Subgroups — #36537635

Order 22784.

774 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	#26031190	C₂	1	2	yes	maximal, complemented
2	C₂×#19151975	C₂	1	1	yes	maximal, characteristic
3	#26019736	C₂	1	4	yes	maximal, complemented
4	#26019612	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018921	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2960	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	—	2	1	no	—
9	C₂²×D₇₁₂	—	2	1	no	—
10	C₂×#15668009	C₂²	1	2	yes	—
11	#19151975	—	2	1	no	—
12	#19151975	—	2	1	no	—
13	#19151975	C₂²	1	2	yes	—
14	#19151975	C₂²	1	2	yes	—
15	#19151972	—	2	2	no	—
16	#19151946	—	2	2	no	—
17	#19151798	C₂²	1	8	yes	complemented
18	#19151748	—	2	4	no	—
19	#19151651	C₂²	1	2	yes	—
20	#19151351	C₂²	1	2	yes	complemented
21	#19151349	C₂²	1	2	yes	complemented
22	#19151341	C₂²	1	8	yes	complemented
23	#533×C₈₉	C₂²	1	1	yes	characteristic
24	#446×C₈₉	—	2	1	no	—
25	#446×C₈₉	C₂²	1	2	yes	complemented
26	C₈×Q₈×C₈₉	—	2	1	no	—
27	C₈×Q₈×C₈₉	C₂²	1	2	yes	complemented
28	#424×C₈₉	—	2	1	no	—
29	#423×C₈₉	C₂²	1	1	yes	characteristic
30	#405×C₈₉	—	2	1	no	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×D₃₅₆	—	2	2	no	—
33	C₂²×D₃₅₆	—	2	2	no	—
34	C₂²×D₃₅₆	—	2	2	no	—
35	C₂×#13083561	—	2	8	no	—
36	C₂×D₇₁₂	—	2	2	no	—
37	C₂×D₇₁₂	—	2	2	no	—
38	C₂×D₇₁₂	—	2	2	no	—
39	C₂×D₇₁₂	—	2	2	no	—
40	C₂×D₇₁₂	—	4	4	no	—
41	C₂×D₇₁₂	—	4	4	no	—
42	#15668009	—	2	2	no	—
43	#15668009	—	2	2	no	—
44	#15668009	—	2	2	no	—
45	#15668009	C₂³	1	4	yes	—
46	C₄⋊#10623494	C₂³	1	4	yes	—
47	#15667932	—	2	8	no	—
48	C₂²⋊#10623494	—	2	2	no	—
49	C₄×#10623494	—	2	2	no	—
50	C₂×M₁₆×C₈₉	—	2	1	no	—
51	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
55	#125×C₈₉	—	2	1	no	—
56	C₂²⋊Q₈×C₈₉	—	2	1	no	—
57	C₂²⋊Q₈×C₈₉	D₄	1	2	yes	—
58	C₄×Q₈×C₈₉	—	2	1	no	—
59	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
60	#117×C₈₉	—	2	1	no	—
61	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₈₉	—	2	2	no	—
63	C₄⋊C₈×C₈₉	—	2	2	no	—
64	C₄⋊C₈×C₈₉	—	2	1	no	—
65	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
66	#97×C₈₉	—	2	2	no	—
67	C₄×C₈×C₈₉	—	2	2	no	—
68	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×D₈₉	—	4	2	no	—
73	C₂×D₃₅₆	—	2	4	no	—
74	C₂×D₃₅₆	—	2	4	no	—
75	C₂×D₃₅₆	—	2	4	no	—
76	C₂×D₃₅₆	—	4	4	no	—
77	C₂×D₃₅₆	—	4	4	no	—
78	C₂×D₃₅₆	—	4	2	no	—
79	C₂×D₃₅₆	—	4	2	no	—
80	C₂×D₃₅₆	—	4	2	no	—
81	C₂×D₃₅₆	—	8	4	no	—
82	C₂×D₃₅₆	—	8	4	no	—
83	#13083561	—	8	8	no	—
84	D₇₁₂	—	4	4	no	—
85	D₇₁₂	—	4	4	no	—
86	D₇₁₂	—	4	4	no	—
87	D₇₁₂	—	4	4	no	—
88	C₂×#10623494	—	2	4	no	—
89	C₂×Q₈×C₈₉	—	2	1	no	—
90	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
94	M₁₆×C₈₉	—	4	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	SD₁₆	1	4	yes	complemented
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	2	2	no	—
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄×C₈₉	—	2	2	no	—
109	C₄²×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₈₉	#41	1	1	yes	characteristic
113	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂³×C₈₉	#121	1	1	yes	characteristic
115	C₂²×D₈₉	—	4	4	no	—
116	C₂²×D₈₉	—	8	4	no	—
117	C₂²×D₈₉	—	8	4	no	—
