Subgroups — #36445128

Order 22784.

420 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028897	C₂	1	1	yes	maximal, characteristic
2	#26018635	C₂	1	1	yes	maximal, characteristic
3	#26018632	C₂	1	1	yes	maximal, characteristic
4	#26018536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018090	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083574	C₂²	1	1	yes	characteristic
9	#19151660	—	2	2	no	—
10	#19151651	C₂²	1	1	yes	characteristic
11	#19151649	—	2	1	no	—
12	#19151649	—	2	1	no	—
13	#19151626	C₄	1	2	yes	—
14	#19151566	C₄	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
17	#19151330	C₄	1	2	yes	—
18	#19151328	C₄	1	2	yes	—
19	#19151327	C₄	1	2	yes	—
20	#19151325	C₄	1	2	yes	—
21	#19151320	—	2	2	no	—
22	#19151320	—	2	1	no	—
23	C₈×#10623494	—	2	1	no	—
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10623494	—	2	1	no	—
29	C₂²×#10623494	C₂×C₄	1	2	yes	—
30	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
32	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
33	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₈₉	—	2	1	no	—
35	C₄×C₂²⋊D₈₉	—	2	1	no	—
36	#15667971	C₂×C₄	1	1	yes	characteristic
37	#15667969	—	2	1	no	—
38	C₄²×D₈₉	—	2	1	no	—
39	C₄⋊#10623494	C₂×C₄	1	2	yes	—
40	#15667941	—	2	2	no	—
41	#15667941	—	2	2	no	—
42	C₂²⋊#10623494	—	2	1	no	—
43	#15667924	—	2	2	no	—
44	#15667924	—	2	2	no	—
45	C₄×#10623494	—	2	1	no	—
46	C₄×#10623494	—	2	1	no	—
47	C₄×#10623494	C₂×C₄	1	2	yes	—
48	C₄×#10623494	D₄	1	1	yes	characteristic
49	C₄×#10623494	D₄	1	1	yes	characteristic
50	C₄×#10623494	D₄	1	2	yes	—
51	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₈₉	—	2	1	no	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₈₉	—	2	1	no	—
56	#97×C₈₉	—	2	1	no	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
61	#13083574	—	2	2	no	—
62	#13083574	—	2	2	no	—
63	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
64	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×#10623494	—	2	4	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
75	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
76	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	C₂²⋊Dic₈₉	—	2	1	no	—
81	#13083551	—	2	1	no	—
82	#13083551	—	2	1	no	—
83	C₄×Dic₈₉	—	2	1	no	—
84	C₄×Dic₈₉	—	2	1	no	—
85	C₄×Dic₈₉	—	2	1	no	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
89	M₁₆×C₈₉	—	2	2	no	—
90	M₁₆×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	4	2	no	—
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₈₉	#117	1	1	yes	characteristic
100	C₂²×D₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#131	1	2	yes	—
110	C₂×C₄×C₈₉	#131	1	2	yes	—
111	C₂²⋊D₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	2	1	no	—
113	C₂×Dic₈₉	—	2	1	no	—
114	C₂×Dic₈₉	—	2	1	no	—
115	D₃₅₆	—	2	1	no	—
116	D₃₅₆	—	2	1	no	—
117	C₄×D₈₉	—	4	2	no	—
118	C₄×D₈₉	—	4	2	no	—
119	C₄×D₈₉	—	4	2	no	—
120	Dic₁₇₈	—	2	1	no	—
121	Dic₁₇₈	—	2	1	no	—
122	C₈×C₈₉	—	4	2	no	—
123	C₈×C₈₉	—	4	2	no	—
124	C₈×C₈₉	—	4	2	no	—
125	#10623494	—	2	4	no	—
126	#10623494	—	2	2	no	—
127	#10623494	—	2	2	no	—
128	#10623494	—	4	2	no	—
129	C₂²×C₈₉	—	2	1	no	—
130	C₂²×C₈₉	—	2	1	no	—
131	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₈₉	#433	1	2	yes	—
133	C₂×D₈₉	—	4	1	no	—
134	C₂×D₈₉	—	4	1	no	—
135	C₂×D₈₉	—	4	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	#405	1	1	yes	characteristic
139	C₄×C₈₉	#405	1	1	yes	characteristic
140	C₄×C₈₉	#408	1	2	yes	—
141	C₄×C₈₉	#443	1	2	yes	—
142	C₄×C₈₉	#444	1	2	yes	—
143	Dic₈₉	—	4	1	no	—
144	Dic₈₉	—	4	1	no	—
145	Dic₈₉	—	4	1	no	—
146	#8189	—	89	1	no	maximal, Sylow
147	C₂×C₈₉	—	2	2	no	—
148	C₂×C₈₉	#1750	1	1	yes	characteristic
149	C₂×C₈₉	#1764	1	1	yes	characteristic
150	C₂×C₈₉	#1765	1	1	yes	characteristic
151	D₈₉	—	8	1	no	—
152	#2922	—	89	1	no	—
153	#1476	—	89	1	no	—
154	#1476	—	89	1	no	—
155	#1450	—	89	1	no	—
156	C₂×#322	—	89	1	no	—
157	#1278	—	89	1	no	—
158	#1278	D₈₉	1	1	yes	characteristic, complemented
159	C₈₉	#8189	1	1	yes	characteristic, complemented, Sylow
160	C₄×#41	—	89	1	no	—
161	#433	—	178	2	no	—
162	#431	—	89	1	no	—
