Subgroups — #36711718

Order 22784.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032682	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26022481	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022463	C₂	1	2	yes	maximal, complemented
5	#26022361	C₂	1	2	yes	maximal, complemented
6	#26022303	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022187	C₂	1	2	yes	maximal, complemented
8	C₄×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022083	C₂	1	1	yes	maximal, characteristic
10	#26021970	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
13	#19152322	—	2	1	no	—
14	Q₈×D₃₅₆	—	2	1	no	—
15	C₂×Dic₇₁₂	C₂²	1	2	yes	—
16	C₂×#15667958	C₂²	1	1	yes	characteristic
17	C₂×#15667951	C₂²	1	2	yes	—
18	#19151978	C₂²	1	1	yes	characteristic
19	#19151941	—	2	2	no	—
20	#19151880	—	2	1	no	—
21	#19151840	C₂²	1	2	yes	—
22	#19151840	C₂²	1	2	yes	—
23	#19151808	—	2	1	no	—
24	#19151755	—	2	2	no	—
25	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
26	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
27	#19151635	C₂²	1	2	yes	—
28	#19151423	C₂²	1	1	yes	characteristic
29	#19151403	C₂²	1	2	yes	—
30	#19151402	C₂²	1	1	yes	characteristic
31	#19151401	C₂²	1	2	yes	—
32	#19151401	C₂²	1	2	yes	—
33	#19151401	C₂²	1	1	yes	characteristic
34	#19151377	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	#367×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂×Q₈×D₈₉	—	2	2	no	—
42	C₂×Dic₃₅₆	—	2	1	no	—
43	C₂×Dic₃₅₆	C₂³	1	2	yes	—
44	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
45	C₂×#13083564	—	2	2	no	—
46	Q₁₆×D₈₉	—	2	4	no	—
47	Q₁₆×D₈₉	—	4	2	no	—
48	C₂×C₈×D₈₉	C₂³	1	2	yes	—
49	#15668012	—	2	1	no	—
50	#15668012	—	4	1	no	—
51	#15668007	—	4	2	no	—
52	#15668001	—	4	1	no	—
53	#15667992	—	2	2	no	—
54	#15667992	—	2	2	no	—
55	#15667992	—	4	2	no	—
56	Q₈×Dic₈₉	—	4	1	no	—
57	C₄×Dic₁₇₈	—	2	1	no	—
58	C₄×D₃₅₆	—	4	1	no	—
59	C₄×D₃₅₆	D₄	1	1	yes	characteristic
60	#15667970	—	2	2	no	—
61	Dic₇₁₂	—	4	2	no	—
62	#15667958	—	2	2	no	—
63	#15667951	—	4	2	no	—
64	C₂×#13083549	C₂³	1	1	yes	characteristic
65	C₂×#13083549	C₂³	1	1	yes	characteristic
66	#15667946	C₂³	1	1	yes	characteristic
67	#15667943	—	2	2	no	—
68	C₄⋊#10623494	D₄	1	1	yes	characteristic
69	#15667937	—	2	1	no	—
70	#15667935	—	2	2	no	—
71	#15667935	—	2	2	no	—
72	#15667924	D₄	1	2	yes	—
73	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₈₉	—	2	1	no	—
75	C₄×Q₈×C₈₉	—	2	1	no	—
76	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
77	#107×C₈₉	—	2	1	no	—
78	#107×C₈₉	C₂³	1	2	yes	—
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#103×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	1	no	—
83	C₂×Dic₁₇₈	—	2	2	no	—
84	C₂×Dic₁₇₈	—	4	2	no	—
85	Q₈×D₈₉	—	4	4	no	—
86	Q₈×D₈₉	—	8	2	no	—
87	C₂×D₃₅₆	D₈	1	1	yes	characteristic
88	C₂×C₄×D₈₉	—	4	2	no	—
89	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
90	Dic₃₅₆	—	2	2	no	—
91	Dic₃₅₆	—	2	1	no	—
92	Dic₃₅₆	—	2	2	no	—
93	Dic₃₅₆	—	2	1	no	—
94	Dic₃₅₆	—	8	1	no	—
95	#13083564	—	4	4	no	—
96	C₂×#10623494	C₂×D₄	1	2	yes	—
97	C₈×D₈₉	—	2	2	no	—
98	C₈×D₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	2	1	no	—
100	C₄⋊Dic₈₉	—	4	1	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	C₄⋊Dic₈₉	—	4	2	no	—
103	C₄⋊Dic₈₉	—	4	2	no	—
104	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
