Subgroups — #36445364

Order 22784.

488 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668004	C₂	1	1	yes	maximal, characteristic
2	#26020001	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151339	C₂	1	2	yes	maximal
4	#26018134	C₂	1	2	yes	maximal
5	C₄×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668004	C₂²	1	1	yes	characteristic
7	C₂×#15667923	C₂²	1	2	yes	—
8	#19151496	C₄	1	1	yes	characteristic
9	#19151496	C₄	1	1	yes	characteristic
10	#19151482	—	2	1	no	—
11	#19151482	—	2	1	no	—
12	#19151464	C₂²	1	1	yes	characteristic
13	C₄×#13083553	—	2	1	no	—
14	C₄×#13083553	—	2	1	no	—
15	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
16	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
17	#19151353	—	2	1	no	—
18	#19151353	—	2	1	no	—
19	#19151339	C₄	1	8	yes	complemented
20	C₂×#97×C₈₉	—	2	1	no	—
21	C₂×#97×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	—	2	1	no	—
23	C₄³×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×#10623494	—	2	2	no	—
25	#15668004	—	2	2	no	—
26	#15668004	C₂×C₄	1	4	yes	—
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×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083553	—	2	1	no	—
32	C₂×#13083553	—	2	1	no	—
33	C₂×#13083553	—	2	1	no	—
34	C₂×#13083553	—	2	1	no	—
35	C₂×C₄×Dic₈₉	—	2	1	no	—
36	C₂×C₄×Dic₈₉	—	2	1	no	—
37	#15667923	—	2	2	no	—
38	#15667923	C₂×C₄	1	4	yes	—
39	#15667919	—	2	1	no	—
40	#15667919	—	2	1	no	—
41	#15667919	—	2	1	no	—
42	#15667919	—	2	1	no	—
43	#15667919	—	2	1	no	—
44	#15667919	—	2	1	no	—
45	C₂²×C₈×C₈₉	—	2	2	no	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	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	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	#97×C₈₉	—	2	4	no	—
54	#97×C₈₉	—	2	2	no	—
55	#97×C₈₉	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₈₉	—	2	4	no	—
57	C₂²×Dic₈₉	—	2	1	no	—
58	C₂²×Dic₈₉	—	2	1	no	—
59	C₂²×Dic₈₉	—	2	1	no	—
60	C₂²×Dic₈₉	—	2	1	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	4	2	no	—
63	C₂×#10623494	—	4	2	no	—
64	C₄⋊Dic₈₉	—	2	1	no	—
65	C₄⋊Dic₈₉	—	2	1	no	—
66	C₄⋊Dic₈₉	—	4	1	no	—
67	C₄⋊Dic₈₉	—	4	1	no	—
68	C₄⋊Dic₈₉	C₄²	1	2	yes	—
69	C₄⋊Dic₈₉	C₄²	1	2	yes	—
70	#13083553	—	2	4	no	—
71	#13083553	—	2	4	no	—
72	#13083553	—	4	2	no	—
73	#13083553	—	4	2	no	—
74	C₄×Dic₈₉	—	4	2	no	—
75	C₄×Dic₈₉	—	4	2	no	—
76	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	—	2	4	no	—
86	C₄²×C₈₉	—	2	2	no	—
87	C₄²×C₈₉	—	2	2	no	—
88	C₄²×C₈₉	—	2	2	no	—
89	C₄²×C₈₉	—	4	1	no	—
90	C₄²×C₈₉	—	4	1	no	—
91	C₄²×C₈₉	—	4	1	no	—
92	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
93	C₄²×C₈₉	#41	1	2	yes	—
94	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	#101	1	2	yes	—
111	C₂×C₄×C₈₉	#104	1	2	yes	—
112	C₂×C₄×C₈₉	#104	1	2	yes	—
113	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
117	C₂×Dic₈₉	—	2	2	no	—
118	C₂×Dic₈₉	—	2	2	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	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	C₈×C₈₉	—	2	8	no	—
130	#10623494	—	4	4	no	—
131	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
133	C₂²×C₈₉	#413	1	1	yes	characteristic
134	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
135	C₂²×C₈₉	C₂×#104	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	2	no	—
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	4	1	no	—
146	Dic₈₉	—	4	2	no	—
147	Dic₈₉	—	4	2	no	—
148	Dic₈₉	—	8	1	no	—
149	Dic₈₉	—	8	1	no	—
150	#8227	—	89	1	no	maximal, Sylow
151	C₂×C₈₉	#1528	1	1	yes	characteristic
152	C₂×C₈₉	#1529	1	1	yes	characteristic
153	C₂×C₈₉	#1531	1	1	yes	characteristic
154	C₂×C₈₉	#1532	1	1	yes	characteristic
155	C₂×C₈₉	C₄×#101	1	1	yes	characteristic
