Subgroups — #36707037

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152006	C₂	1	2	yes	maximal
2	C₂×#19151998	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151995	C₂	1	1	yes	maximal, characteristic
4	#26026376	C₂	1	2	yes	maximal
5	#26020421	C₂	1	2	yes	maximal
6	#26020419	C₂	1	2	yes	maximal
7	#26020413	C₂	1	1	yes	maximal, characteristic
8	#26020409	C₂	1	1	yes	maximal, characteristic
9	#26020407	C₂	1	1	yes	maximal, characteristic
10	#26019938	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668011	C₂²	1	2	yes	—
13	C₂×#15668003	C₂²	1	2	yes	—
14	#19152006	—	2	8	no	—
15	#19151998	—	2	4	no	—
16	#19151995	—	2	4	no	—
17	C₂×#15667945	C₂²	1	2	yes	—
18	C₂×#15667945	C₂²	1	2	yes	—
19	C₂×#15667944	C₂²	1	1	yes	characteristic
20	C₂×#15667944	C₂²	1	1	yes	characteristic
21	C₂×#15667944	C₂²	1	2	yes	—
22	C₂×#15667943	C₂²	1	2	yes	—
23	C₂×#15667943	C₂²	1	2	yes	—
24	C₂×#15667923	C₂²	1	1	yes	characteristic
25	C₂×#15667922	C₂²	1	1	yes	characteristic
26	C₂×#15667922	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
29	#19151532	—	2	4	no	—
30	#19151519	—	2	4	no	—
31	#19151468	C₂²	1	2	yes	—
32	#19151467	C₂²	1	2	yes	—
33	#19151463	C₂²	1	2	yes	—
34	#19151444	C₂²	1	1	yes	characteristic
35	C₂×#107×C₈₉	C₂²	1	2	yes	—
36	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	#382×C₈₉	C₂²	1	2	yes	—
40	C₂²×Dic₁₇₈	—	2	2	no	—
41	C₂²×#10623494	C₂³	1	1	yes	characteristic
42	C₂²×#10623494	C₂³	1	1	yes	characteristic
43	#15668011	—	2	4	no	—
44	#15668011	—	2	4	no	—
45	#15668003	—	2	4	no	—
46	#15668003	—	2	4	no	—
47	C₂×C₄⋊Dic₈₉	—	2	2	no	—
48	C₂×C₄⋊Dic₈₉	—	2	4	no	—
49	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
51	C₂×#13083553	—	2	2	no	—
52	C₂×#13083553	—	2	2	no	—
53	C₂×#13083553	—	2	4	no	—
54	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
56	#15667945	—	2	4	no	—
57	#15667945	—	2	4	no	—
58	#15667944	—	2	2	no	—
59	#15667944	—	2	2	no	—
60	#15667944	—	2	4	no	—
61	#15667943	—	2	4	no	—
62	#15667943	—	2	4	no	—
63	#15667923	Q₈	1	4	yes	complemented
64	#15667922	D₄	1	4	yes	—
65	#15667922	Q₈	1	4	yes	complemented
66	C₈×Dic₈₉	D₄	1	4	yes	—
67	C₈×Dic₈₉	Q₈	1	4	yes	complemented
68	#15667919	—	2	4	no	—
69	#15667919	D₄	1	4	yes	—
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
74	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	—	2	4	no	—
76	#106×C₈₉	—	2	2	no	—
77	#106×C₈₉	—	2	2	no	—
78	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
79	#95×C₈₉	—	2	4	no	—
80	C₂×Dic₁₇₈	—	4	4	no	—
81	C₂×Dic₁₇₈	—	8	4	no	—
82	C₂²×Dic₈₉	—	2	2	no	—
83	C₂²×Dic₈₉	—	2	2	no	—
84	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
85	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
86	C₂×#10623494	C₂×D₄	1	2	yes	—
87	C₂×#10623494	C₂×D₄	1	2	yes	—
88	C₂×#10623494	C₂×Q₈	1	2	yes	—
89	C₂×#10623494	C₂×Q₈	1	2	yes	—
90	C₂×#10623494	C₂×Q₈	1	2	yes	—
91	C₂×#10623494	C₂×Q₈	1	2	yes	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₄⋊Dic₈₉	—	4	4	no	—
97	C₄⋊Dic₈₉	—	8	4	no	—
98	#13083553	—	4	4	no	—
99	#13083553	—	4	4	no	—
100	#13083553	—	8	4	no	—
101	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
102	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