105	#13083553	—	4	1	no	—
106	#13083551	—	4	2	no	—
107	#13083551	—	4	2	no	—
108	#13083551	—	4	2	no	—
109	#13083551	D₈	1	2	yes	—
110	C₄×Dic₈₉	—	4	1	no	—
111	C₄×Dic₈₉	—	4	2	no	—
112	#13083549	—	2	1	no	—
113	#13083549	—	2	1	no	—
114	C₂×Q₈×C₈₉	—	2	1	no	—
115	C₂×Q₈×C₈₉	—	4	1	no	—
116	Q₁₆×C₈₉	—	2	1	no	—
117	Q₁₆×C₈₉	#41	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	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	—	4	1	no	—
124	C₄²×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₈₉	—	2	2	no	—
127	C₂²×D₈₉	C₂×D₈	1	2	yes	—
128	Q₈×C₈₉	—	2	2	no	—
129	Q₈×C₈₉	—	4	1	no	—
130	Q₈×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	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	2	no	—
140	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
141	D₃₅₆	—	2	2	no	—
142	C₄×D₈₉	—	2	2	no	—
143	C₄×D₈₉	—	2	2	no	—
144	C₄×D₈₉	—	8	4	no	—
145	Dic₁₇₈	—	4	2	no	—
146	Dic₁₇₈	—	4	1	no	—
147	Dic₁₇₈	—	4	2	no	—
148	Dic₁₇₈	—	4	1	no	—
149	Dic₁₇₈	—	8	4	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
153	#10623494	—	2	2	no	—
154	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₈₉	—	2	2	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	4	2	no	—
160	C₄×C₈₉	—	8	2	no	—
161	C₄×C₈₉	—	8	1	no	—
162	C₄×C₈₉	#510	1	2	yes	—
163	C₄×C₈₉	#546	1	1	yes	characteristic
164	C₄×C₈₉	#577	1	1	yes	characteristic
165	Dic₈₉	—	2	2	no	—
166	Dic₈₉	—	8	1	no	—
167	Dic₈₉	—	8	2	no	—
168	Dic₈₉	—	8	1	no	—
169	#33895	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	#3294	1	1	yes	characteristic
171	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
172	C₂×C₈₉	#3417	1	1	yes	characteristic
173	D₈₉	—	4	2	no	—
174	#3342	—	89	1	no	—
175	D₄×Q₁₆	—	89	1	no	—
176	#2241	—	89	1	no	—
177	#2237	—	89	2	no	—
178	#2225	D₈₉	1	1	yes	characteristic, complemented
179	#2219	—	89	2	no	—
180	#2209	—	89	1	no	—
181	#2188	—	89	2	no	—
182	C₄×Q₃₂	—	89	1	no	—
183	#2169	—	89	1	no	—
184	#2141	—	89	2	no	—
185	C₈₉	#33895	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₁₆	—	89	2	no	—
187	#556	—	178	1	no	—
188	D₄×Q₈	—	178	1	no	—
189	C₂×Q₃₂	—	89	1	no	—
190	C₂×Q₃₂	—	89	2	no	—
191	C₂×M₃₂	—	89	2	no	—
192	#500	C₂×D₈₉	1	1	yes	characteristic
193	#494	—	89	1	no	—
194	#485	—	178	2	no	—
195	#475	—	178	1	no	—
196	#467	—	89	2	no	—
197	#467	—	89	2	no	—
198	#462	—	178	1	no	—
199	#451	—	178	2	no	—
200	C₄×Q₁₆	—	89	1	no	—
201	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
202	C₈×D₄	—	89	1	no	—
203	#427	—	89	2	no	—
204	#367	C₂×D₈₉	1	2	yes	—
205	#366	—	89	1	no	—
206	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
207	#358	—	89	1	no	—
208	#358	—	89	2	no	—
209	#358	—	89	2	no	—
210	#358	—	89	1	no	—
211	#358	—	89	2	no	—
212	#358	C₂×D₈₉	1	2	yes	—
213	C₂²⋊C₁₆	—	89	2	no	—
214	C₄×C₁₆	—	89	1	no	—
215	C₂²×Q₈	—	178	2	no	—
216	C₂×Q₁₆	—	89	2	no	—
217	C₂×Q₁₆	—	89	1	no	—
218	C₂×Q₁₆	—	356	2	no	—
219	C₂×Q₁₆	—	178	2	no	—
220	C₂×Q₁₆	—	178	4	no	—
221	C₂×Q₁₆	—	178	1	no	—
222	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
223	C₂²×C₈	—	89	2	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	356	1	no	—
226	C₄⋊Q₈	—	178	1	no	—
227	#126	—	356	2	no	—
228	#124	—	356	1	no	—
229	C₂²⋊Q₈	—	356	2	no	—
230	C₂²⋊Q₈	—	178	2	no	—
231	C₂²⋊Q₈	—	178	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	356	1	no	—
234	C₄×Q₈	—	178	1	no	—
