Subgroups — #36442845

Order 22784.

384 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028634	C₂	1	1	yes	maximal, characteristic
2	#26020395	C₂	1	1	yes	maximal, characteristic
3	#26018077	C₂	1	2	yes	maximal
4	#26018074	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151504	C₄	1	1	yes	characteristic
7	#19151504	C₄	1	1	yes	characteristic
8	#19151504	C₂²	1	1	yes	characteristic
9	#19151496	C₄	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	#19151482	—	2	1	no	—
15	#19151466	—	2	1	no	—
16	#19151353	—	2	1	no	—
17	#19151353	—	2	1	no	—
18	#19151353	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
20	C₂×C₄×C₈×C₈₉	—	2	1	no	—
21	C₄³×C₈₉	C₂²	1	1	yes	characteristic
22	#336×C₈₉	C₂²	1	2	yes	—
23	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10623494	—	2	2	no	—
25	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×#13083553	—	2	1	no	—
29	C₂×#13083553	—	2	1	no	—
30	C₂×#13083553	—	2	1	no	—
31	#15667919	—	2	1	no	—
32	#15667919	—	2	1	no	—
33	#15667919	—	2	1	no	—
34	#15667919	—	2	1	no	—
35	#15667919	—	2	1	no	—
36	#15667919	—	2	1	no	—
37	#15667919	—	2	1	no	—
38	#15667919	—	2	1	no	—
39	#15667919	—	2	1	no	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	—	2	4	no	—
49	C₄×C₈×C₈₉	—	4	2	no	—
50	C₂×Dic₁₇₈	—	2	2	no	—
51	C₂×Dic₁₇₈	—	4	1	no	—
52	C₂×Dic₁₇₈	—	4	1	no	—
53	C₂²×Dic₈₉	—	2	1	no	—
54	C₂²×Dic₈₉	—	2	1	no	—
55	C₂²×Dic₈₉	—	2	1	no	—
56	C₂²×Dic₈₉	—	2	1	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₂×#10623494	—	4	2	no	—
60	C₄⋊Dic₈₉	—	2	2	no	—
61	C₄⋊Dic₈₉	—	4	1	no	—
62	C₄⋊Dic₈₉	—	4	1	no	—
63	#13083553	—	4	2	no	—
64	#13083553	—	4	2	no	—
65	#13083553	—	4	2	no	—
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	2	4	no	—
74	C₂×C₈×C₈₉	—	4	2	no	—
75	C₂×C₈×C₈₉	—	4	2	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	4	1	no	—
80	C₄²×C₈₉	—	4	1	no	—
81	C₄²×C₈₉	—	4	1	no	—
82	C₄²×C₈₉	SD₁₆	1	2	yes	—
83	C₄²×C₈₉	Q₁₆	1	2	yes	—
84	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
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	#103	1	2	yes	—
101	C₂×C₄×C₈₉	#103	1	2	yes	—
102	C₂×C₄×C₈₉	#104	1	2	yes	—
103	C₂×C₄×C₈₉	#131	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#135	1	2	yes	—
107	C₂×Dic₈₉	—	4	1	no	—
108	C₂×Dic₈₉	—	4	1	no	—
109	C₂×Dic₈₉	—	4	1	no	—
110	C₂×Dic₈₉	—	4	1	no	—
111	C₂×Dic₈₉	—	4	1	no	—
112	C₂×Dic₈₉	—	4	1	no	—
113	C₂×Dic₈₉	—	4	1	no	—
114	C₂×Dic₈₉	—	4	1	no	—
115	C₂×Dic₈₉	—	4	1	no	—
116	C₂×Dic₈₉	—	4	1	no	—
117	C₂×Dic₈₉	—	4	1	no	—
118	C₂×Dic₈₉	—	4	1	no	—
119	Dic₁₇₈	—	8	2	no	—
120	C₈×C₈₉	—	4	4	no	—
121	#10623494	—	8	2	no	—
122	C₂²×C₈₉	#408	1	1	yes	characteristic
123	C₂²×C₈₉	#410	1	1	yes	characteristic
124	C₂²×C₈₉	#413	1	1	yes	characteristic
125	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
126	C₂²×C₈₉	#416	1	1	yes	characteristic
127	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
128	C₂²×C₈₉	#421	1	1	yes	characteristic
129	C₄×C₈₉	—	2	2	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	2	no	—
136	C₄×C₈₉	—	4	1	no	—
137	Dic₈₉	—	8	1	no	—
138	Dic₈₉	—	8	1	no	—
139	Dic₈₉	—	8	1	no	—
140	Dic₈₉	—	8	1	no	—
141	#7673	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	#1473	1	1	yes	characteristic
143	C₂×C₈₉	#1481	1	1	yes	characteristic
144	C₂×C₈₉	#1494	1	1	yes	characteristic
145	C₂×C₈₉	#1503	1	1	yes	characteristic
146	C₂×C₈₉	#1521	1	1	yes	characteristic
147	C₂×C₈₉	#1529	1	1	yes	characteristic
148	C₂×C₈₉	#1542	1	1	yes	characteristic
149	#2863	—	89	1	no	—
150	#1839	—	89	1	no	—
151	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