163	#431	—	178	1	no	—
164	#431	—	178	1	no	—
165	#431	C₂×D₈₉	1	1	yes	characteristic
166	#424	—	89	2	no	—
167	#408	—	89	2	no	—
168	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
169	C₂×C₄×C₈	—	89	1	no	—
170	C₂×C₄×C₈	—	89	1	no	—
171	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
172	Q₈⋊C₈	—	89	2	no	—
173	Q₈⋊C₈	—	89	2	no	—
174	D₄⋊C₈	—	89	2	no	—
175	D₄⋊C₈	—	89	2	no	—
176	#322	—	178	2	no	—
177	#322	—	178	1	no	—
178	C₈²	—	178	1	no	—
179	C₂×#41	—	89	1	no	—
180	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	89	2	no	—
183	C₂²×C₈	—	178	1	no	—
184	C₄×Q₈	—	89	1	no	—
185	C₄×Q₈	—	89	1	no	—
186	C₄×D₄	—	89	1	no	—
187	C₄×D₄	—	89	1	no	—
188	C₄×D₄	—	178	1	no	—
189	C₄×D₄	—	178	1	no	—
190	#117	—	89	1	no	—
191	#117	—	178	1	no	—
192	C₂×C₄²	—	178	1	no	—
193	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
194	C₄⋊C₈	—	89	2	no	—
195	C₄⋊C₈	—	178	2	no	—
196	C₄⋊C₈	—	178	2	no	—
197	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	178	2	no	—
200	C₂²⋊C₈	—	178	2	no	—
201	C₂²⋊C₈	—	178	1	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	89	2	no	—
207	C₄×C₈	—	89	2	no	—
208	C₄×C₈	—	89	1	no	—
209	C₄×C₈	—	89	1	no	—
210	C₄×C₈	—	178	1	no	—
211	C₄×C₈	—	178	1	no	—
212	C₄×C₈	C₄×D₈₉	1	2	yes	—
213	#41	—	178	2	no	—
214	#41	—	178	2	no	—
215	C₂×Q₈	—	89	1	no	—
216	C₂×D₄	—	89	1	no	—
217	C₂×D₄	—	178	1	no	—
218	C₂²×C₄	—	178	1	no	—
219	C₂²×C₄	—	178	1	no	—
220	C₂²×C₄	—	178	1	no	—
221	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
222	C₂²×C₄	#13083573	1	1	yes	characteristic
223	C₂²×C₄	#13083575	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	89	2	no	—
231	C₂×C₈	—	89	2	no	—
232	C₂×C₈	—	178	2	no	—
233	C₂×C₈	—	178	4	no	—
234	C₂×C₈	—	178	2	no	—
235	C₂×C₈	—	178	2	no	—
236	C₂×C₈	—	178	2	no	—
237	C₄⋊C₄	—	89	1	no	—
238	C₄⋊C₄	—	89	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	—	178	1	no	—
241	C₂²⋊C₄	—	178	1	no	—
242	C₂²⋊C₄	—	178	1	no	—
243	C₂²⋊C₄	—	178	1	no	—
244	C₄²	—	178	1	no	—
245	C₄²	—	178	1	no	—
246	C₄²	—	178	1	no	—
247	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
249	C₄²	D₄×D₈₉	1	1	yes	characteristic
250	C₄²	D₄×D₈₉	1	1	yes	characteristic
251	C₂³	—	178	1	no	—
252	C₂³	#15667970	1	1	yes	characteristic
253	Q₈	—	178	1	no	—
254	Q₈	—	178	1	no	—
255	D₄	—	356	2	no	—
256	D₄	—	178	1	no	—
257	D₄	—	178	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	356	2	no	—
261	C₂×C₄	—	356	2	no	—
262	C₂×C₄	—	356	2	no	—
263	C₂×C₄	—	178	1	no	—
264	C₂×C₄	—	178	1	no	—
265	C₂×C₄	—	178	1	no	—
266	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₄	#15667972	1	1	yes	characteristic
269	C₂×C₄	#15667972	1	1	yes	characteristic
270	C₂×C₄	#15667974	1	1	yes	characteristic
271	C₂×C₄	#15667974	1	1	yes	characteristic
272	C₂×C₄	#15668018	1	2	yes	—
273	C₂×C₄	#15668020	1	2	yes	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	356	2	no	—
278	C₈	—	178	4	no	—
279	C₈	—	178	2	no	—
280	C₈	—	178	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	356	1	no	—
284	C₂²	—	356	1	no	—
285	C₂²	—	356	1	no	—
286	C₂²	#19151445	1	1	yes	characteristic
287	C₂²	#19151658	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	356	1	no	—
291	C₄	—	356	1	no	—
292	C₄	—	356	1	no	—
293	C₄	#19151552	1	1	yes	characteristic
294	C₄	#19151555	1	1	yes	characteristic
295	C₄	#19151563	1	2	yes	—
296	C₄	#19151714	1	2	yes	—
297	C₄	#19151717	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	712	1	no	—
300	C₂	#26020017	1	1	yes	characteristic
301	C₂	#26020088	1	1	yes	characteristic
302	C₂	#26020093	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1278	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1278	D₈₉
hypocenter	C₈₉	#8189
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #26018536	C₂	semidirect
5. C₂×#19151320	C₂	semidirect
6. #26018090	C₂	semidirect
7. C₈₉×#1278	C₂	semidirect
158. #1278	D₈₉	semidirect
159. C₈₉	#8189	semidirect

Maximal subgroups

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

subgroup	index
#26028897	2
#26018635	2
#26018632	2
#26018536	2
C₂×#19151320	2
#26018090	2
C₈₉×#1278	2
#8189	89