Subgroups — #36442522

Order 22784.

406 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022729	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	#26018077	C₂	1	2	yes	maximal
4	#26018074	C₂	1	2	yes	maximal
5	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#10623494	—	2	1	no	—
8	#19151504	C₄	1	1	yes	characteristic
9	#19151504	C₂²	1	1	yes	characteristic
10	#19151496	C₄	1	1	yes	characteristic
11	#19151496	C₂²	1	1	yes	characteristic
12	#19151482	—	2	1	no	—
13	#19151482	—	2	1	no	—
14	C₄×#13083553	—	2	1	no	—
15	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
16	#19151444	—	2	1	no	—
17	#19151353	—	2	1	no	—
18	#19151353	C₂²	1	2	yes	—
19	C₄³×C₈₉	C₂²	1	1	yes	characteristic
20	#336×C₈₉	—	2	1	no	—
21	#336×C₈₉	—	2	1	no	—
22	#336×C₈₉	C₂²	1	2	yes	—
23	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10623494	—	2	2	no	—
25	C₄×Dic₁₇₈	—	2	2	no	—
26	C₄×Dic₁₇₈	C₈	1	4	yes	complemented
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×#13083553	—	2	1	no	—
31	C₂×#13083553	—	2	1	no	—
32	C₂×#13083553	—	2	1	no	—
33	C₂×C₄×Dic₈₉	—	2	1	no	—
34	C₂×C₄×Dic₈₉	—	2	1	no	—
35	C₄⋊#10623494	—	4	2	no	—
36	#15667919	—	2	1	no	—
37	#15667919	—	2	1	no	—
38	#15667919	—	2	1	no	—
39	#15667919	—	2	1	no	—
40	#15667919	—	2	1	no	—
41	#15667919	—	2	1	no	—
42	#15667919	—	2	1	no	—
43	C₂²×C₈×C₈₉	—	2	2	no	—
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₇₈	—	2	1	no	—
52	C₂×Dic₁₇₈	—	4	1	no	—
53	C₂×Dic₁₇₈	—	4	1	no	—
54	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
55	C₂²×Dic₈₉	—	2	1	no	—
56	C₂²×Dic₈₉	—	2	1	no	—
57	C₂²×Dic₈₉	—	2	1	no	—
58	C₂²×Dic₈₉	—	2	1	no	—
59	C₂×#10623494	—	4	2	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₄⋊Dic₈₉	—	2	1	no	—
63	C₄⋊Dic₈₉	—	4	1	no	—
64	C₄⋊Dic₈₉	—	4	1	no	—
65	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
66	#13083553	—	2	4	no	—
67	#13083553	—	4	2	no	—
68	#13083553	—	4	1	no	—
69	#13083553	—	4	1	no	—
70	C₄×Dic₈₉	—	2	4	no	—
71	C₄×Dic₈₉	—	4	2	no	—
72	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	—	4	2	no	—
80	C₂×C₈×C₈₉	—	4	2	no	—
81	C₂×C₈×C₈₉	—	4	2	no	—
82	C₄²×C₈₉	—	2	1	no	—
83	C₄²×C₈₉	—	2	1	no	—
84	C₄²×C₈₉	—	2	2	no	—
85	C₄²×C₈₉	—	4	1	no	—
86	C₄²×C₈₉	—	4	1	no	—
87	C₄²×C₈₉	—	4	1	no	—
88	C₄²×C₈₉	—	4	1	no	—
89	C₄²×C₈₉	C₂×C₈	1	2	yes	—
90	C₄²×C₈₉	M₁₆	1	2	yes	—
91	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#137	1	2	yes	—
114	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
115	C₂×Dic₈₉	—	2	2	no	—
116	C₂×Dic₈₉	—	2	2	no	—
117	C₂×Dic₈₉	—	4	1	no	—
118	C₂×Dic₈₉	—	4	1	no	—
119	C₂×Dic₈₉	—	4	1	no	—
