Subgroups — #36712474

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	#26033256	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033242	C₂	1	1	yes	maximal, characteristic
3	#26022520	C₂	1	2	yes	maximal
4	#26022316	C₂	1	2	yes	maximal
5	#26022300	C₂	1	2	yes	maximal, complemented
6	#26022299	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022294	C₂	1	1	yes	maximal, characteristic
8	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022123	C₂	1	2	yes	maximal, complemented
10	C₄×#15667957	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2247	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152323	—	2	1	no	—
13	#19152322	—	2	1	no	—
14	C₂×#15667958	C₂²	1	2	yes	—
15	C₂×#15667957	C₂²	1	1	yes	characteristic
16	C₂×#15667955	C₂²	1	2	yes	—
17	#19151983	C₂²	1	2	yes	—
18	#19151976	C₂²	1	2	yes	—
19	#19151822	—	2	2	no	—
20	#19151806	—	2	2	no	—
21	#19151805	—	2	1	no	—
22	#19151800	—	2	1	no	—
23	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
26	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
27	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
28	#19151426	C₂²	1	1	yes	characteristic
29	C₄⋊#13083549	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151403	C₂²	1	1	yes	characteristic
32	#19151402	C₂²	1	2	yes	—
33	#19151402	C₂²	1	2	yes	—
34	#19151400	C₂²	1	1	yes	characteristic
35	#19151373	C₂²	1	2	yes	—
36	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
37	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
38	#494×C₈₉	C₂²	1	1	yes	characteristic
39	#494×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×#13083564	—	2	2	no	—
43	C₂×#13083561	—	2	2	no	—
44	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
45	#15668010	—	2	2	no	—
46	#15668007	—	4	2	no	—
47	#15668003	—	4	1	no	—
48	#15668002	—	4	2	no	—
49	#15668001	—	2	2	no	—
50	#15667992	—	4	1	no	—
51	Q₈×Dic₈₉	—	2	2	no	—
52	Q₈×Dic₈₉	—	2	2	no	—
53	C₄×Dic₁₇₈	—	2	1	no	—
54	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
55	#15667977	—	4	1	no	—
56	C₄×D₃₅₆	—	2	1	no	—
57	#15667970	—	4	1	no	—
58	#15667958	—	4	2	no	—
59	#15667957	—	2	2	no	—
60	#15667955	—	4	2	no	—
61	C₂×#13083549	C₂³	1	2	yes	—
62	#15667946	C₂³	1	1	yes	characteristic
63	#15667946	C₂³	1	1	yes	characteristic
64	C₄⋊#10623494	D₄	1	1	yes	characteristic
65	#15667941	D₄	1	2	yes	—
66	#15667937	—	2	1	no	—
67	#15667936	—	2	2	no	—
68	#15667936	—	2	2	no	—
69	#15667934	—	2	1	no	—
70	C₈×Dic₈₉	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₈₉	—	2	1	no	—
72	C₂×Q₁₆×C₈₉	—	2	1	no	—
73	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
74	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₈₉	—	2	1	no	—
76	C₄⋊Q₈×C₈₉	—	2	1	no	—
77	Q₃₂×C₈₉	—	2	4	no	—
78	Q₃₂×C₈₉	—	4	2	no	—
79	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	1	no	—
83	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
84	C₂×D₃₅₆	—	2	1	no	—
85	C₂×C₄×D₈₉	—	4	1	no	—
86	Dic₃₅₆	—	2	1	no	—
87	Dic₃₅₆	—	2	1	no	—
88	#13083564	—	8	2	no	—
89	#13083561	—	8	2	no	—
90	D₇₁₂	—	2	1	no	—
91	D₇₁₂	—	2	1	no	—
92	C₂×#10623494	C₂×D₄	1	2	yes	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	4	2	no	—
