Subgroups — #36712471

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033258	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033241	C₂	1	1	yes	maximal, characteristic
3	#26022523	C₂	1	2	yes	maximal
4	#26022505	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022315	C₂	1	2	yes	maximal
6	#26022303	C₂	1	2	yes	maximal, complemented
7	#26022303	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022298	C₂	1	1	yes	maximal, characteristic
9	#26022124	C₂	1	2	yes	maximal, complemented
10	C₄×#15667957	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152327	—	2	1	no	—
13	#19152321	—	2	1	no	—
14	C₂×Dic₇₁₂	C₂²	1	2	yes	—
15	C₂×Dic₇₁₂	C₂²	1	2	yes	—
16	C₂×Dic₇₁₂	C₂²	1	2	yes	—
17	C₂×#15667957	C₂²	1	1	yes	characteristic
18	C₂×#15667957	C₂²	1	2	yes	—
19	#19151982	C₂²	1	2	yes	—
20	#19151978	C₂²	1	1	yes	characteristic
21	#19151978	C₂²	1	1	yes	characteristic
22	#19151978	C₂²	1	2	yes	—
23	#19151821	—	2	2	no	—
24	#19151809	—	2	1	no	—
25	#19151809	—	2	2	no	—
26	#19151804	—	2	1	no	—
27	C₄×Dic₃₅₆	C₂²	1	2	yes	—
28	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₅₆	C₂²	1	2	yes	—
30	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
31	#19151426	C₂²	1	1	yes	characteristic
32	#19151403	C₂²	1	2	yes	—
33	#19151403	C₂²	1	1	yes	characteristic
34	#19151403	C₂²	1	2	yes	—
35	#19151400	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
39	#346×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	—	2	2	no	—
42	C₂×Dic₃₅₆	—	2	1	no	—
43	C₂×Dic₃₅₆	—	2	1	no	—
44	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₅₆	C₂³	1	2	yes	—
46	C₂×Dic₃₅₆	C₂³	1	2	yes	—
47	C₂×#13083563	—	2	2	no	—
48	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
49	#15668012	—	2	1	no	—
50	#15668012	—	2	1	no	—
51	#15668012	—	2	2	no	—
52	#15668007	—	4	2	no	—
53	#15668003	—	4	1	no	—
54	#15668003	—	4	2	no	—
55	#15668000	—	2	2	no	—
56	#15667988	—	4	1	no	—
57	Q₈×Dic₈₉	—	4	1	no	—
58	C₄×Dic₁₇₈	—	2	1	no	—
59	C₄×Dic₁₇₈	—	2	2	no	—
60	C₄×Dic₁₇₈	—	2	2	no	—
61	C₄×D₃₅₆	—	2	1	no	—
62	#15667970	—	4	1	no	—
63	Dic₇₁₂	—	2	4	no	—
64	Dic₇₁₂	—	4	2	no	—
65	Dic₇₁₂	—	4	2	no	—
66	#15667957	—	2	2	no	—
67	#15667957	—	4	2	no	—
68	#15667946	C₂³	1	1	yes	characteristic
69	#15667946	C₂³	1	1	yes	characteristic
70	#15667937	—	2	2	no	—
71	#15667937	—	2	2	no	—
72	#15667937	—	2	1	no	—
73	#15667934	—	2	1	no	—
74	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
77	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
79	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
80	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₈₉	C₂³	1	2	yes	—
82	C₂×Dic₁₇₈	—	2	2	no	—
83	C₂×Dic₁₇₈	—	2	1	no	—
84	C₂×Dic₁₇₈	—	2	2	no	—
85	C₂×D₃₅₆	—	2	1	no	—
86	C₂×C₄×D₈₉	—	4	1	no	—
87	Dic₃₅₆	—	2	1	no	—
88	Dic₃₅₆	—	2	2	no	—
89	Dic₃₅₆	—	2	2	no	—
90	Dic₃₅₆	—	2	2	no	—
91	Dic₃₅₆	—	2	1	no	—
92	Dic₃₅₆	—	2	2	no	—
93	Dic₃₅₆	—	8	2	no	—
94	Dic₃₅₆	—	8	1	no	—
95	Dic₃₅₆	—	8	1	no	—
96	#13083563	—	8	2	no	—
97	D₇₁₂	—	2	1	no	—
98	D₇₁₂	—	2	1	no	—
99	C₄⋊Dic₈₉	—	2	1	no	—
100	C₄⋊Dic₈₉	—	2	1	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	C₄⋊Dic₈₉	—	4	2	no	—
103	C₄⋊Dic₈₉	—	4	1	no	—
104	C₄⋊Dic₈₉	—	4	2	no	—
105	C₄⋊Dic₈₉	—	4	1	no	—
106	C₄⋊Dic₈₉	—	4	1	no	—
107	#13083553	—	4	2	no	—
108	#13083553	—	4	2	no	—
109	#13083553	—	4	1	no	—
110	#13083551	—	4	2	no	—