152	#1272	—	89	2	no	—
153	#1271	—	89	2	no	—
154	C₈₉	#7673	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄×C₈	—	2	1	no	—
157	#391	—	89	1	no	—
158	#391	—	89	1	no	—
159	#391	—	89	1	no	—
160	#389	—	89	1	no	—
161	#389	—	89	1	no	—
162	#389	—	89	1	no	—
163	#387	—	178	1	no	—
164	#387	—	178	1	no	—
165	#387	—	178	1	no	—
166	#383	—	178	1	no	—
167	C₄³	C₂×D₈₉	1	1	yes	characteristic
168	#336	—	89	2	no	—
169	#336	—	178	1	no	—
170	#336	—	178	1	no	—
171	#336	C₂×D₈₉	1	2	yes	—
172	C₂²×Q₈	—	89	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	178	2	no	—
175	C₂×C₄⋊C₄	—	89	1	no	—
176	C₂×C₄⋊C₄	—	89	1	no	—
177	C₂×C₄⋊C₄	—	89	1	no	—
178	C₂×C₄⋊C₄	—	178	1	no	—
179	C₂×C₄⋊C₄	—	178	1	no	—
180	C₂×C₄⋊C₄	—	178	1	no	—
181	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
183	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
184	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄×C₈	—	4	2	no	—
190	#95	—	178	1	no	—
191	#95	—	178	1	no	—
192	#95	—	178	1	no	—
193	#95	—	178	1	no	—
194	#95	—	178	1	no	—
195	#95	—	178	1	no	—
196	#95	—	178	1	no	—
197	#95	—	178	1	no	—
198	#95	—	178	1	no	—
199	C₂×Q₈	—	356	1	no	—
200	C₂×Q₈	—	356	1	no	—
201	C₂×Q₈	—	178	2	no	—
202	C₂²×C₄	—	178	1	no	—
203	C₂²×C₄	—	178	1	no	—
204	C₂²×C₄	—	178	1	no	—
205	C₂²×C₄	—	178	1	no	—
206	C₂²×C₄	#13083551	1	1	yes	characteristic
207	C₂²×C₄	#13083551	1	1	yes	characteristic
208	C₂²×C₄	#13083551	1	1	yes	characteristic
209	C₂²×C₄	#13083551	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	356	2	no	—
217	C₂×C₈	—	356	2	no	—
218	C₂×C₈	—	356	2	no	—
219	C₄⋊C₄	—	356	1	no	—
220	C₄⋊C₄	—	356	1	no	—
221	C₄⋊C₄	—	356	2	no	—
222	C₄⋊C₄	—	356	2	no	—
223	C₄⋊C₄	—	356	2	no	—
224	C₄⋊C₄	—	178	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	1	no	—
231	C₄²	#13083562	1	2	yes	—
232	C₄²	#13083564	1	2	yes	—
233	C₂³	C₂×#13083551	1	1	yes	characteristic
234	Q₈	—	712	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	356	1	no	—
251	C₂×C₄	—	356	1	no	—
252	C₂×C₄	—	356	1	no	—
253	C₂×C₄	—	356	1	no	—
254	C₂×C₄	—	356	1	no	—
255	C₂×C₄	—	356	1	no	—
256	C₂×C₄	—	356	1	no	—
257	C₂×C₄	—	356	1	no	—
258	C₂×C₄	—	356	1	no	—
259	C₂×C₄	—	356	1	no	—
260	C₂×C₄	—	356	1	no	—
261	C₂×C₄	—	356	1	no	—
262	C₂×C₄	#15667935	1	2	yes	—
263	C₂×C₄	#15667935	1	2	yes	—
264	C₂×C₄	#15667940	1	2	yes	—
265	C₂×C₄	#15668018	1	2	yes	—
266	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
267	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
268	C₂×C₄	#15668034	1	2	yes	—
269	C₈	—	4	4	no	—
270	C₈	—	712	2	no	—
271	C₂²	#19151564	1	1	yes	characteristic
272	C₂²	#19151571	1	1	yes	characteristic
273	C₂²	#19151580	1	1	yes	characteristic
274	C₂²	C₂×#15667935	1	1	yes	characteristic
275	C₂²	#19151595	1	1	yes	characteristic
276	C₂²	C₂×#15667940	1	1	yes	characteristic
277	C₂²	#19151614	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	712	1	no	—
287	C₄	—	712	1	no	—
288	C₄	—	712	1	no	—
289	C₄	—	712	1	no	—
290	C₂	#26018619	1	1	yes	characteristic
291	C₂	#26018654	1	1	yes	characteristic
292	C₂	#26018713	1	1	yes	characteristic
293	C₂	#26018750	1	1	yes	characteristic
294	C₂	#26018830	1	1	yes	characteristic
295	C₂	#26018857	1	1	yes	characteristic
296	C₂	#26018910	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#7673
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
154. C₈₉	#7673	semidirect

Maximal subgroups

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

subgroup	index
#26028634	2
#26020395	2
#26018077	2
#26018074	2
C₄×C₄⋊C₈×C₈₉	2
#7673	89