120	C₂×Dic₈₉	—	4	1	no	—
121	C₂×Dic₈₉	—	4	1	no	—
122	C₂×Dic₈₉	—	4	1	no	—
123	C₂×Dic₈₉	—	4	1	no	—
124	C₂×Dic₈₉	—	4	1	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	Dic₁₇₈	—	4	2	no	—
128	Dic₁₇₈	—	4	2	no	—
129	C₈×C₈₉	—	8	2	no	—
130	#10623494	—	8	2	no	—
131	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₈₉	#410	1	1	yes	characteristic
133	C₂²×C₈₉	#413	1	1	yes	characteristic
134	C₂²×C₈₉	#417	1	1	yes	characteristic
135	C₂²×C₈₉	#419	1	1	yes	characteristic
136	C₂²×C₈₉	#421	1	1	yes	characteristic
137	C₂²×C₈₉	#421	1	1	yes	characteristic
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	4	1	no	—
146	C₄×C₈₉	—	4	1	no	—
147	Dic₈₉	—	4	2	no	—
148	Dic₈₉	—	4	2	no	—
149	Dic₈₉	—	8	1	no	—
150	Dic₈₉	—	8	1	no	—
151	#7596	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	#1458	1	1	yes	characteristic
153	C₂×C₈₉	#1477	1	1	yes	characteristic
154	C₂×C₈₉	#1479	1	1	yes	characteristic
155	C₂×C₈₉	#1512	1	1	yes	characteristic
156	C₂×C₈₉	#1515	1	1	yes	characteristic
157	C₂×C₈₉	#1535	1	1	yes	characteristic
158	C₂×C₈₉	#1559	1	1	yes	characteristic
159	#2295	—	89	1	no	—
160	#1839	D₈₉	1	1	yes	characteristic
161	#1831	—	89	1	no	—
162	#1272	—	89	2	no	—
163	#1271	—	89	2	no	—
164	C₈₉	#7596	1	1	yes	characteristic, complemented, Sylow
165	C₂×C₄×Q₈	—	89	1	no	—
166	C₂×C₄⋊C₈	—	178	1	no	—
167	#391	—	89	1	no	—
168	#391	—	89	1	no	—
169	#389	—	89	1	no	—
170	#389	—	89	1	no	—
171	#387	—	178	1	no	—
172	#387	—	178	1	no	—
173	C₄×C₄⋊C₄	—	89	1	no	—
174	C₄×C₄⋊C₄	—	178	1	no	—
175	#376	—	178	1	no	—
176	C₄³	C₂×D₈₉	1	1	yes	characteristic
177	#336	—	2	1	no	—
178	#336	—	2	1	no	—
179	#336	—	89	2	no	—
180	#336	—	178	1	no	—
181	#336	C₂×D₈₉	1	2	yes	—
182	C₂²×Q₈	—	89	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	178	2	no	—
185	C₄×Q₈	—	89	4	no	—
186	C₄×Q₈	—	178	2	no	—
187	C₂×C₄⋊C₄	—	89	1	no	—
188	C₂×C₄⋊C₄	—	89	1	no	—
189	C₂×C₄⋊C₄	—	89	1	no	—
190	C₂×C₄⋊C₄	—	178	1	no	—
191	C₂×C₄⋊C₄	—	178	1	no	—
192	C₂×C₄⋊C₄	—	178	1	no	—
193	C₂×C₄²	—	178	1	no	—
194	C₂×C₄²	—	178	1	no	—
195	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
197	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
198	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
202	C₄⋊C₈	—	356	2	no	—
203	#95	—	178	1	no	—
204	#95	—	178	1	no	—
205	#95	—	178	1	no	—
206	#95	—	178	1	no	—
207	#95	—	178	1	no	—
208	#95	—	178	1	no	—
209	#95	—	178	1	no	—
210	C₂×Q₈	—	89	2	no	—
211	C₂×Q₈	—	356	1	no	—
212	C₂×Q₈	—	356	1	no	—
213	C₂×Q₈	—	178	1	no	—
214	C₂²×C₄	—	178	1	no	—
215	C₂²×C₄	—	178	1	no	—
216	C₂²×C₄	—	178	1	no	—
