Subgroups — #36712160

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 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, complemented
2	D₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022490	C₂	1	2	yes	maximal, complemented
4	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022402	C₂	1	2	yes	maximal, complemented
6	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022309	C₂	1	2	yes	maximal
8	#26022157	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022117	C₂	1	2	yes	maximal, complemented
10	C₄×#15667953	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2208	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152322	—	2	1	no	—
13	D₄×Dic₁₇₈	—	2	1	no	—
14	C₂×#15667953	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	#19151878	—	2	1	no	—
19	#19151876	—	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	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
25	D₈×Dic₈₉	C₂²	1	2	yes	—
26	#19151424	C₂²	1	2	yes	—
27	#19151423	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	2	yes	—
29	#19151402	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	#19151400	C₂²	1	2	yes	—
32	#19151399	C₂²	1	1	yes	characteristic
33	#19151398	C₂²	1	1	yes	characteristic
34	#19151373	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	—
37	#494×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂²×Dic₁₇₈	—	4	1	no	—
42	C₂×#13083561	—	2	2	no	—
43	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
44	#15668012	—	2	1	no	—
45	#15668011	—	2	2	no	—
46	#15668007	—	4	2	no	—
47	#15668001	—	2	2	no	—
48	#15667992	—	4	1	no	—
49	#15667991	—	4	1	no	—
50	#15667989	—	4	2	no	—
51	Q₈×Dic₈₉	—	2	2	no	—
52	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
53	C₄×D₃₅₆	—	2	1	no	—
54	D₄×Dic₈₉	—	2	2	no	—
55	#15667970	—	4	1	no	—
56	#15667953	—	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	1	yes	characteristic
61	#15667946	C₂³	1	2	yes	—
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	#15667934	—	2	1	no	—
67	#15667933	—	2	2	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×D₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	SD₃₂×C₈₉	—	2	4	no	—
75	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#102×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	—	8	4	no	—
82	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂²×Dic₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	—	4	1	no	—
86	#13083561	—	8	2	no	—
87	D₇₁₂	—	2	1	no	—
88	D₇₁₂	—	2	1	no	—
89	C₂×#10623494	C₂×D₄	1	2	yes	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	4	1	no	—
93	C₄⋊Dic₈₉	—	4	2	no	—
94	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
95	#13083553	—	4	2	no	—
96	#13083553	D₈	1	2	yes	—
97	C₂²⋊Dic₈₉	—	4	2	no	—
98	#13083551	—	4	1	no	—
99	#13083551	—	4	2	no	—
100	#13083551	—	4	2	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
103	#13083549	—	2	1	no	—
104	#13083549	—	2	1	no	—
105	C₂×Q₈×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	2	1	no	—
107	C₂²×C₄×C₈₉	—	4	1	no	—
108	Q₁₆×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	2	2	no	—
110	Q₁₆×C₈₉	—	8	1	no	—
111	D₈×C₈₉	—	2	1	no	—
112	D₈×C₈₉	#41	1	2	yes	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₄⋊C₄×C₈₉	—	4	1	no	—
118	C₂²⋊C₄×C₈₉	—	4	1	no	—
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₁₆×C₈₉	—	2	2	no	—
121	C₂³×C₈₉	—	4	1	no	—
122	C₂²×D₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	4	2	no	—
124	Q₈×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	2	2	no	—
126	D₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	8	2	no	—
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
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	C₂×Dic₈₉	—	8	4	no	—
136	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
137	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
138	D₃₅₆	—	4	1	no	—
139	D₃₅₆	—	4	1	no	—
140	C₄×D₈₉	—	8	2	no	—
141	Dic₁₇₈	—	2	2	no	—
142	Dic₁₇₈	—	8	4	no	—
143	Dic₁₇₈	—	8	2	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	#10623494	—	2	2	no	—
148	C₂²×C₈₉	—	4	2	no	—
149	C₂²×C₈₉	—	8	1	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₈₉	—	8	1	no	—
153	C₂×D₈₉	—	8	1	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	8	2	no	—
157	C₄×C₈₉	—	8	1	no	—
158	C₄×C₈₉	#509	1	2	yes	—
159	C₄×C₈₉	#548	1	1	yes	characteristic
160	C₄×C₈₉	#577	1	1	yes	characteristic
161	Dic₈₉	—	2	2	no	—
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	8	2	no	—
164	Dic₈₉	—	8	1	no	—
165	#33934	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	8	2	no	—
167	C₂×C₈₉	#3358	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
169	C₂×C₈₉	#3417	1	1	yes	characteristic
170	D₈₉	—	16	1	no	—
171	#3393	—	89	1	no	—
172	Q₈×D₈	—	89	1	no	—
173	#2243	—	89	2	no	—
174	#2241	—	89	1	no	—
