Subgroups — #36447581

Order 22784.

434 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151675	C₂	1	1	yes	maximal, characteristic
2	#26020389	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018998	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018678	C₂	1	1	yes	maximal, characteristic
5	#26018676	C₂	1	1	yes	maximal, characteristic
6	#26018149	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083557	—	2	1	no	—
9	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
10	#19151675	—	2	1	no	—
11	#19151675	—	2	1	no	—
12	#19151675	C₄	1	2	yes	—
13	#19151675	C₄	1	2	yes	—
14	#19151651	C₂²	1	1	yes	characteristic
15	#19151622	C₂²	1	1	yes	characteristic
16	C₂×#15667924	—	2	1	no	—
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	#19151362	—	2	1	no	—
19	#19151362	—	2	1	no	—
20	#19151350	—	2	1	no	—
21	#19151348	—	2	1	no	—
22	#19151327	—	2	2	no	—
23	#19151325	—	2	2	no	—
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	#423×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×D₃₅₆	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₈₉	—	2	1	no	—
29	C₂×#13083558	—	2	1	no	—
30	C₂×#13083557	—	2	2	no	—
31	C₂×#13083557	—	2	2	no	—
32	C₂×#13083557	—	4	2	no	—
33	C₂×#13083557	—	4	2	no	—
34	C₂²×#10623494	D₄	1	1	yes	characteristic
35	C₂²×#10623494	D₄	1	1	yes	characteristic
36	C₄×D₃₅₆	—	2	1	no	—
37	C₄×D₃₅₆	—	2	1	no	—
38	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
39	C₄×D₃₅₆	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₈₉	Q₈	1	1	yes	characteristic
41	C₂×#13083551	—	2	1	no	—
42	C₄⋊#10623494	—	2	1	no	—
43	C₄⋊#10623494	—	2	1	no	—
44	C₄⋊#10623494	—	2	1	no	—
45	C₂²⋊#10623494	—	2	1	no	—
46	#15667924	—	2	2	no	—
47	#15667924	—	2	2	no	—
48	C₄×#10623494	—	2	1	no	—
49	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
50	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
51	C₄×#10623494	D₄	1	1	yes	characteristic
52	C₄×#10623494	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	—	2	1	no	—
54	C₂²×C₈×C₈₉	—	2	1	no	—
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	—	2	1	no	—
57	C₄⋊C₈×C₈₉	—	2	1	no	—
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₈₉	—	2	1	no	—
62	C₄×C₈×C₈₉	—	2	1	no	—
63	C₂³×D₈₉	—	2	1	no	—
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	4	2	no	—
66	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	2	no	—
69	C₂×C₄×D₈₉	—	2	2	no	—
70	C₂×C₄×D₈₉	—	4	1	no	—
71	C₂×C₄×D₈₉	—	4	1	no	—
72	C₂×C₄×D₈₉	—	4	1	no	—
73	C₂×C₄×D₈₉	—	4	1	no	—
74	#13083558	—	4	1	no	—
75	#13083558	—	4	1	no	—
76	#13083557	—	4	4	no	—
77	#13083557	—	4	4	no	—
78	C₂×#10623494	—	2	1	no	—
79	C₂×#10623494	—	2	1	no	—
80	C₂×#10623494	—	2	1	no	—
81	C₂×#10623494	—	2	1	no	—
82	C₂×#10623494	—	4	1	no	—
83	C₂×#10623494	—	4	1	no	—
84	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
90	#13083551	—	2	2	no	—
91	#13083551	—	2	2	no	—
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
95	M₁₆×C₈₉	—	4	1	no	—
96	M₁₆×C₈₉	—	4	1	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	—	4	1	no	—
99	C₂×C₈×C₈₉	—	4	1	no	—
100	C₂×C₈×C₈₉	—	4	1	no	—
101	C₂×C₈×C₈₉	—	4	1	no	—
102	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₉	#41	1	1	yes	characteristic