96	C₄⋊Dic₈₉	—	4	2	no	—
97	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
98	#13083553	—	4	1	no	—
99	#13083553	—	4	2	no	—
100	#13083553	—	4	2	no	—
101	#13083553	D₈	1	2	yes	—
102	#13083551	—	4	1	no	—
103	#13083551	—	4	2	no	—
104	#13083551	—	4	2	no	—
105	C₄×Dic₈₉	—	4	1	no	—
106	C₄×Dic₈₉	—	4	2	no	—
107	C₄×Dic₈₉	—	4	2	no	—
108	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
109	#13083549	—	2	2	no	—
110	C₂×Q₈×C₈₉	—	2	2	no	—
111	C₂×Q₈×C₈₉	—	2	2	no	—
112	Q₁₆×C₈₉	—	2	2	no	—
113	Q₁₆×C₈₉	—	2	2	no	—
114	Q₁₆×C₈₉	—	2	2	no	—
115	Q₁₆×C₈₉	—	2	2	no	—
116	Q₁₆×C₈₉	—	8	1	no	—
117	Q₁₆×C₈₉	—	8	1	no	—
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	—	4	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	—	4	1	no	—
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₈₉	—	2	1	no	—
126	C₁₆×C₈₉	#41	1	2	yes	—
127	C₂²×D₈₉	—	4	1	no	—
128	Q₈×C₈₉	—	4	2	no	—
129	Q₈×C₈₉	—	4	2	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	Q₈×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	—	4	1	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	4	1	no	—
140	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
141	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
142	D₃₅₆	—	4	1	no	—
143	D₃₅₆	—	4	1	no	—
144	C₄×D₈₉	—	8	2	no	—
145	Dic₁₇₈	—	2	2	no	—
146	Dic₁₇₈	—	4	1	no	—
147	Dic₁₇₈	—	4	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
151	#10623494	—	2	2	no	—
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	1	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	—	8	2	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	—	2	2	no	—
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	8	1	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	D₈₉	1	1	yes	characteristic, complemented
176	#2211	—	89	2	no	—
177	#2209	—	89	2	no	—
178	#2209	—	89	1	no	—
179	#2208	—	89	1	no	—
180	#2183	—	89	1	no	—
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	C₂×D₈₉	1	2	yes	—
188	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
189	C₂×SD₃₂	—	89	2	no	—
190	C₂×SD₃₂	—	89	1	no	—
191	#495	—	89	2	no	—
192	#494	—	89	2	no	—
193	#494	C₂×D₈₉	1	1	yes	characteristic
194	#494	C₂×D₈₉	1	1	yes	characteristic
195	#464	—	178	2	no	—
196	#462	—	178	1	no	—
197	#462	—	178	2	no	—
198	#461	—	178	1	no	—
199	C₈×Q₈	—	89	1	no	—
200	C₄×Q₁₆	—	89	2	no	—
201	C₄×Q₁₆	—	89	2	no	—
202	C₄×Q₁₆	—	89	1	no	—
203	C₄×D₈	—	89	1	no	—
204	#367	—	89	1	no	—
205	C₄⋊C₁₆	—	89	2	no	—
206	C₄⋊C₁₆	—	89	1	no	—
207	#358	—	89	2	no	—
208	#358	—	89	2	no	—
209	#358	—	89	1	no	—
210	#357	—	89	1	no	—
211	#346	—	89	2	no	—
212	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	89	1	no	—
216	C₂×Q₁₆	—	178	2	no	—
217	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
218	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
219	C₂×SD₁₆	—	178	2	no	—
220	C₂×D₈	—	89	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	178	2	no	—
224	#126	—	356	2	no	—
225	#125	—	356	2	no	—
226	#125	—	356	1	no	—
227	#124	—	178	2	no	—
228	C₂²⋊Q₈	—	356	1	no	—
229	C₄×Q₈	—	89	1	no	—
230	C₄×Q₈	—	356	1	no	—
231	C₄×Q₈	—	178	2	no	—
