Subgroups — #36453380

Order 22784.

568 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668012	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151328	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×C₄⋊#10623494	—	2	1	no	—
8	C₂×C₄×#10623494	C₂²	1	2	yes	—
9	#19151504	C₄	1	2	yes	—
10	#19151496	—	2	1	no	—
11	#19151496	—	2	1	no	—
12	#19151468	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₈₉	—	2	1	no	—
14	C₄×C₄⋊Dic₈₉	—	2	1	no	—
15	#19151353	—	2	1	no	—
16	#19151328	—	2	8	no	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	—	2	1	no	—
19	C₄³×C₈₉	C₂²	1	1	yes	characteristic
20	#336×C₈₉	—	2	1	no	—
21	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
22	C₂²×#10623494	—	2	2	no	—
23	#15668012	—	2	2	no	—
24	#15668012	C₈	1	4	yes	complemented
25	C₄×Dic₁₇₈	—	2	4	no	—
26	C₄×Dic₁₇₈	—	2	4	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
32	C₂×#13083553	—	2	2	no	—
33	C₂×C₄×Dic₈₉	—	2	2	no	—
34	C₄⋊#10623494	—	2	4	no	—
35	C₄×#10623494	—	2	4	no	—
36	#15667919	—	2	2	no	—
37	#15667919	—	2	2	no	—
38	C₂²×C₈×C₈₉	—	2	2	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	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	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₈₉	—	2	2	no	—
47	C₄⋊C₈×C₈₉	D₄	1	4	yes	—
48	C₄×C₈×C₈₉	—	4	2	no	—
49	C₂×Dic₁₇₈	—	2	4	no	—
50	C₂×Dic₁₇₈	—	2	4	no	—
51	C₂×Dic₁₇₈	—	2	2	no	—
52	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
53	C₂²×Dic₈₉	—	2	2	no	—
54	C₂²×Dic₈₉	—	2	2	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	4	2	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	2	4	no	—
60	C₄⋊Dic₈₉	—	2	4	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₄⋊Dic₈₉	—	4	2	no	—
63	#13083553	—	4	2	no	—
64	#13083553	—	4	2	no	—
65	C₄×Dic₈₉	—	4	4	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	C₂×C₈	1	2	yes	—
81	C₄²×C₈₉	M₁₆	1	2	yes	—
82	C₄²×C₈₉	SD₁₆	1	2	yes	—
83	C₄²×C₈₉	Q₁₆	1	2	yes	—
84	C₄²×C₈₉	C₂×D₄	1	2	yes	—
85	C₄²×C₈₉	#41	1	2	yes	—
86	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₈₉	#103	1	2	yes	—
99	C₂×C₄×C₈₉	#103	1	2	yes	—
100	C₂×C₄×C₈₉	#104	1	2	yes	—
101	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₈₉	#121	1	2	yes	—
103	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#131	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#137	1	2	yes	—
111	C₂×Dic₈₉	—	2	4	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	4	2	no	—
115	C₂×Dic₈₉	—	4	2	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	Dic₁₇₈	—	2	8	no	—
118	Dic₁₇₈	—	4	4	no	—
119	C₈×C₈₉	—	4	4	no	—
120	#10623494	—	4	4	no	—
121	C₂²×C₈₉	#390	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₈₉	#408	1	1	yes	characteristic
124	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
125	C₂²×C₈₉	#419	1	1	yes	characteristic
126	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
127	C₂²×C₈₉	#421	1	1	yes	characteristic
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	4	1	no	—
133	C₄×C₈₉	Q₈⋊C₈	1	4	yes	complemented
134	C₄×C₈₉	C₈×D₄	1	2	yes	—
135	C₄×C₈₉	#436	1	2	yes	—
136	C₄×C₈₉	#461	1	2	yes	—
137	C₄×C₈₉	#462	1	2	yes	—
138	Dic₈₉	—	4	4	no	—
139	Dic₈₉	—	8	2	no	—
140	#9673	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
142	C₂×C₈₉	#1479	1	1	yes	characteristic
143	C₂×C₈₉	#1481	1	1	yes	characteristic
144	C₂×C₈₉	#1485	1	1	yes	characteristic
145	C₂×C₈₉	#1954	1	1	yes	characteristic
146	C₂×C₈₉	#1980	1	1	yes	characteristic
147	C₂×C₈₉	#1981	1	1	yes	characteristic
148	C₄×C₄⋊Q₈	—	89	1	no	—
149	C₄×C₄⋊C₈	—	89	1	no	—
150	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
151	C₂×Q₈⋊C₈	—	89	4	no	—
152	C₈₉	#9673	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄⋊Q₈	—	89	1	no	—
154	C₂×C₄×Q₈	—	89	2	no	—
155	C₂×C₄⋊C₈	—	178	1	no	—
156	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
157	C₂×C₄×C₈	—	2	1	no	—