106	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×D₈₉	—	2	2	no	—
108	C₂²×D₈₉	—	4	2	no	—
109	C₂²×D₈₉	—	4	2	no	—
110	C₂²×D₈₉	—	4	1	no	—
111	C₂²×D₈₉	—	4	1	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	4	1	no	—
114	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
122	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×Dic₈₉	—	2	2	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	D₃₅₆	—	2	4	no	—
128	D₃₅₆	—	4	2	no	—
129	C₄×D₈₉	—	4	2	no	—
130	C₄×D₈₉	—	4	2	no	—
131	C₄×D₈₉	—	4	2	no	—
132	C₄×D₈₉	—	4	2	no	—
133	C₈×C₈₉	—	4	2	no	—
134	C₈×C₈₉	—	8	1	no	—
135	C₈×C₈₉	—	8	1	no	—
136	#10623494	—	2	1	no	—
137	#10623494	—	2	1	no	—
138	#10623494	—	2	1	no	—
139	#10623494	—	2	1	no	—
140	#10623494	—	8	1	no	—
141	#10623494	—	8	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	#380	1	1	yes	characteristic
145	C₂²×C₈₉	#478	1	1	yes	characteristic
146	C₂²×C₈₉	#478	1	1	yes	characteristic
147	C₂×D₈₉	—	4	2	no	—
148	C₂×D₈₉	—	4	2	no	—
149	C₂×D₈₉	—	4	2	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₂×D₈₉	—	8	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	#416	1	1	yes	characteristic
156	C₄×C₈₉	#418	1	1	yes	characteristic
157	C₄×C₈₉	#430	1	2	yes	—
158	C₄×C₈₉	#463	1	1	yes	characteristic
159	C₄×C₈₉	#463	1	1	yes	characteristic
160	Dic₈₉	—	4	2	no	—
161	#8664	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	#1804	1	1	yes	characteristic
165	C₂×C₈₉	#1811	1	1	yes	characteristic
166	C₂×C₈₉	#1814	1	1	yes	characteristic
167	D₈₉	—	8	2	no	—
168	C₂×#436	—	89	1	no	—
169	#1837	—	89	1	no	—
170	#1564	—	89	1	no	—
171	#1486	—	89	1	no	—
172	#1486	—	89	1	no	—
173	#1301	—	89	1	no	—
174	#1301	D₈₉	1	1	yes	characteristic, complemented
175	C₈₉	#8664	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	—	178	1	no	—
177	C₂×C₄×D₄	—	89	1	no	—
178	#436	—	89	2	no	—
179	#436	—	89	2	no	—
180	#436	—	178	1	no	—
181	#436	—	178	1	no	—
182	#431	—	89	1	no	—
183	#431	C₂×D₈₉	1	1	yes	characteristic
184	#423	—	89	1	no	—
185	#423	C₂×D₈₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
187	C₂×C₂²⋊C₈	—	178	1	no	—
188	C₂×C₄×C₈	—	89	1	no	—
189	#341	—	178	1	no	—
190	#341	—	178	1	no	—
191	#335	—	178	1	no	—
192	#334	—	178	1	no	—
193	D₄⋊C₈	—	178	2	no	—
194	D₄⋊C₈	—	178	2	no	—
195	C₂²×D₄	—	89	1	no	—
196	C₂³×C₄	—	178	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	356	2	no	—
199	C₂×M₁₆	—	356	2	no	—
200	C₂×M₁₆	—	178	2	no	—
201	C₂×M₁₆	—	178	2	no	—
202	C₂×M₁₆	—	178	1	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	89	1	no	—
205	C₂²×C₈	—	89	1	no	—
206	C₄×D₄	—	89	2	no	—
207	C₄×D₄	—	89	2	no	—
208	C₄×D₄	—	178	1	no	—
209	C₄×D₄	—	178	1	no	—
210	C₂×C₄⋊C₄	—	89	1	no	—
211	C₂×C₂²⋊C₄	—	178	1	no	—
212	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	178	1	no	—
218	C₄⋊C₈	—	178	1	no	—
219	C₄⋊C₈	—	178	1	no	—
220	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	178	2	no	—
223	C₂²⋊C₈	—	178	2	no	—
224	C₂²⋊C₈	—	178	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	89	1	no	—
227	C₄×C₈	—	89	1	no	—
228	C₄×C₈	—	89	1	no	—
229	C₄×C₈	—	89	1	no	—
230	C₄×C₈	—	178	1	no	—
231	C₂⁴	—	178	1	no	—
232	C₂×D₄	—	89	2	no	—
233	C₂×D₄	—	356	2	no	—
234	C₂×D₄	—	178	1	no	—
235	C₂²×C₄	—	356	1	no	—
236	C₂²×C₄	—	356	1	no	—
237	C₂²×C₄	—	356	1	no	—
