Subgroups — #36712493

Order 22784.

510 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	#26033265	C₂	1	1	yes	maximal, characteristic
2	#26033258	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022523	C₂	1	2	yes	maximal, complemented
4	#26022508	C₂	1	1	yes	maximal, characteristic
5	#26022315	C₂	1	2	yes	maximal, complemented
6	#26022303	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022298	C₂	1	2	yes	maximal, complemented
8	#26022293	C₂	1	1	yes	maximal, characteristic
9	#26022118	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152333	—	2	1	no	—
13	#19152327	—	2	1	no	—
14	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
15	C₂×Dic₇₁₂	C₂²	1	2	yes	—
16	C₂×#15667957	C₂²	1	2	yes	—
17	C₂×#15667957	C₂²	1	2	yes	—
18	C₂×#15667957	C₂²	1	2	yes	—
19	#19151982	C₂²	1	2	yes	—
20	#19151978	C₂²	1	1	yes	characteristic
21	#19151978	C₂²	1	2	yes	—
22	#19151975	C₂²	1	1	yes	characteristic
23	#19151814	—	2	2	no	—
24	#19151809	—	2	1	no	—
25	#19151809	—	2	2	no	—
26	#19151799	—	2	1	no	—
27	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
28	C₄×Dic₃₅₆	C₂²	1	2	yes	—
29	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
30	C₄×D₇₁₂	C₂²	1	2	yes	—
31	#19151424	C₂²	1	1	yes	characteristic
32	#19151403	C₂²	1	1	yes	characteristic
33	#19151403	C₂²	1	2	yes	—
34	#19151403	C₂²	1	1	yes	characteristic
35	#19151400	C₂²	1	2	yes	—
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	1	no	—
42	C₂×Dic₃₅₆	—	2	2	no	—
43	C₂×Dic₃₅₆	C₂³	1	2	yes	—
44	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
46	C₂×#13083563	—	2	2	no	—
47	C₂×D₇₁₂	—	2	1	no	—
48	C₂×D₇₁₂	C₂³	1	2	yes	—
49	#15668012	—	2	2	no	—
50	#15668012	—	2	1	no	—
51	#15668009	—	2	1	no	—
52	#15668003	—	4	1	no	—
53	#15668003	—	4	2	no	—
54	#15668001	—	4	1	no	—
55	#15668000	—	2	2	no	—
56	#15667997	—	4	2	no	—
57	Q₈×Dic₈₉	—	4	1	no	—
58	C₄×Dic₁₇₈	—	2	1	no	—
59	C₄×Dic₁₇₈	—	2	2	no	—
60	C₄×Dic₁₇₈	—	2	1	no	—
61	C₄×D₃₅₆	—	2	2	no	—
62	#15667974	—	4	1	no	—
63	Dic₇₁₂	—	2	2	no	—
64	Dic₇₁₂	—	4	2	no	—
65	#15667957	—	2	4	no	—
66	#15667957	—	4	2	no	—
67	#15667957	—	4	2	no	—
68	#15667946	C₂³	1	1	yes	characteristic
69	#15667946	C₂³	1	1	yes	characteristic
70	#15667937	—	2	1	no	—
71	#15667937	—	2	2	no	—
72	#15667937	—	2	1	no	—
73	#15667934	—	2	2	no	—
74	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
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	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	1	no	—
83	C₂×Dic₁₇₈	—	2	2	no	—
84	C₂×Dic₁₇₈	—	2	1	no	—
85	C₂×D₃₅₆	—	2	2	no	—
86	C₂×D₃₅₆	—	4	1	no	—
87	C₂×D₃₅₆	—	4	1	no	—
88	C₂×C₄×D₈₉	—	4	2	no	—
89	Dic₃₅₆	—	2	2	no	—
90	Dic₃₅₆	—	2	1	no	—
91	Dic₃₅₆	—	2	1	no	—
92	Dic₃₅₆	—	2	1	no	—
93	Dic₃₅₆	—	2	2	no	—
94	Dic₃₅₆	—	2	1	no	—
95	Dic₃₅₆	—	8	1	no	—
96	Dic₃₅₆	—	8	2	no	—
97	#13083563	—	8	2	no	—
98	D₇₁₂	—	2	2	no	—
99	D₇₁₂	—	2	2	no	—
100	D₇₁₂	—	8	1	no	—
101	C₄⋊Dic₈₉	—	2	1	no	—
102	C₄⋊Dic₈₉	—	2	1	no	—
103	C₄⋊Dic₈₉	—	4	2	no	—
104	C₄⋊Dic₈₉	—	4	1	no	—
105	C₄⋊Dic₈₉	—	4	2	no	—
106	C₄⋊Dic₈₉	—	4	1	no	—
107	#13083553	—	4	1	no	—
108	#13083553	—	4	2	no	—
109	#13083553	—	4	1	no	—
110	#13083551	—	4	2	no	—
111	#13083551	—	4	2	no	—
112	#13083551	—	4	2	no	—
113	C₄×Dic₈₉	—	4	1	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	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