158	C₂×C₄×C₈	—	89	2	no	—
159	#391	—	89	2	no	—
160	#389	—	178	1	no	—
161	#389	—	178	1	no	—
162	#384	—	89	1	no	—
163	C₄×C₄⋊C₄	—	178	1	no	—
164	C₄×C₄⋊C₄	—	178	1	no	—
165	C₄³	C₂×D₈₉	1	1	yes	characteristic
166	#336	—	2	1	no	—
167	#336	—	178	1	no	—
168	Q₈⋊C₈	—	178	8	no	—
169	C₂²×Q₈	—	89	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	178	2	no	—
172	C₄⋊Q₈	—	89	4	no	—
173	C₄⋊Q₈	—	178	2	no	—
174	C₄×Q₈	—	178	4	no	—
175	C₄×Q₈	—	178	4	no	—
176	C₂×C₄⋊C₄	—	89	2	no	—
177	C₂×C₄⋊C₄	—	178	1	no	—
178	C₂×C₄⋊C₄	—	178	1	no	—
179	C₂×C₄⋊C₄	—	178	2	no	—
180	C₂×C₄⋊C₄	—	178	1	no	—
181	C₂×C₄⋊C₄	—	178	1	no	—
182	C₂×C₄²	—	178	2	no	—
183	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
185	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
186	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	178	4	no	—
192	C₄⋊C₈	D₃₅₆	1	4	yes	—
193	C₄×C₈	—	4	2	no	—
194	C₄×C₈	—	178	4	no	—
195	#95	—	178	2	no	—
196	#95	—	178	2	no	—
197	C₂×Q₈	—	89	4	no	—
198	C₂×Q₈	—	178	4	no	—
199	C₂×Q₈	—	178	4	no	—
200	C₂×Q₈	—	178	2	no	—
201	C₂²×C₄	—	178	2	no	—
202	C₂²×C₄	—	178	2	no	—
203	C₂²×C₄	#13083551	1	1	yes	characteristic
204	C₂²×C₄	#13083551	1	1	yes	characteristic
205	C₂²×C₄	#13083551	1	1	yes	characteristic
206	C₂²×C₄	#13083551	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	356	2	no	—
214	C₂×C₈	—	356	2	no	—
215	C₂×C₈	—	178	4	no	—
216	C₄⋊C₄	—	356	2	no	—
217	C₄⋊C₄	—	356	2	no	—
218	C₄⋊C₄	—	356	2	no	—
219	C₄⋊C₄	—	356	2	no	—
220	C₄⋊C₄	—	178	4	no	—
221	C₄⋊C₄	—	178	4	no	—
222	C₄⋊C₄	—	178	4	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	356	4	no	—
228	C₄²	C₈×D₈₉	1	2	yes	—
229	C₄²	#13083557	1	2	yes	—
230	C₄²	#13083562	1	2	yes	—
231	C₄²	#13083564	1	2	yes	—
232	C₄²	C₂×D₃₅₆	1	2	yes	—
233	C₄²	#13083574	1	2	yes	—
234	C₂³	C₂×#13083551	1	1	yes	characteristic
235	Q₈	—	356	4	no	—
236	Q₈	—	178	8	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	—	356	2	no	—
247	C₂×C₄	—	356	2	no	—
248	C₂×C₄	—	356	2	no	—
249	C₂×C₄	—	356	2	no	—
250	C₂×C₄	—	356	2	no	—
251	C₂×C₄	—	178	4	no	—
252	C₂×C₄	#15667924	1	2	yes	—
253	C₂×C₄	#15667924	1	2	yes	—
254	C₂×C₄	#15667935	1	2	yes	—
255	C₂×C₄	#15667935	1	2	yes	—
256	C₂×C₄	#15667938	1	2	yes	—
257	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
258	C₂×C₄	#15667986	1	2	yes	—
259	C₂×C₄	#15668000	1	1	yes	characteristic
260	C₂×C₄	#15668009	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
262	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
263	C₂×C₄	#15668018	1	2	yes	—
264	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
266	C₂×C₄	#15668042	1	2	yes	—
267	C₈	—	4	4	no	—
268	C₈	—	356	4	no	—
269	C₂²	#19151500	1	1	yes	characteristic
270	C₂²	C₂×#15667924	1	1	yes	characteristic
271	C₂²	#19151564	1	1	yes	characteristic
272	C₂²	C₂×#15667935	1	1	yes	characteristic
273	C₂²	#19151606	1	1	yes	characteristic
274	C₂²	C₂×#15667938	1	1	yes	characteristic
275	C₂²	#19151614	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	356	4	no	—
282	C₄	—	712	2	no	—
283	C₄	#19151328	1	4	yes	complemented
284	C₄	C₈×D₃₅₆	1	2	yes	—
285	C₄	#19151677	1	2	yes	—
286	C₄	#19151803	1	2	yes	—
287	C₄	#19151808	1	2	yes	—
288	C₂	C₂×#19151328	1	1	yes	characteristic
289	C₂	#26018646	1	1	yes	characteristic
290	C₂	#26018654	1	1	yes	characteristic
291	C₂	#26018672	1	1	yes	characteristic
292	C₂	#26021046	1	1	yes	characteristic
293	C₂	#26021171	1	1	yes	characteristic
294	C₂	#26021175	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
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	#9673
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
24. #15668012	C₈	semidirect
133. C₄×C₈₉	Q₈⋊C₈	semidirect
152. C₈₉	#9673	semidirect
283. C₄	#19151328	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668012	2
C₄×C₄⋊#10623494	2
C₂×#19151328	2
C₄×C₄⋊C₈×C₈₉	2
#9673	89