Subgroups — #36445539

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028965	C₂	1	1	yes	maximal, characteristic
2	#26018902	C₂	1	1	yes	maximal, characteristic
3	#26018746	C₂	1	1	yes	maximal, characteristic
4	#26018544	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018155	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018149	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	—	2	1	no	—
9	#19152148	C₂²	1	1	yes	characteristic
10	#19151677	—	2	1	no	—
11	#19151677	C₄	1	2	yes	complemented
12	C₈×D₃₅₆	—	2	1	no	—
13	C₈×D₃₅₆	C₄	1	2	yes	complemented
14	#19151651	C₂²	1	1	yes	characteristic
15	#19151651	C₂²	1	1	yes	characteristic
16	#19151622	C₂²	1	1	yes	characteristic
17	#19151564	—	2	1	no	—
18	#19151345	C₄	1	2	yes	complemented
19	#19151341	C₄	1	2	yes	complemented
20	#19151331	C₄	1	4	yes	—
21	#19151320	—	2	1	no	—
22	#19151320	—	2	1	no	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	#336×C₈₉	—	2	1	no	—
28	C₂×#13083574	—	2	1	no	—
29	#15668018	—	4	2	no	—
30	#15668018	—	4	2	no	—
31	C₂×#13083558	—	2	1	no	—
32	C₂×#13083557	—	2	2	no	—
33	C₂×C₈×D₈₉	—	2	2	no	—
34	#15668012	C₂×C₄	1	1	yes	characteristic
35	#15668009	C₂×C₄	1	1	yes	characteristic
36	#15668000	—	2	1	no	—
37	#15667986	—	2	1	no	—
38	#15667986	C₂×C₄	1	2	yes	—
39	C₄×D₃₅₆	—	2	1	no	—
40	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
41	C₄⋊#10623494	—	2	1	no	—
42	C₄⋊#10623494	—	2	1	no	—
43	C₄⋊#10623494	C₂×C₄	1	2	yes	—
44	C₄⋊#10623494	C₂×C₄	1	2	yes	—
45	C₂²⋊#10623494	—	2	1	no	—
46	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
47	#15667924	—	2	2	no	—
48	#15667924	—	2	2	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₄×#10623494	—	2	1	no	—
51	C₂²×C₈×C₈₉	—	2	1	no	—
52	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₄²×C₈₉	—	2	1	no	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₈₉	—	2	1	no	—
58	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₈₉	—	4	2	no	—
60	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
64	#95×C₈₉	C₂×C₄	1	2	yes	—
65	#13083574	—	4	1	no	—
66	#13083574	—	4	1	no	—
67	#13083574	—	4	1	no	—
68	#13083574	—	4	1	no	—
69	C₂×Dic₁₇₈	—	2	1	no	—
70	C₂×D₃₅₆	—	2	1	no	—
71	C₂×D₃₅₆	—	2	1	no	—
72	C₂×D₃₅₆	C₄²	1	2	yes	—
73	C₂×C₂²⋊D₈₉	—	2	2	no	—
74	C₂×C₄×D₈₉	—	2	2	no	—
75	#13083558	—	4	1	no	—
76	#13083558	—	4	1	no	—
77	#13083557	—	4	4	no	—
78	C₂×#10623494	—	2	2	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₂×#10623494	—	2	2	no	—
81	C₈×D₈₉	—	4	4	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	C₄²	1	2	yes	—
84	#13083551	—	2	2	no	—
85	#13083551	—	2	2	no	—
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	1	no	—
92	C₂×C₈×C₈₉	—	2	1	no	—
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	4	1	no	—
96	C₂×C₈×C₈₉	—	4	1	no	—
97	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	4	no	—
102	C₄²×C₈₉	—	4	1	no	—
103	C₄²×C₈₉	—	4	1	no	—
104	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	#41	1	1	yes	characteristic
108	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	4	1	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	AGL(1,ℤ/8)	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×C₄×C₈₉	#137	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	4	2	no	—
126	C₂²⋊D₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	2	2	no	—
128	D₃₅₆	—	4	2	no	—
129	D₃₅₆	—	4	1	no	—
130	D₃₅₆	—	4	1	no	—
131	D₃₅₆	—	4	2	no	—
132	C₄×D₈₉	—	4	2	no	—
133	C₄×D₈₉	—	4	2	no	—
134	Dic₁₇₈	—	4	1	no	—
135	Dic₁₇₈	—	4	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	4	2	no	—
141	#10623494	—	4	2	no	—
142	#10623494	—	4	2	no	—
143	#10623494	—	4	2	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₈₉	#440	1	1	yes	characteristic
148	C₂²×C₈₉	#442	1	1	yes	characteristic
149	C₂×D₈₉	—	4	2	no	—
150	C₂×D₈₉	—	4	2	no	—
151	C₂×D₈₉	—	4	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	4	4	no	—
156	C₄×C₈₉	#417	1	1	yes	characteristic
157	C₄×C₈₉	#418	1	1	yes	characteristic
158	C₄×C₈₉	#440	1	1	yes	characteristic
159	C₄×C₈₉	#442	1	1	yes	characteristic
160	C₄×C₈₉	#443	1	2	yes	—
161	Dic₈₉	—	4	2	no	—
162	#8257	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	—	2	1	no	—
165	C₂×C₈₉	#1757	1	1	yes	characteristic
166	C₂×C₈₉	#1762	1	1	yes	characteristic
167	C₂×C₈₉	#1764	1	1	yes	characteristic
168	D₈₉	—	8	2	no	—
169	#2931	—	89	1	no	—
170	C₈×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
171	#1540	—	89	1	no	—
172	#1502	—	89	1	no	—
173	#1452	—	89	1	no	—
174	#1303	—	89	1	no	—
175	#1301	—	89	1	no	—
176	C₈₉	#8257	1	1	yes	characteristic, complemented, Sylow
177	C₂×#131	—	178	1	no	—
178	#532	—	89	1	no	—