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	2	yes	—
123	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₈₉	—	2	1	no	—
125	C₁₆×C₈₉	—	2	2	no	—
126	C₁₆×C₈₉	#41	1	2	yes	—
127	C₂²×D₈₉	—	4	2	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	1	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	D₃₅₆	—	4	2	no	—
140	D₃₅₆	—	4	2	no	—
141	D₃₅₆	—	8	2	no	—
142	D₃₅₆	—	16	1	no	—
143	C₄×D₈₉	—	16	2	no	—
144	Dic₁₇₈	—	4	2	no	—
145	Dic₁₇₈	—	4	1	no	—
146	Dic₁₇₈	—	4	1	no	—
147	Dic₁₇₈	—	4	1	no	—
148	Dic₁₇₈	—	4	2	no	—
149	Dic₁₇₈	—	4	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	2	no	—
152	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₈₉	—	8	2	no	—
156	C₂×D₈₉	—	8	2	no	—
157	C₂×D₈₉	—	8	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	2	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	#510	1	2	yes	—
162	C₄×C₈₉	#550	1	1	yes	characteristic
163	C₄×C₈₉	#576	1	1	yes	characteristic
164	Dic₈₉	—	8	1	no	—
165	Dic₈₉	—	8	2	no	—
166	Dic₈₉	—	8	1	no	—
167	Dic₈₉	—	8	1	no	—
168	Dic₈₉	—	8	1	no	—
169	#33967	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	#3391	1	1	yes	characteristic
171	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
172	C₂×C₈₉	#3416	1	1	yes	characteristic
173	D₈₉	—	16	2	no	—
174	#3396	—	89	1	no	—
175	#3395	—	89	1	no	—
176	#2251	—	89	2	no	—
177	#2248	—	89	1	no	—
178	#2211	—	89	2	no	—
179	#2209	—	89	1	no	—
180	#2208	—	89	2	no	—
181	#2207	—	89	1	no	—
182	#2183	D₈₉	1	1	yes	characteristic, complemented
183	#2175	—	89	2	no	—
184	C₄×Q₃₂	—	89	1	no	—
185	C₈₉	#33967	1	1	yes	characteristic, complemented, Sylow
186	#559	—	178	1	no	—
187	#557	—	178	1	no	—
188	C₂×Q₃₂	—	89	2	no	—
189	C₂×Q₃₂	—	89	1	no	—
190	C₂×SD₃₂	—	89	2	no	—
191	C₂×SD₃₂	—	89	2	no	—
192	C₂×SD₃₂	—	89	2	no	—
193	#495	—	89	2	no	—
194	#494	—	89	2	no	—
195	#494	—	89	1	no	—
196	#493	—	89	1	no	—
197	#463	—	178	2	no	—
198	#462	—	178	1	no	—
199	#462	—	178	2	no	—
200	#460	—	178	1	no	—
201	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
202	C₄×Q₁₆	—	89	1	no	—
203	C₄×Q₁₆	—	89	2	no	—
204	C₄×Q₁₆	—	89	1	no	—
205	C₄×D₈	—	89	2	no	—
206	#366	—	89	1	no	—
207	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
209	#358	—	89	1	no	—
210	#358	—	89	1	no	—
211	#358	—	89	2	no	—
212	#357	—	89	2	no	—
213	#346	C₂×D₈₉	1	2	yes	—
214	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
215	C₂×Q₁₆	—	89	2	no	—
216	C₂×Q₁₆	—	89	1	no	—
217	C₂×Q₁₆	—	89	1	no	—
218	C₂×Q₁₆	—	178	2	no	—
219	C₂×Q₁₆	—	178	1	no	—
220	C₂×SD₁₆	—	178	2	no	—
221	C₂×D₈	—	89	2	no	—
222	C₂×D₈	—	178	1	no	—
223	C₄⋊Q₈	—	178	2	no	—
224	C₄⋊Q₈	—	178	1	no	—
225	#127	—	178	1	no	—
226	#125	—	356	1	no	—
227	#125	—	356	2	no	—
228	#124	—	356	1	no	—
229	#124	—	178	2	no	—
230	#123	—	356	2	no	—
231	C₄×Q₈	—	356	1	no	—
232	C₄×Q₈	—	178	2	no	—
233	C₄×Q₈	—	178	1	no	—
234	C₄×Q₈	—	178	1	no	—
235	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
236	C₄×D₄	—	356	1	no	—
237	C₄×D₄	—	178	2	no	—
238	Q₃₂	—	356	2	no	—
239	Q₃₂	—	178	2	no	—
240	SD₃₂	—	356	2	no	—
241	SD₃₂	—	356	2	no	—
242	SD₃₂	—	178	4	no	—
243	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
245	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
246	#107	—	89	1	no	—