175	#2227	—	89	2	no	—
176	#2224	—	89	1	no	—
177	#2210	—	89	2	no	—
178	#2208	D₈₉	1	1	yes	characteristic, complemented
179	#2183	—	89	1	no	—
180	#2175	—	89	2	no	—
181	C₄×D₁₆	—	89	1	no	—
182	C₈₉	#33934	1	1	yes	characteristic, complemented, Sylow
183	#556	—	178	1	no	—
184	D₄×Q₈	—	178	1	no	—
185	C₂×SD₃₂	C₂×D₈₉	1	2	yes	—
186	C₂×D₁₆	—	89	1	no	—
187	#500	—	89	2	no	—
188	#500	—	89	1	no	—
189	#495	—	89	2	no	—
190	#494	C₂×D₈₉	1	1	yes	characteristic
191	#474	—	178	2	no	—
192	#474	—	178	1	no	—
193	#464	—	178	2	no	—
194	#461	—	178	1	no	—
195	C₈×Q₈	—	89	1	no	—
196	C₄×Q₁₆	—	89	2	no	—
197	C₄×D₈	—	89	2	no	—
198	C₄×D₈	—	89	1	no	—
199	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
200	#366	—	89	2	no	—
201	#366	—	89	1	no	—
202	C₄⋊C₁₆	—	89	2	no	—
203	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
204	#358	—	89	2	no	—
205	#358	—	89	2	no	—
206	#358	C₂×D₈₉	1	2	yes	—
207	#357	—	89	1	no	—
208	#357	—	89	2	no	—
209	#357	—	89	1	no	—
210	#346	—	89	2	no	—
211	C₄×C₁₆	—	89	1	no	—
212	C₂²×Q₈	—	356	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
215	C₂×SD₁₆	—	178	2	no	—
216	C₂×D₈	—	89	1	no	—
217	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	178	2	no	—
220	C₄⋊Q₈	—	178	1	no	—
221	#126	—	356	2	no	—
222	#124	—	178	2	no	—
223	C₂²⋊Q₈	—	356	2	no	—
224	C₂²⋊Q₈	—	356	1	no	—
225	C₂²⋊Q₈	—	356	1	no	—
226	C₄×Q₈	—	89	1	no	—
227	C₄×Q₈	—	178	2	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	178	2	no	—
230	C₄×D₄	—	178	1	no	—
231	#117	—	356	1	no	—
232	SD₃₂	—	2	4	no	—
233	D₁₆	—	178	2	no	—
234	C₂×C₁₆	—	89	1	no	—
235	C₂×C₁₆	—	89	1	no	—
236	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
237	#107	—	89	2	no	—
238	#107	—	89	1	no	—
239	#107	—	178	2	no	—
240	#107	—	178	1	no	—
241	#107	C₂²×D₈₉	1	1	yes	characteristic
242	C₄⋊C₈	—	89	2	no	—
243	C₄⋊C₈	—	89	1	no	—
244	#103	—	178	2	no	—
245	#102	—	2	1	no	—
246	#102	—	178	2	no	—
247	#102	—	178	1	no	—
248	C₄×C₈	—	89	2	no	—
249	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	89	1	no	—
252	C₂×Q₈	—	356	2	no	—
253	C₂×Q₈	—	356	1	no	—
254	C₂×Q₈	—	712	4	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	356	2	no	—
259	C₂²×C₄	—	356	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	8	1	no	—
263	SD₁₆	—	712	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	178	1	no	—
266	D₈	—	178	1	no	—
267	D₈	#13083573	1	2	yes	—
268	C₂×C₈	—	89	2	no	—
269	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	89	2	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	356	2	no	—
278	C₄⋊C₄	—	356	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	356	2	no	—
283	C₂²⋊C₄	—	356	2	no	—
284	C₂²⋊C₄	—	356	2	no	—
285	C₂²⋊C₄	—	356	1	no	—
286	C₄²	—	89	2	no	—
287	C₄²	—	356	2	no	—
288	C₄²	D₄×D₈₉	1	1	yes	characteristic
289	C₁₆	—	2	2	no	—
290	C₁₆	—	178	1	no	—
291	C₁₆	—	178	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	356	1	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	178	2	no	—
297	Q₈	—	712	4	no	—
298	Q₈	—	712	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	356	1	no	—
302	D₄	—	356	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	89	2	no	—
307	C₂×C₄	—	89	2	no	—
308	C₂×C₄	—	356	1	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	—	712	4	no	—
311	C₂×C₄	—	712	2	no	—
312	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
313	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	178	2	no	—
317	C₈	C₂×#13083573	1	1	yes	characteristic
318	C₈	#15668059	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	712	1	no	—
323	C₂²	—	712	1	no	—
324	C₂²	—	712	1	no	—
325	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	178	2	no	—
330	C₄	—	178	2	no	—
331	C₄	—	712	2	no	—
332	C₄	—	712	1	no	—
333	C₄	#19152035	1	2	yes	—
334	C₄	#19152254	1	1	yes	characteristic
335	C₄	#19152408	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	1424	1	no	—
338	C₂	#26032882	1	1	yes	characteristic
339	C₂	C₂×#19152035	1	1	yes	characteristic
340	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×#2208	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15667953	C₂²
hypercenter	#2208	D₈₉
hypocenter	C₈₉	#33934
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2208

Complements

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

N	H ≅ G/N	type
1. #26033242	C₂	semidirect
2. D₈×Dic₁₇₈	C₂	semidirect
3. #26022490	C₂	semidirect
4. #26022479	C₂	semidirect
5. #26022402	C₂	semidirect
6. #26022390	C₂	semidirect
8. #26022157	C₂	semidirect
9. #26022117	C₂	semidirect
11. C₈₉×#2208	C₂	semidirect
178. #2208	D₈₉	semidirect
182. C₈₉	#33934	semidirect

Maximal subgroups

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

subgroup	index
#26033242	2
D₈×Dic₁₇₈	2
#26022490	2
#26022479	2
#26022402	2
#26022390	2
#26022309	2
#26022157	2
#26022117	2
C₄×#15667953	2
C₈₉×#2208	2
#33934	89