Subgroups — #36712182

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033242	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022490	C₂	1	2	yes	maximal
4	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022400	C₂	1	1	yes	maximal, characteristic
6	#26022392	C₂	1	2	yes	maximal, complemented
7	#26022316	C₂	1	2	yes	maximal
8	#26022157	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022123	C₂	1	2	yes	maximal, complemented
10	C₄×#15667955	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₇₈	—	2	1	no	—
13	#19152322	—	2	1	no	—
14	C₂×#15667955	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	C₂²	1	2	yes	—
17	#19151983	C₂²	1	2	yes	—
18	#19151893	—	2	1	no	—
19	#19151891	—	2	2	no	—
20	#19151822	—	2	2	no	—
21	#19151800	—	2	1	no	—
22	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
24	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
25	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
26	#19151426	C₂²	1	2	yes	—
27	#19151425	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	2	yes	—
29	#19151402	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	#19151402	C₂²	1	1	yes	characteristic
32	#19151400	C₂²	1	2	yes	—
33	#19151398	C₂²	1	1	yes	characteristic
34	#19151373	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
37	#494×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083561	—	2	2	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	1	no	—
44	#15668012	—	4	1	no	—
45	#15668011	—	2	2	no	—
46	#15668011	—	4	2	no	—
47	#15668007	—	4	2	no	—
48	#15668001	—	2	2	no	—
49	#15667992	—	4	1	no	—
50	Q₈×Dic₈₉	—	2	2	no	—
51	Q₈×Dic₈₉	—	2	2	no	—
52	C₄×Dic₁₇₈	—	4	1	no	—
53	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
54	C₄×D₃₅₆	—	2	1	no	—
55	#15667970	—	4	1	no	—
56	#15667955	—	2	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	#15667946	—	2	1	no	—
60	#15667946	C₂³	1	2	yes	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667945	—	2	2	no	—
63	C₄⋊#10623494	D₄	1	1	yes	characteristic
64	#15667941	D₄	1	2	yes	—
65	#15667936	—	2	2	no	—
66	#15667936	—	2	2	no	—
67	#15667934	—	2	1	no	—
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₈₉	—	2	1	no	—
70	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₈₉	—	2	1	no	—
73	C₄×Q₈×C₈₉	—	2	1	no	—
74	Q₃₂×C₈₉	—	2	4	no	—
75	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#103×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×Dic₁₇₈	—	4	2	no	—
81	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×C₄×D₈₉	—	4	1	no	—
84	#13083561	—	8	2	no	—
85	D₇₁₂	—	2	1	no	—
86	D₇₁₂	—	2	1	no	—
87	C₂×#10623494	C₂×D₄	1	2	yes	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
94	#13083553	—	4	2	no	—
95	#13083553	D₈	1	2	yes	—
96	#13083551	—	4	1	no	—
97	#13083551	—	4	2	no	—
98	#13083551	—	4	2	no	—
99	C₄×Dic₈₉	—	4	2	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
102	#13083549	—	2	1	no	—
103	#13083549	—	2	1	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	C₂×Q₈×C₈₉	—	2	2	no	—
106	Q₁₆×C₈₉	—	2	2	no	—
107	Q₁₆×C₈₉	—	2	1	no	—
108	Q₁₆×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	8	1	no	—
110	Q₁₆×C₈₉	#41	1	2	yes	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	1	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	—	4	1	no	—
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₂²×D₈₉	—	4	1	no	—
121	Q₈×C₈₉	—	2	2	no	—
122	Q₈×C₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	4	2	no	—
124	Q₈×C₈₉	—	4	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	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	—	4	2	no	—
133	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
134	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
135	D₃₅₆	—	4	1	no	—
136	D₃₅₆	—	4	1	no	—
137	C₄×D₈₉	—	8	2	no	—
138	Dic₁₇₈	—	2	2	no	—
139	Dic₁₇₈	—	16	2	no	—
140	Dic₁₇₈	—	16	1	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	#10623494	—	2	2	no	—
145	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₈₉	—	8	1	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₂×D₈₉	—	8	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	4	2	no	—
151	C₄×C₈₉	—	8	2	no	—
152	C₄×C₈₉	—	8	1	no	—
153	C₄×C₈₉	#510	1	2	yes	—
154	C₄×C₈₉	#548	1	1	yes	characteristic
155	C₄×C₈₉	#577	1	1	yes	characteristic
156	Dic₈₉	—	2	2	no	—
157	Dic₈₉	—	2	2	no	—
158	Dic₈₉	—	8	1	no	—
159	Dic₈₉	—	8	2	no	—
160	#33936	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	#3358	1	1	yes	characteristic
162	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
163	C₂×C₈₉	#3416	1	1	yes	characteristic
164	D₈₉	—	16	1	no	—
165	#3393	—	89	1	no	—
166	Q₈×Q₁₆	—	89	1	no	—