238	C₂²×C₄	—	356	1	no	—
239	C₂²×C₄	—	178	1	no	—
240	C₂²×C₄	—	178	2	no	—
241	C₂²×C₄	—	178	2	no	—
242	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
243	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	#13083574	1	1	yes	characteristic
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	356	4	no	—
248	M₁₆	—	356	4	no	—
249	M₁₆	—	356	1	no	—
250	M₁₆	—	356	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	89	1	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	356	1	no	—
261	C₂×C₈	—	356	1	no	—
262	C₂×C₈	—	178	1	no	—
263	C₂×C₈	—	178	1	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	—	178	1	no	—
266	C₄⋊C₄	—	89	2	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₂²⋊C₄	—	178	2	no	—
269	C₂²⋊C₄	—	178	2	no	—
270	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₄²	D₄×D₈₉	1	1	yes	characteristic
272	C₄²	D₄×D₈₉	1	1	yes	characteristic
273	C₄²	#13083574	1	1	yes	characteristic
274	C₂³	—	356	2	no	—
275	C₂³	—	356	2	no	—
276	C₂³	—	356	1	no	—
277	C₂³	—	356	1	no	—
278	C₂³	—	178	2	no	—
279	C₂³	#15667990	1	1	yes	characteristic
280	D₄	—	356	2	no	—
281	D₄	—	178	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	356	1	no	—
285	C₂×C₄	—	356	1	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	—	356	2	no	—
288	C₂×C₄	—	356	2	no	—
289	C₂×C₄	—	356	2	no	—
290	C₂×C₄	—	178	2	no	—
291	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
293	C₂×C₄	#15667969	1	1	yes	characteristic
294	C₂×C₄	#15667985	1	1	yes	characteristic
295	C₂×C₄	#15667985	1	1	yes	characteristic
296	C₂×C₄	#15667990	1	1	yes	characteristic
297	C₂×C₄	#15668033	1	1	yes	characteristic
298	C₂×C₄	#15668035	1	1	yes	characteristic
299	C₂×C₄	#15668036	1	1	yes	characteristic
300	C₂×C₄	#15668039	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	8	1	no	—
303	C₈	—	8	1	no	—
304	C₈	—	178	1	no	—
305	C₈	—	178	1	no	—
306	C₈	—	178	1	no	—
307	C₈	—	178	1	no	—
308	C₈	—	712	1	no	—
309	C₈	—	712	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	356	2	no	—
313	C₂²	—	356	2	no	—
314	C₂²	—	356	2	no	—
315	C₂²	—	712	2	no	—
316	C₂²	—	712	2	no	—
317	C₂²	#19151457	1	1	yes	characteristic
318	C₂²	#19151897	1	1	yes	characteristic
319	C₂²	#19151901	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	356	2	no	—
324	C₄	#19151593	1	1	yes	characteristic
325	C₄	#19151601	1	1	yes	characteristic
326	C₄	#19151645	1	2	yes	—
327	C₄	#19151812	1	1	yes	characteristic
328	C₄	#19151816	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	712	2	no	—
332	C₂	#26020261	1	1	yes	characteristic
333	C₂	#26020293	1	1	yes	characteristic
334	C₂	#26020305	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667985
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1301	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂²×D₃₅₆	Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1301	D₈₉
hypocenter	C₈₉	#8664
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #26020389	C₂	semidirect
3. #26018998	C₂	semidirect
6. #26018149	C₂	semidirect
7. C₈₉×#1301	C₂	semidirect
174. #1301	D₈₉	semidirect
175. C₈₉	#8664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151675	2
#26020389	2
#26018998	2
#26018678	2
#26018676	2
#26018149	2
C₈₉×#1301	2
#8664	89