217	C₂²×C₄	—	178	1	no	—
218	C₂²×C₄	#13083551	1	1	yes	characteristic
219	C₂²×C₄	#13083551	1	1	yes	characteristic
220	C₂²×C₄	#13083551	1	1	yes	characteristic
221	C₂²×C₄	#13083551	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	356	2	no	—
229	C₂×C₈	—	356	2	no	—
230	C₂×C₈	—	356	2	no	—
231	C₄⋊C₄	—	89	2	no	—
232	C₄⋊C₄	—	356	1	no	—
233	C₄⋊C₄	—	356	1	no	—
234	C₄⋊C₄	—	356	2	no	—
235	C₄⋊C₄	—	356	1	no	—
236	C₄⋊C₄	—	356	1	no	—
237	C₄⋊C₄	—	178	1	no	—
238	C₄⋊C₄	—	178	4	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	356	2	no	—
247	C₄²	—	178	4	no	—
248	C₄²	C₈×D₈₉	1	2	yes	—
249	C₄²	#13083557	1	2	yes	—
250	C₂³	C₂×#13083551	1	1	yes	characteristic
251	Q₈	—	356	2	no	—
252	Q₈	—	356	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	356	1	no	—
270	C₂×C₄	—	356	1	no	—
271	C₂×C₄	—	356	1	no	—
272	C₂×C₄	—	356	1	no	—
273	C₂×C₄	—	356	1	no	—
274	C₂×C₄	—	356	1	no	—
275	C₂×C₄	—	356	1	no	—
276	C₂×C₄	—	356	1	no	—
277	C₂×C₄	—	356	1	no	—
278	C₂×C₄	—	356	1	no	—
279	C₂×C₄	—	178	2	no	—
280	C₂×C₄	—	178	2	no	—
281	C₂×C₄	#15667924	1	2	yes	—
282	C₂×C₄	#15667924	1	2	yes	—
283	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
284	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
285	C₂×C₄	#15668040	1	1	yes	characteristic
286	C₂×C₄	#15668042	1	2	yes	—
287	C₂×C₄	#15668045	1	1	yes	characteristic
288	C₈	—	8	2	no	—
289	C₈	—	712	2	no	—
290	C₂²	C₂×#15667924	1	1	yes	characteristic
291	C₂²	#19151571	1	1	yes	characteristic
292	C₂²	#19151580	1	1	yes	characteristic
293	C₂²	#19151600	1	1	yes	characteristic
294	C₂²	#19151606	1	1	yes	characteristic
295	C₂²	#19151614	1	1	yes	characteristic
296	C₂²	#19151616	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	356	2	no	—
307	C₄	—	356	2	no	—
308	C₄	—	712	1	no	—
309	C₄	—	712	1	no	—
310	C₂	#26018564	1	1	yes	characteristic
311	C₂	#26018640	1	1	yes	characteristic
312	C₂	#26018646	1	1	yes	characteristic
313	C₂	#26018791	1	1	yes	characteristic
314	C₂	#26018802	1	1	yes	characteristic
315	C₂	#26018884	1	1	yes	characteristic
316	C₂	#26018983	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1839	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1839	D₈₉
hypocenter	C₈₉	#7596
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1839

Complements

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

N	H ≅ G/N	type
26. C₄×Dic₁₇₈	C₈	semidirect
164. C₈₉	#7596	semidirect

Maximal subgroups

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

subgroup	index
#26022729	2
#26020369	2
#26018077	2
#26018074	2
C₈₉×#1839	2
#7596	89