232	C₄×Q₈	—	178	2	no	—
233	C₄×Q₈	—	178	1	no	—
234	C₄×D₄	—	178	1	no	—
235	#117	—	356	1	no	—
236	Q₃₂	—	2	4	no	—
237	Q₃₂	—	4	2	no	—
238	Q₃₂	—	356	2	no	—
239	SD₃₂	—	356	2	no	—
240	SD₃₂	—	178	2	no	—
241	C₂×C₁₆	—	89	2	no	—
242	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
244	#107	—	89	1	no	—
245	#107	—	89	1	no	—
246	C₄⋊C₈	—	89	2	no	—
247	C₄⋊C₈	—	89	1	no	—
248	#103	—	178	2	no	—
249	#103	—	178	2	no	—
250	#103	—	178	1	no	—
251	#102	—	178	1	no	—
252	C₄×C₈	—	89	2	no	—
253	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	89	1	no	—
257	C₂×Q₈	—	178	1	no	—
258	C₂×D₄	—	178	1	no	—
259	C₂²×C₄	—	356	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	8	1	no	—
265	Q₁₆	—	8	1	no	—
266	Q₁₆	—	178	1	no	—
267	Q₁₆	—	178	1	no	—
268	Q₁₆	—	712	2	no	—
269	SD₁₆	—	712	2	no	—
270	D₈	—	178	1	no	—
271	D₈	—	178	1	no	—
272	C₂×C₈	—	89	2	no	—
273	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	89	1	no	—
280	C₄⋊C₄	—	89	2	no	—
281	C₄⋊C₄	—	356	2	no	—
282	C₄⋊C₄	—	356	2	no	—
283	C₄⋊C₄	—	356	2	no	—
284	C₄⋊C₄	—	356	2	no	—
285	C₄⋊C₄	—	356	1	no	—
286	C₄⋊C₄	—	178	1	no	—
287	C₄⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	356	2	no	—
289	C₂²⋊C₄	—	356	2	no	—
290	C₂²⋊C₄	—	356	1	no	—
291	C₄²	—	89	2	no	—
292	C₄²	—	356	2	no	—
293	C₄²	—	356	2	no	—
294	C₄²	—	356	1	no	—
295	C₄²	D₄×D₈₉	1	1	yes	characteristic
296	C₁₆	—	2	1	no	—
297	C₁₆	—	178	2	no	—
298	C₁₆	#13083573	1	2	yes	—
299	C₂³	—	356	1	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	356	1	no	—
305	Q₈	—	356	1	no	—
306	Q₈	—	178	2	no	—
307	D₄	—	356	1	no	—
308	D₄	—	356	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	89	2	no	—
312	C₂×C₄	—	89	2	no	—
313	C₂×C₄	—	356	1	no	—
314	C₂×C₄	—	356	1	no	—
315	C₂×C₄	—	356	1	no	—
316	C₂×C₄	—	712	2	no	—
317	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
318	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	178	2	no	—
322	C₈	C₂×#13083573	1	1	yes	characteristic
323	C₈	#15668057	1	1	yes	characteristic
324	C₂²	—	712	1	no	—
325	C₂²	—	712	1	no	—
326	C₂²	—	712	1	no	—
327	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	178	2	no	—
332	C₄	—	178	2	no	—
333	C₄	—	712	1	no	—
334	C₄	—	712	1	no	—
335	C₄	—	712	1	no	—
336	C₄	#19152040	1	2	yes	—
337	C₄	#19152268	1	1	yes	characteristic
338	C₄	#19152399	1	1	yes	characteristic
339	C₂	—	1424	1	no	—
340	C₂	#26033225	1	1	yes	characteristic
341	C₂	C₂×#19152040	1	1	yes	characteristic
342	C₂	#26033401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2247	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#2247	D₈₉
hypocenter	C₈₉	#33965
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

Complements

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

N	H ≅ G/N	type
1. #26033256	C₂	semidirect
5. #26022300	C₂	semidirect
6. #26022299	C₂	semidirect
8. #26022160	C₂	semidirect
9. #26022123	C₂	semidirect
11. C₈₉×#2247	C₂	semidirect
175. #2247	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
#26033256	2
#26033242	2
#26022520	2
#26022316	2
#26022300	2
#26022299	2
#26022294	2
#26022160	2
#26022123	2
C₄×#15667957	2
C₈₉×#2247	2
#33965	89