247	#107	—	89	1	no	—
248	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
249	C₄⋊C₈	D₃₅₆	1	2	yes	—
250	#103	—	178	2	no	—
251	#103	—	178	1	no	—
252	#103	—	178	1	no	—
253	#102	—	178	2	no	—
254	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₈₉	1	2	yes	—
256	C₂×Q₈	—	178	1	no	—
257	C₂×Q₈	—	178	2	no	—
258	C₂×Q₈	—	178	1	no	—
259	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
260	C₂×D₄	—	356	1	no	—
261	C₂×D₄	—	356	1	no	—
262	C₂×D₄	—	178	2	no	—
263	C₂²×C₄	—	356	2	no	—
264	Q₁₆	—	178	2	no	—
265	Q₁₆	—	178	1	no	—
266	Q₁₆	—	178	1	no	—
267	Q₁₆	—	178	1	no	—
268	Q₁₆	—	178	2	no	—
269	Q₁₆	—	178	1	no	—
270	Q₁₆	—	712	1	no	—
271	Q₁₆	—	712	2	no	—
272	SD₁₆	—	712	2	no	—
273	D₈	—	178	2	no	—
274	D₈	—	178	2	no	—
275	D₈	—	712	1	no	—
276	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
277	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	—	356	2	no	—
280	C₄⋊C₄	—	356	1	no	—
281	C₄⋊C₄	—	356	1	no	—
282	C₄⋊C₄	—	356	2	no	—
283	C₄⋊C₄	—	356	1	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	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
289	C₄⋊C₄	D₇₁₂	1	2	yes	—
290	C₂²⋊C₄	—	356	2	no	—
291	C₂²⋊C₄	—	356	2	no	—
292	C₂²⋊C₄	—	356	2	no	—
293	C₄²	—	356	1	no	—
294	C₄²	—	356	2	no	—
295	C₄²	—	356	1	no	—
296	C₄²	C₂×D₃₅₆	1	2	yes	—
297	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	#13083575	1	2	yes	—
301	C₂³	—	356	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	356	1	no	—
304	Q₈	—	356	2	no	—
305	Q₈	—	356	1	no	—
306	Q₈	—	356	2	no	—
307	Q₈	—	356	1	no	—
308	Q₈	—	356	1	no	—
309	D₄	—	356	2	no	—
310	D₄	—	356	2	no	—
311	D₄	—	1424	1	no	—
312	D₄	—	712	2	no	—
313	C₂×C₄	—	356	1	no	—
314	C₂×C₄	—	356	1	no	—
315	C₂×C₄	—	356	1	no	—
316	C₂×C₄	—	356	2	no	—
317	C₂×C₄	—	356	1	no	—
318	C₂×C₄	—	1424	2	no	—
319	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
320	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
321	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
322	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	C₂×#13083575	1	1	yes	characteristic
327	C₈	#15668058	1	1	yes	characteristic
328	C₂²	—	712	2	no	—
329	C₂²	—	712	2	no	—
330	C₂²	—	712	2	no	—
331	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	712	1	no	—
336	C₄	—	712	2	no	—
337	C₄	—	712	1	no	—
338	C₄	—	712	1	no	—
339	C₄	—	712	1	no	—
340	C₄	#19152041	1	2	yes	—
341	C₄	#19152270	1	1	yes	characteristic
342	C₄	#19152403	1	1	yes	characteristic
343	C₂	—	1424	2	no	—
344	C₂	#26033227	1	1	yes	characteristic
345	C₂	C₂×#19152041	1	1	yes	characteristic
346	C₂	#26033400	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	#19151975	C₂²
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33967
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
2. #26033258	C₂	semidirect
3. #26022523	C₂	semidirect
5. #26022315	C₂	semidirect
6. #26022303	C₂	semidirect
7. #26022298	C₂	semidirect
9. #26022118	C₂	semidirect
10. C₄×Dic₇₁₂	C₂	semidirect
11. C₈₉×#2183	C₂	semidirect
182. #2183	D₈₉	semidirect
185. C₈₉	#33967	semidirect

Maximal subgroups

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

subgroup	index
#26033265	2
#26033258	2
#26022523	2
#26022508	2
#26022315	2
#26022303	2
#26022298	2
#26022293	2
#26022118	2
C₄×Dic₇₁₂	2
C₈₉×#2183	2
#33967	89