167	#2243	—	89	2	no	—
168	#2241	—	89	1	no	—
169	#2226	—	89	1	no	—
170	#2225	—	89	2	no	—
171	#2211	—	89	2	no	—
172	#2209	D₈₉	1	1	yes	characteristic, complemented
173	#2183	—	89	1	no	—
174	#2176	—	89	2	no	—
175	C₄×SD₃₂	—	89	1	no	—
176	C₈₉	#33936	1	1	yes	characteristic, complemented, Sylow
177	Q₈²	—	178	1	no	—
178	#556	—	178	1	no	—
179	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
180	C₂×SD₃₂	—	89	1	no	—
181	#500	—	89	2	no	—
182	#500	—	89	1	no	—
183	#495	—	89	2	no	—
184	#494	C₂×D₈₉	1	1	yes	characteristic
185	#477	—	178	2	no	—
186	#477	—	178	1	no	—
187	#464	—	178	2	no	—
188	#461	—	178	1	no	—
189	C₈×Q₈	—	89	1	no	—
190	C₄×Q₁₆	—	89	2	no	—
191	C₄×Q₁₆	—	89	2	no	—
192	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
193	C₄×D₈	—	89	1	no	—
194	#367	—	89	2	no	—
195	#367	—	89	1	no	—
196	C₄⋊C₁₆	—	89	2	no	—
197	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
198	#358	—	89	1	no	—
199	#358	—	89	2	no	—
200	#358	—	89	2	no	—
201	#358	C₂×D₈₉	1	2	yes	—
202	#357	—	89	2	no	—
203	#357	—	89	1	no	—
204	#346	—	89	2	no	—
205	C₄×C₁₆	—	89	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
208	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
209	C₂×SD₁₆	—	178	2	no	—
210	C₂×D₈	—	89	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	356	1	no	—
213	C₄⋊Q₈	—	356	2	no	—
214	C₄⋊Q₈	—	178	2	no	—
215	C₄⋊Q₈	—	178	1	no	—
216	#126	—	356	2	no	—
217	#124	—	178	2	no	—
218	C₂²⋊Q₈	—	356	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	89	1	no	—
221	C₄×Q₈	—	356	1	no	—
222	C₄×Q₈	—	178	2	no	—
223	C₄×Q₈	—	178	2	no	—
224	C₄×D₄	—	178	1	no	—
225	#117	—	356	1	no	—
226	Q₃₂	—	2	4	no	—
227	SD₃₂	—	178	2	no	—
228	C₂×C₁₆	—	89	1	no	—
229	C₂×C₁₆	—	89	1	no	—
230	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
231	#107	—	89	2	no	—
232	#107	—	89	1	no	—
233	#107	—	178	2	no	—
234	#107	—	178	1	no	—
235	#107	C₂²×D₈₉	1	1	yes	characteristic
236	C₄⋊C₈	—	89	2	no	—
237	C₄⋊C₈	—	89	1	no	—
238	#103	—	2	1	no	—
239	#103	—	178	2	no	—
240	#103	—	178	2	no	—
241	#102	—	178	1	no	—
242	C₄×C₈	—	89	2	no	—
243	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	89	1	no	—
247	C₂×Q₈	—	356	2	no	—
248	C₂×Q₈	—	356	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂²×C₄	—	356	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	2	1	no	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	8	1	no	—
255	Q₁₆	#13083573	1	2	yes	—
256	SD₁₆	—	712	2	no	—
257	D₈	—	178	1	no	—
258	D₈	—	178	1	no	—
259	C₂×C₈	—	89	2	no	—
260	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	89	2	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	356	2	no	—
270	C₄⋊C₄	—	356	2	no	—
271	C₄⋊C₄	—	356	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	2	no	—
274	C₂²⋊C₄	—	356	2	no	—
275	C₂²⋊C₄	—	356	2	no	—
276	C₂²⋊C₄	—	356	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	89	2	no	—
279	C₄²	—	356	2	no	—
280	C₄²	—	356	2	no	—
281	C₄²	D₄×D₈₉	1	1	yes	characteristic
282	C₁₆	—	2	2	no	—
283	C₁₆	—	178	1	no	—
284	C₁₆	—	178	1	no	—
285	C₂³	—	356	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	1424	1	no	—
291	Q₈	—	1424	2	no	—
292	Q₈	—	178	2	no	—
293	D₄	—	356	1	no	—
294	D₄	—	356	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	89	2	no	—
299	C₂×C₄	—	89	2	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	712	2	no	—
303	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
304	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	178	2	no	—
308	C₈	C₂×#13083573	1	1	yes	characteristic
309	C₈	#15668059	1	1	yes	characteristic
310	C₂²	—	712	1	no	—
311	C₂²	—	712	1	no	—
312	C₂²	—	712	1	no	—
313	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	178	2	no	—
319	C₄	—	178	2	no	—
320	C₄	—	712	1	no	—
321	C₄	—	712	2	no	—
322	C₄	#19152040	1	2	yes	—
323	C₄	#19152254	1	1	yes	characteristic
324	C₄	#19152408	1	1	yes	characteristic
325	C₂	—	1424	1	no	—
326	C₂	#26032882	1	1	yes	characteristic
327	C₂	C₂×#19152040	1	1	yes	characteristic
328	C₂	#26033397	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×#2209	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#2209	D₈₉
hypocenter	C₈₉	#33936
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₇₈	C₂	semidirect
4. #26022479	C₂	semidirect
6. #26022392	C₂	semidirect
8. #26022157	C₂	semidirect
9. #26022123	C₂	semidirect
11. C₈₉×#2209	C₂	semidirect
172. #2209	D₈₉	semidirect
176. C₈₉	#33936	semidirect

Maximal subgroups

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

subgroup	index
#26033242	2
Q₁₆×Dic₁₇₈	2
#26022490	2
#26022479	2
#26022400	2
#26022392	2
#26022316	2
#26022157	2
#26022123	2
C₄×#15667955	2
C₈₉×#2209	2
#33936	89