103	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
104	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₈₉	—	2	2	no	—
106	C₂²×C₄×C₈₉	—	2	2	no	—
107	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
114	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
120	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
121	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	—	2	4	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	4	no	—
126	C₂×C₄×C₈₉	—	4	4	no	—
127	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
134	C₂×Dic₈₉	—	2	4	no	—
135	C₂×Dic₈₉	—	2	4	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	4	4	no	—
139	C₂×Dic₈₉	—	4	4	no	—
140	C₂×Dic₈₉	#127	1	4	yes	—
141	C₂×Dic₈₉	C₄⋊Q₈	1	4	yes	—
142	Dic₁₇₈	—	8	8	no	—
143	C₈×C₈₉	C₄⋊Q₈	1	4	yes	—
144	C₈×C₈₉	C₄⋊Q₈	1	4	yes	—
145	#10623494	C₄⋊Q₈	1	4	yes	—
146	#10623494	C₄⋊Q₈	1	4	yes	—
147	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
148	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂²×C₈₉	#577	1	4	yes	—
151	C₄×C₈₉	—	2	4	no	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	—	8	2	no	—
154	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	Dic₈₉	—	2	4	no	—
159	Dic₈₉	—	2	4	no	—
160	Dic₈₉	—	8	2	no	—
161	Dic₈₉	—	8	2	no	—
162	#33318	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	#2868	1	1	yes	characteristic
164	C₂×C₈₉	#3154	1	2	yes	—
165	C₂×C₈₉	#3163	1	2	yes	—
166	C₂×C₈₉	#3163	1	2	yes	—
167	C₂×#501	—	89	2	no	—
168	C₂×#499	—	89	1	no	—
169	C₂×#498	—	89	1	no	—
170	#2660	—	89	2	no	—
171	#1847	—	89	2	no	—
172	#1847	—	89	2	no	—
173	#1846	—	89	1	no	—
174	#1846	D₈₉	1	1	yes	characteristic
175	#1844	—	89	1	no	—
176	#1844	—	89	1	no	—
177	#1726	—	89	1	no	—
178	C₈₉	#33318	1	1	yes	characteristic, complemented, Sylow
179	C₂×C₄⋊Q₈	—	89	2	no	—
180	C₂×#125	—	89	2	no	—
181	#501	—	178	8	no	—
182	#499	—	178	4	no	—
183	#498	—	178	4	no	—
184	C₂×#107	—	89	2	no	—
185	C₂×#107	—	89	2	no	—
186	C₂×#107	C₂×D₈₉	1	2	yes	—
187	C₂×#106	—	89	1	no	—
188	C₂×#106	—	89	1	no	—
189	C₂×#106	—	89	2	no	—
190	C₂×#106	—	89	2	no	—
191	C₂×#106	—	89	2	no	—
192	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
193	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
194	C₂×#97	—	89	1	no	—
195	C₂×#97	—	89	1	no	—
196	C₂×#97	—	89	1	no	—
197	C₂×C₄×C₈	—	89	1	no	—
198	C₂×C₄×C₈	—	89	1	no	—
199	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
200	#398	—	178	4	no	—
201	#395	—	178	4	no	—
202	#384	—	89	2	no	—
203	#384	—	89	2	no	—
204	#382	—	89	2	no	—
205	#382	C₂×D₈₉	1	2	yes	—
206	#376	—	89	1	no	—
207	C₂²×Q₈	—	178	2	no	—
208	C₂²×C₈	—	89	1	no	—
209	C₂²×C₈	—	89	1	no	—
210	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
212	C₄⋊Q₈	—	178	4	no	—
213	C₄⋊Q₈	—	178	4	no	—
214	#125	—	178	4	no	—
215	#125	—	178	4	no	—
216	C₂×C₄⋊C₄	—	89	2	no	—
217	C₂×C₄⋊C₄	—	89	2	no	—
218	C₂×C₄⋊C₄	—	178	2	no	—
219	C₂×C₄⋊C₄	—	178	2	no	—
220	C₂×C₄⋊C₄	—	178	2	no	—
221	C₂×C₄⋊C₄	—	178	4	no	—
222	C₂×C₄⋊C₄	—	178	4	no	—
223	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
225	C₂×C₄²	—	89	1	no	—
226	C₂×C₄²	—	89	1	no	—
227	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