156	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
157	C₂×C₈₉	#1759	1	1	yes	characteristic
158	C₄×#125	—	89	1	no	—
159	#1745	—	89	1	no	—
160	C₄×#97	D₈₉	1	1	yes	characteristic
161	C₂×#330	—	89	2	no	—
162	#1295	—	89	2	no	—
163	C₈₉	#8227	1	1	yes	characteristic, complemented, Sylow
164	C₂×#125	—	89	1	no	—
165	C₂×#97	—	2	1	no	—
166	C₂×#97	—	89	2	no	—
167	C₂×#97	C₂×D₈₉	1	2	yes	—
168	C₂×C₄×C₈	—	2	1	no	—
169	#389	—	89	1	no	—
170	#389	—	89	1	no	—
171	#387	—	178	1	no	—
172	#387	—	178	1	no	—
173	#382	—	89	1	no	—
174	C₄×C₄⋊C₄	—	89	1	no	—
175	C₄×C₄⋊C₄	—	89	1	no	—
176	C₄×C₄⋊C₄	—	178	1	no	—
177	C₄×C₄⋊C₄	—	178	1	no	—
178	C₄³	C₂×D₈₉	1	1	yes	characteristic
179	#336	—	178	1	no	—
180	#336	—	178	1	no	—
181	#330	—	89	8	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	178	2	no	—
184	#125	—	89	4	no	—
185	#125	—	178	2	no	—
186	C₂×C₄⋊C₄	—	89	1	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×C₄⋊C₄	—	178	1	no	—
194	C₂×C₄²	—	178	1	no	—
195	C₂×C₄²	—	178	1	no	—
196	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
198	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
199	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
203	#97	—	2	4	no	—
204	#97	—	2	2	no	—
205	#97	—	89	4	no	—
206	#97	—	178	2	no	—
207	#97	C₄×D₈₉	1	4	yes	—
208	C₄×C₈	—	2	4	no	—
209	#95	—	178	1	no	—
210	#95	—	178	1	no	—
211	#95	—	178	1	no	—
212	#95	—	178	1	no	—
213	#95	—	178	1	no	—
214	#95	—	178	1	no	—
215	C₂²×C₄	—	178	1	no	—
216	C₂²×C₄	—	178	1	no	—
217	C₂²×C₄	—	178	1	no	—
218	C₂²×C₄	—	178	1	no	—
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	#13083551	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	356	2	no	—
230	C₂×C₈	—	356	2	no	—
231	C₂×C₈	—	178	4	no	—
232	C₄⋊C₄	—	89	2	no	—
233	C₄⋊C₄	—	89	2	no	—
234	C₄⋊C₄	—	356	1	no	—
235	C₄⋊C₄	—	356	1	no	—
236	C₄⋊C₄	—	356	2	no	—
237	C₄⋊C₄	—	356	2	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	—	178	4	no	—
241	C₄⋊C₄	—	178	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	356	2	no	—
249	C₄²	—	356	2	no	—
250	C₄²	C₂×C₄×D₈₉	1	2	yes	—
251	C₄²	#13083574	1	2	yes	—
252	C₂³	C₂×#13083551	1	1	yes	characteristic
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	—	356	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	—	178	2	no	—
279	C₂×C₄	—	178	2	no	—
280	C₂×C₄	#15667930	1	2	yes	—
281	C₂×C₄	#15667940	1	2	yes	—
282	C₂×C₄	#15667940	1	2	yes	—
283	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
284	C₂×C₄	#15667969	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
286	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
287	C₈	—	2	8	no	—
288	C₈	—	356	4	no	—
289	C₂²	C₄×#13083551	1	1	yes	characteristic
290	C₂²	C₂×#15667930	1	1	yes	characteristic
291	C₂²	#19151580	1	1	yes	characteristic
292	C₂²	C₂×#15667940	1	1	yes	characteristic
293	C₂²	C₂×#15667940	1	1	yes	characteristic
294	C₂²	#19151614	1	1	yes	characteristic
295	C₂²	#19151614	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	356	2	no	—
305	C₄	—	356	2	no	—
306	C₄	—	712	1	no	—
307	C₄	—	712	1	no	—
308	C₂	#26018853	1	1	yes	characteristic
309	C₂	#26018857	1	1	yes	characteristic
310	C₂	#26018866	1	1	yes	characteristic
311	C₂	#26018870	1	1	yes	characteristic
312	C₂	C₄×#15667930	1	1	yes	characteristic
313	C₂	C₄×#15667940	1	1	yes	characteristic
314	C₂	#26020065	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#97×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#97	D₈₉
hypocenter	C₈₉	#8227
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
19. #19151339	C₄	semidirect
163. C₈₉	#8227	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668004	2
#26020001	2
C₂×#19151339	2
#26018134	2
C₄×#97×C₈₉	2
#8227	89