Subgroups — #36712123

Order 22784.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033241	C₂	1	1	yes	maximal, characteristic
2	#26022486	C₂	1	2	yes	maximal
3	#26022481	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022413	C₂	1	2	yes	maximal, complemented
5	#26022386	C₂	1	1	yes	maximal, characteristic
6	#26022314	C₂	1	2	yes	maximal
7	#26022302	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15667958	C₂	1	2	yes	maximal, complemented
10	C₄×#15667955	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152321	—	2	1	no	—
13	C₂×#15667958	C₂²	1	2	yes	—
14	C₂×#15667955	C₂²	1	1	yes	characteristic
15	#19151982	C₂²	1	2	yes	—
16	#19151978	C₂²	1	1	yes	characteristic
17	#19151821	—	2	2	no	—
18	#19151804	—	2	1	no	—
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
21	#19151425	C₂²	1	1	yes	characteristic
22	#19151423	C₂²	1	2	yes	—
23	C₄⋊#13083549	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151402	C₂²	1	1	yes	characteristic
26	#19151401	C₂²	1	2	yes	—
27	#19151401	C₂²	1	2	yes	—
28	#19151401	C₂²	1	2	yes	—
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	C₄×#13083549	C₂²	1	1	yes	characteristic
32	C₄×#13083549	C₂²	1	2	yes	—
33	#557×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#499×C₈₉	C₂²	1	2	yes	—
36	#479×C₈₉	—	2	2	no	—
37	#475×C₈₉	—	2	1	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	—	2	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	2	yes	—
43	C₂×#13083563	—	2	2	no	—
44	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
45	#15668012	—	2	1	no	—
46	#15668007	—	4	2	no	—
47	#15668000	—	2	2	no	—
48	#15667988	—	4	1	no	—
49	C₄×Dic₁₇₈	—	2	2	no	—
50	C₄×D₃₅₆	—	2	1	no	—
51	#15667970	—	4	1	no	—
52	#15667958	—	2	4	no	—
53	#15667955	—	2	2	no	—
54	C₂×#13083549	C₂³	1	1	yes	characteristic
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667937	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₈₉	—	2	1	no	—
62	C₄⋊Q₈×C₈₉	—	4	1	no	—
63	#125×C₈₉	—	2	2	no	—
64	#125×C₈₉	—	4	2	no	—
65	C₄×Q₈×C₈₉	—	2	1	no	—
66	C₄×Q₈×C₈₉	—	2	2	no	—
67	C₄×Q₈×C₈₉	—	4	1	no	—
68	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
69	#107×C₈₉	—	2	1	no	—
70	#107×C₈₉	C₂³	1	2	yes	—
71	#107×C₈₉	C₂³	1	1	yes	characteristic
72	#106×C₈₉	—	2	2	no	—
73	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
75	#103×C₈₉	—	2	2	no	—
76	#103×C₈₉	—	2	1	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂×Dic₁₇₈	—	2	2	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×C₄×D₈₉	—	4	1	no	—
82	Dic₃₅₆	—	2	2	no	—
83	Dic₃₅₆	—	2	2	no	—
84	Dic₃₅₆	—	8	1	no	—
85	#13083563	—	8	2	no	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	#13083553	—	4	2	no	—
92	#13083551	—	4	2	no	—
93	#13083551	—	4	1	no	—
94	#13083551	—	4	2	no	—
95	C₄×Dic₈₉	—	4	2	no	—
96	#13083549	—	2	1	no	—
97	#13083549	—	2	2	no	—
98	#13083549	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
102	Q₁₆×C₈₉	—	2	1	no	—
103	Q₁₆×C₈₉	#41	1	2	yes	—
104	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₄⋊C₄×C₈₉	—	4	1	no	—
110	C₄⋊C₄×C₈₉	—	4	2	no	—
111	C₄⋊C₄×C₈₉	—	4	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
116	C₄²×C₈₉	—	4	2	no	—
117	C₄²×C₈₉	—	4	1	no	—
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₈₉	C₂×D₄	1	2	yes	—
120	C₂²×D₈₉	—	4	1	no	—
121	Q₈×C₈₉	—	2	2	no	—
122	Q₈×C₈₉	—	2	2	no	—
123	Q₈×C₈₉	—	4	1	no	—
124	Q₈×C₈₉	—	8	2	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	—	4	1	no	—
134	C₂×Dic₈₉	—	4	2	no	—
135	D₃₅₆	—	4	1	no	—
136	D₃₅₆	—	4	1	no	—
137	C₄×D₈₉	—	8	2	no	—
138	Dic₁₇₈	—	4	2	no	—
139	Dic₁₇₈	—	4	2	no	—
140	C₈×C₈₉	—	2	2	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₈₉	—	8	1	no	—
146	C₂×D₈₉	—	8	1	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	2	2	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	4	2	no	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	—	8	1	no	—
154	C₄×C₈₉	#508	1	2	yes	—
155	C₄×C₈₉	#548	1	1	yes	characteristic
156	C₄×C₈₉	#577	1	1	yes	characteristic
157	Dic₈₉	—	8	2	no	—
158	Dic₈₉	—	8	1	no	—
159	#33931	—	89	1	no	maximal, Sylow
160	C₂×C₈₉	#3358	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
162	C₂×C₈₉	#3417	1	1	yes	characteristic
163	D₈₉	—	16	1	no	—
164	#3393	—	89	1	no	—
165	#3383	D₈₉	1	1	yes	characteristic, complemented
166	#2242	—	89	2	no	—
167	#2241	—	89	1	no	—
168	#2229	—	89	2	no	—
169	#2224	—	89	1	no	—
170	#2211	—	89	2	no	—
171	#2209	—	89	1	no	—
172	Q₈×C₁₆	—	89	1	no	—
173	C₄×Q₃₂	—	89	2	no	—