179	#436	—	89	2	no	—
180	#436	—	178	1	no	—
181	C₈×D₄	—	89	2	no	—
182	C₈×D₄	—	178	1	no	—
183	#431	—	89	1	no	—
184	#431	—	89	1	no	—
185	#423	—	89	1	no	—
186	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
187	#408	—	178	1	no	—
188	C₂×C₄×C₈	—	2	1	no	—
189	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
191	#336	—	2	1	no	—
192	#332	—	89	2	no	—
193	#331	—	89	2	no	—
194	#327	—	89	4	no	—
195	#322	—	178	1	no	—
196	#322	—	178	1	no	—
197	C₂×#41	—	178	1	no	—
198	#131	—	356	2	no	—
199	#131	—	356	2	no	—
200	C₂×M₁₆	—	178	1	no	—
201	C₂×M₁₆	—	178	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	178	2	no	—
204	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₄⋊Q₈	—	89	1	no	—
207	#127	—	89	1	no	—
208	#124	—	178	1	no	—
209	#121	—	89	2	no	—
210	#121	—	178	1	no	—
211	C₄×D₄	—	89	2	no	—
212	C₄×D₄	—	178	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	89	2	no	—
218	C₄⋊C₈	—	89	2	no	—
219	C₄⋊C₈	—	178	1	no	—
220	C₄⋊C₈	—	178	1	no	—
221	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
222	C₂²⋊C₈	—	89	2	no	—
223	C₂²⋊C₈	—	178	1	no	—
224	C₂²⋊C₈	—	178	2	no	—
225	C₂²⋊C₈	—	178	2	no	—
226	C₄×C₈	—	4	2	no	—
227	C₄×C₈	—	178	1	no	—
228	C₄×C₈	—	178	1	no	—
229	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
231	C₄×C₈	D₃₅₆	1	1	yes	characteristic
232	C₄×C₈	D₃₅₆	1	1	yes	characteristic
233	#95	C₄×D₈₉	1	2	yes	—
234	#41	—	356	1	no	—
235	#41	—	356	1	no	—
236	#41	—	356	1	no	—
237	#41	—	356	1	no	—
238	C₂×Q₈	—	178	1	no	—
239	C₂×D₄	—	89	2	no	—
240	C₂×D₄	—	178	2	no	—
241	C₂×D₄	—	178	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	178	2	no	—
245	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
247	C₂²×C₄	#13083574	1	1	yes	characteristic
248	M₁₆	—	356	4	no	—
249	M₁₆	—	356	1	no	—
250	M₁₆	—	356	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	356	4	no	—
259	C₂×C₈	—	178	2	no	—
260	C₂×C₈	—	178	2	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	#13083551	1	1	yes	characteristic
263	C₂×C₈	#13083551	1	1	yes	characteristic
264	C₂×C₈	#13083551	1	1	yes	characteristic
265	C₂×C₈	#13083551	1	1	yes	characteristic
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	89	2	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	2	no	—
270	C₂²⋊C₄	—	178	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
276	C₄²	#13083574	1	1	yes	characteristic
277	C₂³	—	178	2	no	—
278	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
279	Q₈	—	356	1	no	—
280	Q₈	—	356	1	no	—
281	D₄	—	356	2	no	—
282	D₄	—	356	2	no	—
283	D₄	—	356	2	no	—
284	D₄	—	356	1	no	—
285	D₄	—	356	1	no	—
286	D₄	—	356	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	356	2	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	178	2	no	—
295	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
296	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
297	C₂×C₄	#15667969	1	1	yes	characteristic
298	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
299	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
300	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
301	C₂×C₄	#15668037	1	1	yes	characteristic
302	C₂×C₄	#15668037	1	1	yes	characteristic
303	C₂×C₄	#15668038	1	1	yes	characteristic
304	C₂×C₄	#15668043	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	356	2	no	—
311	C₈	—	356	2	no	—
312	C₈	—	356	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	356	2	no	—
316	C₂²	—	356	2	no	—
317	C₂²	—	356	2	no	—
318	C₂²	C₄×#13083551	1	1	yes	characteristic
319	C₂²	#19151696	1	1	yes	characteristic
320	C₂²	#19151708	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	356	2	no	—
326	C₄	#19151597	1	1	yes	characteristic
327	C₄	#19151602	1	1	yes	characteristic
328	C₄	#19151696	1	1	yes	characteristic
329	C₄	#19151708	1	1	yes	characteristic
330	C₄	#19151716	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	712	2	no	—
334	C₂	#26020053	1	1	yes	characteristic
335	C₂	#26020080	1	1	yes	characteristic
336	C₂	#26020089	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×C₄⋊C₄×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#8257
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #26018544	C₂	semidirect
5. #26018155	C₂	semidirect
6. #26018149	C₂	semidirect
7. C₈×C₄⋊C₄×C₈₉	C₂	semidirect
11. #19151677	C₄	semidirect
13. C₈×D₃₅₆	C₄	semidirect
18. #19151345	C₄	semidirect
19. #19151341	C₄	semidirect
170. C₈×C₄⋊C₄	D₈₉	semidirect
176. C₈₉	#8257	semidirect

Maximal subgroups

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

subgroup	index
#26028965	2
#26018902	2
#26018746	2
#26018544	2
#26018155	2
#26018149	2
C₈×C₄⋊C₄×C₈₉	2
#8257	89