Subgroups — #36536797

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	#26030291	C₂	1	2	yes	maximal
3	#26019656	C₂	1	2	yes	maximal, complemented
4	#26019646	C₂	1	2	yes	maximal, complemented
5	#26019634	C₂	1	4	yes	maximal, complemented
6	#26019000	C₂	1	1	yes	maximal, characteristic
7	#26018910	C₂	1	2	yes	maximal
8	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152155	C₂²	1	2	yes	—
10	#19152003	C₂²	1	1	yes	characteristic
11	#19152003	C₂²	1	1	yes	characteristic
12	#19152000	—	2	1	no	—
13	#19152000	—	2	1	no	—
14	#19151996	C₂²	1	2	yes	—
15	#19151933	—	2	2	no	—
16	#19151893	C₂²	1	4	yes	—
17	#19151808	C₂²	1	4	yes	—
18	#19151754	—	2	2	no	—
19	C₄×#13083564	—	2	4	no	—
20	#19151651	C₂²	1	2	yes	—
21	#19151636	—	2	1	no	—
22	#19151636	—	2	1	no	—
23	#19151351	C₂²	1	2	yes	—
24	#19151349	C₂²	1	2	yes	—
25	#19151346	C₂²	1	4	yes	—
26	#19151343	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₈₉	—	2	1	no	—
29	C₈×Q₈×C₈₉	—	2	1	no	—
30	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
31	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
33	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#13083564	—	2	4	no	—
37	#15668012	C₂³	1	2	yes	—
38	#15668012	C₂³	1	2	yes	—
39	#15668004	—	2	2	no	—
40	#15667991	—	2	2	no	—
41	#15667991	—	2	2	no	—
42	C₄×Dic₁₇₈	—	2	2	no	—
43	C₄×Dic₁₇₈	—	2	2	no	—
44	#15667971	—	2	2	no	—
45	#15667946	—	2	1	no	—
46	#15667946	—	2	1	no	—
47	#15667946	—	2	1	no	—
48	#15667946	—	2	1	no	—
49	#15667945	—	2	4	no	—
50	#15667944	—	2	2	no	—
51	#15667944	—	2	2	no	—
52	C₄⋊#10623494	C₂³	1	4	yes	—
53	#15667936	—	2	4	no	—
54	#15667935	—	2	4	no	—
55	C₂²⋊#10623494	—	2	2	no	—
56	C₄×#10623494	—	2	2	no	—
57	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	—	2	1	no	—
59	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₈₉	—	2	1	no	—
62	C₄×Q₈×C₈₉	—	2	1	no	—
63	C₄×Q₈×C₈₉	D₄	1	2	yes	—
64	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
66	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₈₉	—	2	1	no	—
68	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₉	—	2	2	no	—
70	C₄⋊C₈×C₈₉	—	2	2	no	—
71	C₄⋊C₈×C₈₉	—	2	1	no	—
72	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
73	C₄×C₈×C₈₉	—	2	2	no	—
74	C₄×C₈×C₈₉	—	2	2	no	—
75	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	2	no	—
80	C₂×Dic₁₇₈	—	2	2	no	—
81	#13083564	—	4	8	no	—
82	C₂×#10623494	—	2	4	no	—
83	C₄⋊Dic₈₉	—	2	2	no	—
84	C₄⋊Dic₈₉	—	2	2	no	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	#13083553	—	4	2	no	—
88	#13083553	—	4	2	no	—
89	C₂²⋊Dic₈₉	—	4	2	no	—
90	C₄×Dic₈₉	—	4	2	no	—
91	C₂×Q₈×C₈₉	—	2	4	no	—
92	C₂×Q₈×C₈₉	—	2	1	no	—
93	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₈₉	—	2	1	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	—	4	2	no	—
102	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
111	C₄²×C₈₉	—	2	2	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂³×C₈₉	#121	1	1	yes	characteristic
118	Q₈×C₈₉	—	2	4	no	—
119	Q₈×C₈₉	—	2	2	no	—
120	Q₈×C₈₉	#135	1	4	yes	—
121	C₂×C₄×C₈₉	—	2	4	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
131	C₂×C₄×C₈₉	#137	1	2	yes	—
132	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₈₉	—	4	2	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	Dic₁₇₈	—	4	4	no	—
138	Dic₁₇₈	—	8	2	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	4	2	no	—
142	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
143	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
144	#10623494	—	4	4	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	#462	1	2	yes	—
148	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
149	C₄×C₈₉	—	2	4	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	4	2	no	—
154	C₄×C₈₉	#462	1	2	yes	—
155	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
156	C₄×C₈₉	C₂×#135	1	2	yes	—
157	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
158	Dic₈₉	—	8	2	no	—
159	Dic₈₉	—	8	2	no	—
160	#19118	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	2	no	—
162	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
163	C₂×C₈₉	#3050	1	1	yes	characteristic
164	C₂×C₈₉	#3067	1	1	yes	characteristic
165	#3160	—	89	1	no	—
166	#3104	—	89	2	no	—
167	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
168	#1676	—	89	2	no	—
169	#1674	—	89	2	no	—
170	#1672	—	89	4	no	—
171	#1564	—	89	1	no	—
