	—
96	C₂×Q₈×C₈₉	—	4	1	no	—
97	C₂×D₄×C₈₉	—	2	1	no	—
98	C₂×D₄×C₈₉	—	4	2	no	—
99	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	—	4	2	no	—
101	C₂²×C₄×C₈₉	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
103	D₈×C₈₉	—	2	1	no	—
104	D₈×C₈₉	—	2	1	no	—
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
115	C₂²⋊C₄×C₈₉	—	4	1	no	—
116	C₂²⋊C₄×C₈₉	—	4	2	no	—
117	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂³×C₈₉	—	4	1	no	—
120	C₂³×C₈₉	C₂×D₈	1	2	yes	—
121	Q₈×C₈₉	—	8	2	no	—
122	D₄×C₈₉	—	4	1	no	—
123	D₄×C₈₉	—	4	1	no	—
124	D₄×C₈₉	—	8	4	no	—
125	D₄×C₈₉	SD₃₂	1	4	yes	complemented
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	8	4	no	—
131	C₂×C₄×C₈₉	—	8	2	no	—
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	—	4	1	no	—
137	C₂×Dic₈₉	—	4	1	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	16	2	no	—
140	Dic₁₇₈	—	4	1	no	—
141	Dic₁₇₈	—	4	1	no	—
142	Dic₁₇₈	—	16	1	no	—
143	C₈×C₈₉	—	2	2	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	8	1	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	—	8	1	no	—
152	C₂²×C₈₉	C₂×SD₃₂	1	4	yes	—
153	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₈₉	—	2	2	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	8	1	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
159	C₄×C₈₉	#547	1	1	yes	characteristic
160	C₄×C₈₉	#578	1	1	yes	characteristic
161	Dic₈₉	—	8	1	no	—
162	Dic₈₉	—	8	2	no	—
163	Dic₈₉	—	8	1	no	—
164	#33902	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	4	no	—
166	C₂×C₈₉	—	16	1	no	—
167	C₂×C₈₉	#3334	1	1	yes	characteristic
168	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
169	C₂×C₈₉	#3416	1	1	yes	characteristic
170	#3338	—	89	1	no	—
171	#3334	D₈₉	1	1	yes	characteristic
172	#2254	—	89	1	no	—
173	#2235	—	89	2	no	—
174	#2233	—	89	2	no	—
175	#2226	—	89	1	no	—
176	#2225	—	89	1	no	—
177	#2217	—	89	2	no	—
178	C₄×SD₃₂	—	89	1	no	—
179	D₄×C₁₆	—	89	1	no	—
180	C₂×#367	—	89	2	no	—
181	C₈₉	#33902	1	1	yes	characteristic, complemented, Sylow
182	#554	—	178	1	no	—
183	#547	—	2	1	no	—
184	C₂×SD₃₂	—	89	1	no	—
185	C₂²×C₁₆	—	89	2	no	—
186	#500	—	89	1	no	—
187	#500	C₂×D₈₉	1	1	yes	characteristic
188	#484	—	178	2	no	—
189	#480	—	2	2	no	—
190	#477	—	178	1	no	—
191	#474	—	2	1	no	—
192	#467	—	89	2	no	—
193	#466	C₂×D₈₉	1	2	yes	—
194	C₄×Q₁₆	—	89	1	no	—
195	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
196	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
197	C₂×#107	—	89	2	no	—
198	C₂×#107	C₂×D₈₉	1	2	yes	—
199	#367	—	89	1	no	—
200	#367	—	89	2	no	—
201	#367	—	89	2	no	—
202	#367	—	89	2	no	—
203	#367	—	89	2	no	—
204	C₄⋊C₁₆	—	89	1	no	—
205	#358	—	89	1	no	—
206	#358	—	89	2	no	—
207	#357	—	89	1	no	—
208	#357	—	89	2	no	—
209	C₂²⋊C₁₆	—	89	2	no	—
210	C₄×C₁₆	—	89	1	no	—
211	C₂×#41	—	4	1	no	—
212	C₂×Q₁₆	—	89	1	no	—
213	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	178	1	no	—
217	#125	—	356	1	no	—
218	#123	—	4	2	no	—
219	C₂²⋊Q₈	—	4	1	no	—
220	C₂²⋊Q₈	—	356	2	no	—
221	C₂²⋊Q₈	—	178	2	no	—
222	#121	—	2	2	no	—
223	C₄×Q₈	—	178	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	356	1	no	—
226	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	178	2	no	—