111	#13083551	—	4	1	no	—
112	#13083551	—	4	2	no	—
113	C₄×Dic₈₉	—	4	2	no	—
114	C₄×Dic₈₉	—	4	2	no	—
115	C₄×Dic₈₉	—	4	1	no	—
116	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	C₂×D₄	1	2	yes	—
124	C₁₆×C₈₉	—	2	2	no	—
125	C₁₆×C₈₉	—	2	1	no	—
126	C₁₆×C₈₉	#41	1	2	yes	—
127	C₂²×D₈₉	—	4	1	no	—
128	Q₈×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	—	4	1	no	—
135	C₂×Dic₈₉	—	4	1	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	D₃₅₆	—	4	1	no	—
140	D₃₅₆	—	4	1	no	—
141	C₄×D₈₉	—	8	2	no	—
142	Dic₁₇₈	—	4	1	no	—
143	Dic₁₇₈	—	4	2	no	—
144	Dic₁₇₈	—	4	2	no	—
145	Dic₁₇₈	—	4	2	no	—
146	Dic₁₇₈	—	4	1	no	—
147	Dic₁₇₈	—	4	2	no	—
148	C₈×C₈₉	—	2	2	no	—
149	C₈×C₈₉	—	2	1	no	—
150	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
151	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₈₉	—	8	1	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₂×D₈₉	—	8	1	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	#510	1	2	yes	—
160	C₄×C₈₉	#550	1	1	yes	characteristic
161	C₄×C₈₉	#576	1	1	yes	characteristic
162	Dic₈₉	—	8	2	no	—
163	Dic₈₉	—	8	1	no	—
164	Dic₈₉	—	8	2	no	—
165	Dic₈₉	—	8	1	no	—
166	Dic₈₉	—	8	1	no	—
167	#33965	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	#3391	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
170	C₂×C₈₉	#3417	1	1	yes	characteristic
171	D₈₉	—	16	1	no	—
172	#3395	—	89	1	no	—
173	#3393	—	89	1	no	—
174	#2251	—	89	2	no	—
175	#2247	—	89	1	no	—
176	#2211	—	89	2	no	—
177	#2209	—	89	2	no	—
178	#2209	—	89	1	no	—
179	#2208	—	89	1	no	—
180	#2183	D₈₉	1	1	yes	characteristic, complemented
181	#2176	—	89	2	no	—
182	C₄×SD₃₂	—	89	1	no	—
183	C₈₉	#33965	1	1	yes	characteristic, complemented, Sylow
184	#557	—	178	1	no	—
185	#556	—	178	1	no	—
186	C₂×Q₃₂	—	89	2	no	—
187	C₂×Q₃₂	—	89	2	no	—
188	C₂×Q₃₂	—	89	2	no	—
189	C₂×SD₃₂	—	89	2	no	—
190	C₂×SD₃₂	—	89	1	no	—
191	#495	—	89	2	no	—
192	#494	—	89	1	no	—
193	#494	—	89	1	no	—
194	#494	—	89	2	no	—
195	#464	—	178	2	no	—
196	#462	—	178	1	no	—
197	#462	—	178	2	no	—
198	#461	—	178	1	no	—
199	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
200	C₄×Q₁₆	—	89	2	no	—
201	C₄×Q₁₆	—	89	1	no	—
202	C₄×Q₁₆	—	89	2	no	—
203	C₄×D₈	—	89	1	no	—
204	#367	—	89	1	no	—
205	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
206	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
207	#358	—	89	1	no	—
208	#358	—	89	2	no	—
209	#358	—	89	2	no	—
210	#357	—	89	1	no	—
211	#346	C₂×D₈₉	1	2	yes	—
212	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
213	C₂×Q₁₆	—	89	1	no	—
214	C₂×Q₁₆	—	89	2	no	—
215	C₂×Q₁₆	—	89	2	no	—
216	C₂×Q₁₆	—	178	1	no	—
217	C₂×Q₁₆	—	178	2	no	—
218	C₂×Q₁₆	—	178	1	no	—
219	C₂×SD₁₆	—	178	2	no	—
220	C₂×D₈	—	89	1	no	—
221	C₄⋊Q₈	—	178	1	no	—
222	C₄⋊Q₈	—	178	1	no	—
223	C₄⋊Q₈	—	178	2	no	—
224	#126	—	356	2	no	—
225	#125	—	356	1	no	—
226	#125	—	356	2	no	—
227	#124	—	178	2	no	—
228	C₂²⋊Q₈	—	356	1	no	—
229	C₄×Q₈	—	356	1	no	—
230	C₄×Q₈	—	178	2	no	—
231	C₄×Q₈	—	178	1	no	—
232	C₄×Q₈	—	178	2	no	—
233	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
234	C₄×D₄	—	178	1	no	—
235	#117	—	356	1	no	—
236	Q₃₂	—	356	2	no	—
237	Q₃₂	—	356	2	no	—
238	Q₃₂	—	178	4	no	—
239	SD₃₂	—	356	2	no	—
240	SD₃₂	—	178	2	no	—
241	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
243	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