118	C₂²×D₈₉	—	8	4	no	—
119	Q₈×C₈₉	—	2	4	no	—
120	Q₈×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	2	2	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	#121	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
133	D₃₅₆	—	4	4	no	—
134	D₃₅₆	—	4	4	no	—
135	D₃₅₆	—	4	8	no	—
136	D₃₅₆	—	8	4	no	—
137	D₃₅₆	—	8	4	no	—
138	D₃₅₆	—	8	4	no	—
139	D₃₅₆	—	8	4	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	4	2	no	—
143	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
144	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
145	#10623494	—	4	4	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	#460	1	2	yes	—
149	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
150	C₂×D₈₉	—	8	4	no	—
151	C₂×D₈₉	—	8	4	no	—
152	C₂×D₈₉	—	8	4	no	—
153	C₂×D₈₉	—	16	8	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	4	2	no	—
157	C₄×C₈₉	—	4	2	no	—
158	C₄×C₈₉	#460	1	2	yes	—
159	C₄×C₈₉	#472	1	2	yes	—
160	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
161	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₄×C₈₉	#576	1	2	yes	—
163	#19203	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	—	2	2	no	—
165	C₂×C₈₉	C₂×#460	1	1	yes	characteristic
166	C₂×C₈₉	#3058	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
168	D₈₉	—	16	4	no	—
169	#3208	—	89	2	no	—
170	C₂×#493	—	89	1	no	—
171	#2960	D₈₉	1	1	yes	characteristic, complemented
172	#1688	—	89	4	no	—
173	#1668	—	89	4	no	—
174	#1564	—	89	1	no	—
175	#1545	—	89	2	no	—
176	C₈₉	#19203	1	1	yes	characteristic, complemented, Sylow
177	C₂²×D₈	—	178	1	no	—
178	C₂²×D₈	—	178	1	no	—
179	#533	C₂×D₈₉	1	1	yes	characteristic
180	C₂×#127	—	89	2	no	—
181	#493	—	89	2	no	—
182	#493	—	89	2	no	—
183	#493	—	178	1	no	—
184	#493	—	178	1	no	—
185	#492	—	178	2	no	—
186	#486	—	178	2	no	—
187	#460	—	89	8	no	—
188	#450	—	178	4	no	—
189	#446	—	2	1	no	—
190	#446	C₂×D₈₉	1	2	yes	complemented
191	C₈×Q₈	—	2	1	no	—
192	C₈×Q₈	C₂×D₈₉	1	2	yes	complemented
193	#431	—	89	2	no	—
194	#424	—	2	1	no	—
195	#423	C₂×D₈₉	1	1	yes	characteristic
196	#405	—	2	1	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	#335	—	89	2	no	—
199	#334	—	89	2	no	—
200	#331	—	89	8	no	—
201	C₂²×D₄	—	178	2	no	—
202	C₂²×D₄	—	178	2	no	—
203	C₂²×D₄	—	178	2	no	—
204	C₂×SD₁₆	—	178	8	no	—
205	C₂×D₈	—	356	4	no	—
206	C₂×D₈	—	356	4	no	—
207	C₂×D₈	—	178	2	no	—
208	C₂×D₈	—	178	2	no	—
209	C₂×D₈	—	178	2	no	—
210	C₂×D₈	—	178	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₄⋊Q₈	D₃₅₆	1	1	yes	characteristic
215	C₄⋊Q₈	D₃₅₆	1	1	yes	characteristic
216	#127	—	89	4	no	—
217	#127	—	178	2	no	—
218	#127	—	178	2	no	—
219	#127	—	178	2	no	—
220	#125	—	2	1	no	—
221	C₂²⋊Q₈	—	2	1	no	—
222	C₂²⋊Q₈	D₃₅₆	1	2	yes	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
225	#117	—	2	1	no	—
226	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	89	4	no	—
231	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
232	#102	—	178	8	no	—
233	C₂²⋊C₈	—	178	2	no	—
234	#97	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	178	2	no	—
237	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₂⁴	—	356	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	C₂×D₃₅₆	1	2	yes	—