229	SD₃₂	—	178	2	no	—
230	C₂×C₁₆	—	89	1	no	—
231	C₂×C₁₆	—	89	1	no	—
232	C₂×C₁₆	—	89	2	no	—
233	C₂×C₁₆	—	178	4	no	—
234	#107	—	2	2	no	—
235	#107	—	2	1	no	—
236	#107	—	89	1	no	—
237	#107	—	89	2	no	—
238	#107	—	178	2	no	—
239	#107	—	178	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	#103	—	178	1	no	—
244	#103	—	178	2	no	—
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	#41	—	8	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	356	1	no	—
252	C₂×Q₈	—	178	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	#13083559	1	1	yes	characteristic
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	356	2	no	—
259	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
260	Q₁₆	—	178	1	no	—
261	Q₁₆	—	178	1	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
268	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	356	2	no	—
274	C₄⋊C₄	—	356	1	no	—
275	C₄⋊C₄	—	356	1	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	2	no	—
279	C₄⋊C₄	#13083559	1	1	yes	characteristic
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	356	2	no	—
283	C₂²⋊C₄	#13083559	1	2	yes	—
284	C₄²	—	356	1	no	—
285	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₁₆	—	89	2	no	—
287	C₁₆	—	178	2	no	—
288	C₁₆	—	178	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	C₂×#13083559	1	2	yes	—
291	Q₈	—	8	2	no	—
292	Q₈	—	356	1	no	—
293	Q₈	—	356	1	no	—
294	Q₈	—	1424	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	D₄	#15667956	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	356	1	no	—
306	C₂×C₄	—	356	1	no	—
307	C₂×C₄	—	356	2	no	—
308	C₂×C₄	—	1424	2	no	—
309	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
310	C₂×C₄	C₂×#13083559	1	2	yes	—
311	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	#41×D₈₉	1	1	yes	characteristic
316	C₈	#15668059	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₂×#15667956	1	4	yes	—
323	C₂²	C₂²×#13083559	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	2	no	—
328	C₄	—	712	1	no	—
329	C₄	—	712	1	no	—
330	C₄	—	712	2	no	—
331	C₄	C₂×#15667956	1	2	yes	—
332	C₄	#19152247	1	1	yes	characteristic
333	C₄	#19152413	1	1	yes	characteristic
334	C₂	—	2	4	no	—
335	C₂	—	16	1	no	—
336	C₂	#26032595	1	1	yes	characteristic
337	C₂	C₂²×#15667956	1	1	yes	characteristic
338	C₂	#26033396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3334	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3334	D₈₉
hypocenter	C₈₉	#33902
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. #26032645	C₂	semidirect
2. #26022532	C₂	semidirect
3. #26022449	C₂	semidirect
4. #26022435	C₂	semidirect
5. #26022399	C₂	semidirect
6. #26022391	C₂	semidirect
7. #26022346	C₂	semidirect
8. C₄×#15667956	C₂	semidirect
9. D₄×#13083549	C₂	semidirect
10. C₂×#19151425	C₂	semidirect
94. #13083549	#41	semidirect
125. D₄×C₈₉	SD₃₂	semidirect
181. C₈₉	#33902	semidirect
298. D₄	#15667956	semidirect

Maximal subgroups

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

subgroup	index
#26032645	2
#26022532	2
#26022449	2
#26022435	2
#26022399	2
#26022391	2
#26022346	2
C₄×#15667956	2
D₄×#13083549	2
C₂×#19151425	2
C₈₉×#3334	2
#33902	89