244	#107	—	89	1	no	—
245	#107	—	89	1	no	—
246	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
247	C₄⋊C₈	D₃₅₆	1	2	yes	—
248	#103	—	178	2	no	—
249	#103	—	178	1	no	—
250	#103	—	178	2	no	—
251	#102	—	178	1	no	—
252	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₈₉	1	2	yes	—
254	C₂×Q₈	—	178	2	no	—
255	C₂×Q₈	—	178	1	no	—
256	C₂×Q₈	—	178	2	no	—
257	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
258	C₂×D₄	—	178	1	no	—
259	C₂²×C₄	—	356	1	no	—
260	Q₁₆	—	178	1	no	—
261	Q₁₆	—	178	2	no	—
262	Q₁₆	—	178	2	no	—
263	Q₁₆	—	178	2	no	—
264	Q₁₆	—	178	1	no	—
265	Q₁₆	—	178	2	no	—
266	Q₁₆	—	712	2	no	—
267	Q₁₆	—	712	1	no	—
268	Q₁₆	—	712	1	no	—
269	SD₁₆	—	712	2	no	—
270	D₈	—	178	1	no	—
271	D₈	—	178	1	no	—
272	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
273	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
274	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	—	356	1	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	356	1	no	—
278	C₄⋊C₄	—	356	2	no	—
279	C₄⋊C₄	—	356	2	no	—
280	C₄⋊C₄	—	356	1	no	—
281	C₄⋊C₄	—	356	2	no	—
282	C₄⋊C₄	—	356	1	no	—
283	C₄⋊C₄	—	356	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
287	C₄⋊C₄	D₇₁₂	1	2	yes	—
288	C₂²⋊C₄	—	356	2	no	—
289	C₂²⋊C₄	—	356	1	no	—
290	C₂²⋊C₄	—	356	2	no	—
291	C₄²	—	356	2	no	—
292	C₄²	—	356	2	no	—
293	C₄²	—	356	1	no	—
294	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
295	C₄²	C₂×D₃₅₆	1	2	yes	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	#13083575	1	2	yes	—
299	C₂³	—	356	1	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	356	2	no	—
302	Q₈	—	356	1	no	—
303	Q₈	—	356	2	no	—
304	Q₈	—	356	1	no	—
305	Q₈	—	356	2	no	—
306	Q₈	—	356	2	no	—
307	D₄	—	356	1	no	—
308	D₄	—	356	1	no	—
309	C₂×C₄	—	356	1	no	—
310	C₂×C₄	—	356	1	no	—
311	C₂×C₄	—	356	2	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	1	no	—
314	C₂×C₄	—	712	2	no	—
315	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
316	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
317	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
318	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	1	no	—
322	C₈	C₂×#13083575	1	1	yes	characteristic
323	C₈	#15668058	1	1	yes	characteristic
324	C₂²	—	712	1	no	—
325	C₂²	—	712	1	no	—
326	C₂²	—	712	1	no	—
327	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	712	2	no	—
332	C₄	—	712	1	no	—
333	C₄	—	712	2	no	—
334	C₄	—	712	1	no	—
335	C₄	—	712	1	no	—
336	C₄	#19152041	1	2	yes	—
337	C₄	#19152270	1	1	yes	characteristic
338	C₄	#19152403	1	1	yes	characteristic
339	C₂	—	1424	1	no	—
340	C₂	#26033227	1	1	yes	characteristic
341	C₂	C₂×#19152041	1	1	yes	characteristic
342	C₂	#26033405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2183	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33965
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
1. #26033258	C₂	semidirect
4. #26022505	C₂	semidirect
6. #26022303	C₂	semidirect
7. #26022303	C₂	semidirect
9. #26022124	C₂	semidirect
11. C₈₉×#2183	C₂	semidirect
180. #2183	D₈₉	semidirect
183. C₈₉	#33965	semidirect

Maximal subgroups

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

subgroup	index
#26033258	2
#26033241	2
#26022523	2
#26022505	2
#26022315	2
#26022303	2
#26022298	2
#26022124	2
C₄×#15667957	2
C₈₉×#2183	2
#33965	89