244	C₂×D₄	—	356	2	no	—
245	C₂×D₄	—	356	2	no	—
246	C₂×D₄	—	356	2	no	—
247	C₂×D₄	—	356	4	no	—
248	C₂×D₄	—	356	4	no	—
249	C₂×D₄	—	178	4	no	—
250	C₂×D₄	—	178	4	no	—
251	C₂×D₄	—	178	4	no	—
252	C₂×D₄	—	712	4	no	—
253	C₂×D₄	—	712	4	no	—
254	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
256	C₂²×C₄	#13083574	1	1	yes	characteristic
257	SD₁₆	—	712	8	no	—
258	D₈	—	356	4	no	—
259	D₈	—	356	4	no	—
260	D₈	—	356	4	no	—
261	D₈	—	356	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	178	4	no	—
266	C₂×C₈	#13083561	1	4	yes	complemented
267	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
280	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
281	C₄²	#13083574	1	1	yes	characteristic
282	C₄²	C₂³×D₈₉	1	1	yes	characteristic
283	C₂³	—	356	4	no	—
284	C₂³	—	712	4	no	—
285	C₂³	—	712	4	no	—
286	C₂³	—	712	4	no	—
287	C₂³	#15667986	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	4	2	no	—
290	D₄	—	356	4	no	—
291	D₄	—	356	8	no	—
292	D₄	—	356	4	no	—
293	D₄	—	712	4	no	—
294	D₄	—	712	4	no	—
295	D₄	—	712	4	no	—
296	D₄	—	712	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	#15667986	1	1	yes	characteristic
302	C₂×C₄	#15667986	1	1	yes	characteristic
303	C₂×C₄	#15667986	1	1	yes	characteristic
304	C₂×C₄	C₂×#13083561	1	2	yes	—
305	C₂×C₄	#15668037	1	2	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
308	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	356	4	no	—
313	C₈	C₂×#13083561	1	2	yes	—
314	C₈	C₂×#13083561	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	1424	8	no	—
318	C₂²	—	712	4	no	—
319	C₂²	—	712	4	no	—
320	C₂²	—	712	4	no	—
321	C₂²	#19151798	1	2	yes	—
322	C₂²	C₂×#15667986	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	#19151798	1	2	yes	—
328	C₄	#19151868	1	2	yes	—
329	C₄	C₂²×#13083561	1	1	yes	characteristic
330	C₄	C₂×#15668037	1	1	yes	characteristic
331	C₄	#19152403	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	1424	4	no	—
334	C₂	C₂×#19151798	1	1	yes	characteristic
335	C₂	#26029900	1	1	yes	characteristic
336	C₂	C₂×#19151868	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667986
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2960	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂×#19151975	C₂
hypercenter	#2960	D₈₉
hypocenter	C₈₉	#19203
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #26031190	C₂	semidirect
3. #26019736	C₂	semidirect
4. #26019612	C₂	semidirect
5. #26019000	C₂	semidirect
6. #26018921	C₂	semidirect
7. C₈₉×#2960	C₂	semidirect
17. #19151798	C₂²	semidirect
20. #19151351	C₂²	semidirect
21. #19151349	C₂²	semidirect
22. #19151341	C₂²	semidirect
25. #446×C₈₉	C₂²	semidirect
27. C₈×Q₈×C₈₉	C₂²	semidirect
97. C₂×C₈×C₈₉	SD₁₆	semidirect
171. #2960	D₈₉	semidirect
176. C₈₉	#19203	semidirect
190. #446	C₂×D₈₉	semidirect
192. C₈×Q₈	C₂×D₈₉	semidirect
266. C₂×C₈	#13083561	semidirect

Maximal subgroups

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

subgroup	index
#26031190	2
C₂×#19151975	2
#26019736	2
#26019612	2
#26019000	2
#26018921	2
C₈₉×#2960	2
#19203	89