228	#107	—	2	4	no	—
229	#107	—	178	4	no	—
230	#107	—	178	4	no	—
231	#106	—	2	2	no	—
232	#106	—	2	2	no	—
233	#106	—	178	4	no	—
234	#106	—	178	4	no	—
235	#106	—	178	2	no	—
236	#106	—	178	2	no	—
237	#106	—	178	4	no	—
238	#97	—	89	4	no	—
239	#97	—	89	4	no	—
240	#97	—	89	4	no	—
241	C₄×C₈	—	89	4	no	—
242	C₄×C₈	—	89	4	no	—
243	C₄×C₈	Dic₁₇₈	1	4	yes	complemented
244	#95	—	2	4	no	—
245	#95	—	89	4	no	—
246	#95	—	178	4	no	—
247	C₂×Q₈	—	356	4	no	—
248	C₂×Q₈	—	712	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	89	2	no	—
252	C₂²×C₄	—	89	2	no	—
253	C₂²×C₄	—	178	2	no	—
254	C₂²×C₄	—	178	2	no	—
255	C₂²×C₄	D₄×D₈₉	1	2	yes	—
256	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
257	C₂×C₈	—	89	2	no	—
258	C₂×C₈	—	89	2	no	—
259	C₂×C₈	—	89	2	no	—
260	C₂×C₈	—	89	2	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	—	89	2	no	—
263	C₂×C₈	D₄×D₈₉	1	2	yes	—
264	C₂×C₈	D₄×D₈₉	1	2	yes	—
265	C₂×C₈	Q₈×D₈₉	1	2	yes	—
266	C₂×C₈	Q₈×D₈₉	1	2	yes	—
267	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
268	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	356	4	no	—
274	C₄⋊C₄	—	356	4	no	—
275	C₄⋊C₄	—	356	4	no	—
276	C₄⋊C₄	—	178	2	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄⋊C₄	—	178	2	no	—
279	C₄⋊C₄	—	178	2	no	—
280	C₄⋊C₄	—	712	4	no	—
281	C₄⋊C₄	—	712	4	no	—
282	C₄²	—	89	2	no	—
283	C₄²	—	89	2	no	—
284	C₄²	—	89	2	no	—
285	C₄²	—	89	2	no	—
286	C₄²	Q₈×D₈₉	1	2	yes	—
287	C₄²	C₂×Dic₁₇₈	1	2	yes	—
288	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
289	Q₈	—	712	8	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	89	4	no	—
296	C₂×C₄	—	89	4	no	—
297	C₂×C₄	—	356	2	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	356	4	no	—
300	C₂×C₄	—	356	4	no	—
301	C₂×C₄	—	178	4	no	—
302	C₂×C₄	—	178	4	no	—
303	C₂×C₄	#15668011	1	4	yes	—
304	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
310	C₈	—	89	4	no	—
311	C₈	—	89	4	no	—
312	C₈	#15668011	1	4	yes	—
313	C₈	#15668011	1	4	yes	—
314	C₂²	#127×D₈₉	1	1	yes	characteristic
315	C₂²	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
316	C₂²	C₂×#15668011	1	1	yes	characteristic
317	C₂²	#19152405	1	4	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	178	4	no	—
322	C₄	—	178	4	no	—
323	C₄	—	712	2	no	—
324	C₄	—	712	2	no	—
325	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
326	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
327	C₄	C₂×#15668011	1	1	yes	characteristic
328	C₄	C₂×#15668011	1	1	yes	characteristic
329	C₂	#26028649	1	1	yes	characteristic
330	C₂	#26030723	1	2	yes	—
331	C₂	#26030781	1	2	yes	—
332	C₂	#26030784	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1846	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
hypercenter	#1846	D₈₉
hypocenter	C₈₉	#33318
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
63. #15667923	Q₈	semidirect
65. #15667922	Q₈	semidirect
67. C₈×Dic₈₉	Q₈	semidirect
78. C₄×C₈×C₈₉	Q₈	semidirect
178. C₈₉	#33318	semidirect
243. C₄×C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152006	2
C₂×#19151998	2
C₂×#19151995	2
#26026376	2
#26020421	2
#26020419	2
#26020413	2
#26020409	2
#26020407	2
#26019938	2
C₈₉×#1846	2
#33318	89