172	#1542	—	89	2	no	—
173	C₈₉	#19118	1	1	yes	characteristic, complemented, Sylow
174	#533	—	89	2	no	—
175	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
176	#500	—	89	1	no	—
177	#500	—	89	1	no	—
178	#499	—	178	1	no	—
179	#499	—	178	1	no	—
180	#498	—	89	2	no	—
181	#484	—	178	2	no	—
182	#477	—	89	4	no	—
183	#462	—	89	4	no	—
184	#451	—	178	2	no	—
185	C₈×Q₈	—	2	1	no	—
186	C₈×Q₈	—	2	1	no	—
187	C₈×Q₈	C₂×D₈₉	1	2	yes	—
188	C₈×Q₈	C₂×D₈₉	1	2	yes	—
189	C₄×Q₁₆	—	178	4	no	—
190	#431	—	89	2	no	—
191	#427	—	178	1	no	—
192	#427	—	178	1	no	—
193	C₂×C₄⋊C₈	—	2	1	no	—
194	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
195	C₂×C₄×C₈	—	2	1	no	—
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	#335	—	89	2	no	—
198	#334	—	89	2	no	—
199	#333	—	89	4	no	—
200	#332	—	89	4	no	—
201	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
202	C₂×Q₁₆	—	178	4	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₄⋊Q₈	—	89	2	no	—
207	C₄⋊Q₈	—	89	2	no	—
208	#125	—	178	2	no	—
209	C₂²⋊Q₈	—	178	2	no	—
210	C₂²⋊Q₈	—	178	2	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	2	1	no	—
213	C₄×Q₈	—	178	2	no	—
214	C₄×Q₈	—	178	2	no	—
215	C₄×Q₈	D₃₅₆	1	2	yes	—
216	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
217	#117	—	178	2	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
222	#107	—	178	1	no	—
223	#107	—	178	1	no	—
224	#107	—	178	1	no	—
225	#107	—	178	1	no	—
226	#107	—	178	4	no	—
227	#106	—	178	2	no	—
228	#106	—	178	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	89	4	no	—
233	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
234	#103	—	178	4	no	—
235	#103	—	178	4	no	—
236	C₂²⋊C₈	—	178	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	—	178	2	no	—
240	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
241	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	178	2	no	—
247	C₂×Q₈	—	178	2	no	—
248	C₂×Q₈	C₂×D₃₅₆	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
253	C₂²×C₄	#13083574	1	1	yes	characteristic
254	Q₁₆	—	356	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	178	4	no	—
259	C₂×C₈	#13083564	1	4	yes	complemented
260	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	356	2	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	178	2	no	—
270	C₄⋊C₄	—	178	2	no	—
271	C₄⋊C₄	—	178	2	no	—
272	C₄⋊C₄	—	178	2	no	—
273	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
274	C₂²⋊C₄	—	356	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	356	2	no	—
278	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
279	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
280	C₄²	#13083574	1	1	yes	characteristic
281	C₄²	C₂³×D₈₉	1	1	yes	characteristic
282	C₂³	#15667986	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	356	4	no	—
286	Q₈	—	712	2	no	—
287	Q₈	#15668034	1	4	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	356	2	no	—
295	C₂×C₄	—	356	2	no	—
296	C₂×C₄	#15667986	1	1	yes	characteristic
297	C₂×C₄	#15667986	1	1	yes	characteristic
298	C₂×C₄	#15667986	1	1	yes	characteristic
299	C₂×C₄	C₂×#13083564	1	2	yes	—
300	C₂×C₄	#15668042	1	2	yes	—
301	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
303	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	2	no	—
307	C₈	—	356	4	no	—
308	C₈	C₂×#13083564	1	2	yes	—
309	C₈	C₂×#13083564	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#19151808	1	2	yes	—
313	C₂²	C₂×#15667986	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	712	2	no	—
320	C₄	—	712	2	no	—
321	C₄	#19151808	1	2	yes	—
322	C₄	C₂²×#13083564	1	1	yes	characteristic
323	C₄	C₂×#15668034	1	2	yes	—
324	C₄	C₂×#15668042	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×#19151808	1	1	yes	characteristic
327	C₂	#26029833	1	1	yes	characteristic
328	C₂	#26029972	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667986
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂³×C₈₉	#121
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#19118
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #26019656	C₂	semidirect
4. #26019646	C₂	semidirect
5. #26019634	C₂	semidirect
102. C₂×C₈×C₈₉	Q₁₆	semidirect
173. C₈₉	#19118	semidirect
259. C₂×C₈	#13083564	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26030291	2
#26019656	2
#26019646	2
#26019634	2
#26019000	2
#26018910	2
C₂×C₈×Q₈×C₈₉	2
#19118	89