235	C₄×D₄	—	89	1	no	—
236	C₄×D₄	—	356	1	no	—
237	#117	—	178	2	no	—
238	Q₃₂	—	356	2	no	—
239	Q₃₂	—	178	2	no	—
240	M₃₂	—	356	2	no	—
241	C₂×C₁₆	—	89	1	no	—
242	C₂×C₁₆	—	89	1	no	—
243	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
244	#107	—	2	1	no	—
245	#107	—	89	1	no	—
246	#107	C₂²×D₈₉	1	2	yes	—
247	#107	C₂²×D₈₉	1	1	yes	characteristic
248	#106	—	178	2	no	—
249	C₄⋊C₈	—	89	1	no	—
250	#103	—	2	1	no	—
251	#103	—	178	2	no	—
252	#103	—	178	2	no	—
253	#103	—	178	1	no	—
254	C₂²⋊C₈	—	89	2	no	—
255	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	356	2	no	—
259	C₂×Q₈	—	356	4	no	—
260	C₂×Q₈	—	178	1	no	—
261	C₂×Q₈	—	178	2	no	—
262	C₂×Q₈	—	712	2	no	—
263	C₂×D₄	—	89	1	no	—
264	C₂²×C₄	—	89	2	no	—
265	C₂²×C₄	—	356	2	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	356	4	no	—
268	Q₁₆	—	178	1	no	—
269	Q₁₆	—	178	2	no	—
270	Q₁₆	—	178	1	no	—
271	Q₁₆	—	178	2	no	—
272	Q₁₆	—	712	1	no	—
273	Q₁₆	#13083575	1	2	yes	—
274	C₂×C₈	—	89	2	no	—
275	C₂×C₈	—	178	2	no	—
276	C₂×C₈	—	178	2	no	—
277	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	89	1	no	—
284	C₄⋊C₄	—	356	2	no	—
285	C₄⋊C₄	—	356	2	no	—
286	C₄⋊C₄	—	356	1	no	—
287	C₄⋊C₄	—	356	1	no	—
288	C₄⋊C₄	—	356	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	89	2	no	—
291	C₂²⋊C₄	—	356	2	no	—
292	C₂²⋊C₄	—	356	2	no	—
293	C₂²⋊C₄	—	356	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	356	2	no	—
296	C₄²	—	356	1	no	—
297	C₄²	D₄×D₈₉	1	1	yes	characteristic
298	C₁₆	—	2	2	no	—
299	C₁₆	—	178	1	no	—
300	C₁₆	—	178	1	no	—
301	C₂³	—	89	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	8	2	no	—
305	Q₈	—	356	2	no	—
306	Q₈	—	356	1	no	—
307	Q₈	—	356	2	no	—
308	Q₈	—	356	1	no	—
309	Q₈	—	712	4	no	—
310	D₄	—	178	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	356	1	no	—
316	C₂×C₄	—	356	1	no	—
317	C₂×C₄	—	356	2	no	—
318	C₂×C₄	—	178	2	no	—
319	C₂×C₄	—	178	2	no	—
320	C₂×C₄	—	712	4	no	—
321	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
322	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	178	2	no	—
326	C₈	C₂×#13083575	1	1	yes	characteristic
327	C₈	#15668058	1	1	yes	characteristic
328	C₂²	—	178	2	no	—
329	C₂²	—	178	2	no	—
330	C₂²	—	178	2	no	—
331	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	178	2	no	—
337	C₄	—	712	2	no	—
338	C₄	—	712	1	no	—
339	C₄	—	712	1	no	—
340	C₄	#19152037	1	2	yes	—
341	C₄	#19152239	1	1	yes	characteristic
342	C₄	#19152407	1	1	yes	characteristic
343	C₂	—	356	2	no	—
344	C₂	#26032062	1	1	yes	characteristic
345	C₂	C₂×#19152037	1	1	yes	characteristic
346	C₂	#26033404	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×#2225	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₃₅₆	D₈
hypercenter	#2225	D₈₉
hypocenter	C₈₉	#33895
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
3. #26022481	C₂	semidirect
4. #26022463	C₂	semidirect
5. #26022361	C₂	semidirect
6. #26022303	C₂	semidirect
7. #26022187	C₂	semidirect
8. C₄×#15667958	C₂	semidirect
10. #26021970	C₂	semidirect
11. C₈₉×#2225	C₂	semidirect
178. #2225	D₈₉	semidirect
185. C₈₉	#33895	semidirect

Maximal subgroups

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

subgroup	index
#26032682	2
Q₁₆×D₃₅₆	2
#26022481	2
#26022463	2
#26022361	2
#26022303	2
#26022187	2
C₄×#15667958	2
#26022083	2
#26021970	2
C₈₉×#2225	2
#33895	89