174	C₄×SD₃₂	—	89	1	no	—
175	C₈₉	#33931	1	1	yes	characteristic, complemented, Sylow
176	#557	—	2	1	no	—
177	#556	—	178	1	no	—
178	C₂×Q₃₂	—	89	2	no	—
179	C₂×SD₃₂	—	89	1	no	—
180	#500	C₂×D₈₉	1	1	yes	characteristic
181	#499	C₂×D₈₉	1	2	yes	—
182	#495	—	89	2	no	—
183	#494	—	89	1	no	—
184	#479	—	2	2	no	—
185	#475	—	2	1	no	—
186	#464	—	178	2	no	—
187	#461	—	178	1	no	—
188	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
189	C₄×Q₁₆	—	89	2	no	—
190	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
191	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
192	C₄×D₈	—	89	1	no	—
193	#367	—	89	1	no	—
194	#366	—	89	2	no	—
195	C₄⋊C₁₆	—	89	1	no	—
196	C₄⋊C₁₆	—	89	2	no	—
197	#358	—	89	1	no	—
198	#358	—	89	2	no	—
199	#358	—	89	2	no	—
200	#358	—	89	2	no	—
201	#357	—	89	1	no	—
202	#357	—	89	2	no	—
203	C₄×C₁₆	—	89	1	no	—
204	C₄×C₁₆	—	89	2	no	—
205	C₂×Q₁₆	—	89	2	no	—
206	C₂×Q₁₆	—	178	1	no	—
207	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
208	C₂×SD₁₆	—	178	2	no	—
209	C₂×D₈	—	89	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	4	1	no	—
212	C₄⋊Q₈	—	178	1	no	—
213	#126	—	356	2	no	—
214	#125	—	2	2	no	—
215	#125	—	4	2	no	—
216	#124	—	178	2	no	—
217	C₂²⋊Q₈	—	356	1	no	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×Q₈	—	178	2	no	—
222	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
223	C₄×D₄	—	178	1	no	—
224	#117	—	356	1	no	—
225	Q₃₂	—	178	4	no	—
226	SD₃₂	—	178	2	no	—
227	C₂×C₁₆	—	89	1	no	—
228	C₂×C₁₆	—	89	2	no	—
229	C₂×C₁₆	—	89	1	no	—
230	#107	—	2	1	no	—
231	#107	—	89	1	no	—
232	#107	C₂²×D₈₉	1	2	yes	—
233	#107	C₂²×D₈₉	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
236	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
237	#103	—	2	2	no	—
238	#103	—	2	1	no	—
239	#103	—	178	2	no	—
240	#102	—	178	1	no	—
241	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₈₉	1	2	yes	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	178	2	no	—
246	C₂×Q₈	#13083559	1	1	yes	characteristic
247	C₂×D₄	—	178	1	no	—
248	C₂²×C₄	—	356	1	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	178	2	no	—
251	Q₁₆	—	178	2	no	—
252	Q₁₆	—	712	1	no	—
253	Q₁₆	#13083575	1	2	yes	—
254	SD₁₆	—	712	2	no	—
255	D₈	—	178	1	no	—
256	D₈	—	178	1	no	—
257	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
259	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	356	1	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	356	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	#13083559	1	1	yes	characteristic
272	C₄⋊C₄	#13083559	1	2	yes	—
273	C₂²⋊C₄	—	356	2	no	—
274	C₂²⋊C₄	—	356	1	no	—
275	C₂²⋊C₄	—	356	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	356	2	no	—
279	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
281	C₁₆	—	178	1	no	—
282	C₁₆	—	178	2	no	—
283	C₁₆	—	178	1	no	—
284	C₂³	—	356	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	8	2	no	—
289	Q₈	—	356	2	no	—
290	Q₈	—	356	2	no	—
291	D₄	—	356	1	no	—
292	D₄	—	356	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	356	1	no	—
297	C₂×C₄	—	356	2	no	—
298	C₂×C₄	—	712	2	no	—
299	C₂×C₄	C₂×#13083559	1	2	yes	—
300	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
301	C₂×C₄	C₂×#13083559	1	2	yes	—
302	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	C₂×#13083575	1	1	yes	characteristic
307	C₈	#15668060	1	1	yes	characteristic
308	C₂²	—	712	1	no	—
309	C₂²	—	712	1	no	—
310	C₂²	—	712	1	no	—
311	C₂²	C₂²×#13083559	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	712	1	no	—
319	C₄	—	712	2	no	—
320	C₄	#19152028	1	2	yes	—
321	C₄	#19152259	1	1	yes	characteristic
322	C₄	#19152406	1	1	yes	characteristic
323	C₂	—	1424	1	no	—
324	C₂	#26032887	1	1	yes	characteristic
325	C₂	C₂×#19152028	1	1	yes	characteristic
326	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3383	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#3383	D₈₉
hypocenter	C₈₉	#33931
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
3. #26022481	C₂	semidirect
4. #26022413	C₂	semidirect
7. #26022302	C₂	semidirect
8. Q₈×#13083549	C₂	semidirect
9. C₄×#15667958	C₂	semidirect
11. C₈₉×#3383	C₂	semidirect
165. #3383	D₈₉	semidirect
175. C₈₉	#33931	semidirect

Maximal subgroups

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

subgroup	index
#26033241	2
#26022486	2
#26022481	2
#26022413	2
#26022386	2
#26022314	2
#26022302	2
Q₈×#13083549	2
C₄×#15667958	2
C₄×#15667955	2
C₈₉×#3383	2
#33931	89