Subgroups — #36444541

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	#26022659	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal, complemented
4	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10623494	—	2	1	no	—
6	C₂×#15667971	—	2	1	no	—
7	C₂×C₄⋊#10623494	—	2	2	no	—
8	C₂×C₄⋊#10623494	—	2	1	no	—
9	C₂×C₄⋊#10623494	C₄	1	4	yes	complemented
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	#19151468	C₂²	1	2	yes	—
12	#19151464	—	2	1	no	—
13	#19151458	—	2	1	no	—
14	#19151458	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₈₉	—	2	1	no	—
16	C₄×C₄⋊Dic₈₉	C₄	1	2	yes	—
17	C₂²×M₁₆×C₈₉	—	2	1	no	—
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	#423×C₈₉	—	2	2	no	—
20	#423×C₈₉	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
23	C₂²×#10623494	—	2	4	no	—
24	C₂²×#10623494	—	2	2	no	—
25	C₂²×#10623494	—	4	4	no	—
26	#15667971	—	4	2	no	—
27	#15667971	—	4	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
32	C₂×C₄×Dic₈₉	—	2	2	no	—
33	C₄⋊#10623494	—	2	4	no	—
34	C₄⋊#10623494	—	2	2	no	—
35	C₄⋊#10623494	C₂×C₄	1	4	yes	—
36	#15667919	—	2	2	no	—
37	#15667919	—	2	2	no	—
38	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₈₉	—	2	4	no	—
42	C₂×M₁₆×C₈₉	—	4	2	no	—
43	C₂×M₁₆×C₈₉	—	4	2	no	—
44	C₂²×C₈×C₈₉	—	2	2	no	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	2	yes	—
51	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	—	2	4	no	—
54	C₄⋊C₈×C₈₉	—	2	2	no	—
55	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
56	C₂²×Dic₈₉	—	2	4	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	4	8	no	—
59	C₂×#10623494	—	4	4	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₄⋊Dic₈₉	—	4	2	no	—
63	C₄⋊Dic₈₉	—	4	2	no	—
64	C₄⋊Dic₈₉	—	4	4	no	—
65	C₂²⋊Dic₈₉	—	4	4	no	—
66	C₄×Dic₈₉	—	4	4	no	—
67	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	—	2	2	no	—
70	C₂²×C₄×C₈₉	—	4	2	no	—
71	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
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	2	yes	—
75	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₈₉	D₈	1	2	yes	—
81	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
82	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
83	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	M₁₆×C₈₉	—	4	8	no	—
88	C₂×C₈×C₈₉	—	2	4	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	—	4	2	no	—
91	C₄²×C₈₉	C₄²	1	2	yes	—
92	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
95	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
96	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
97	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
101	C₂³×C₈₉	—	2	2	no	—
102	C₂³×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	#95	1	2	yes	—
104	C₂³×C₈₉	#102	1	2	yes	—
105	C₂³×C₈₉	#103	1	2	yes	—
106	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	#95	1	2	yes	—
116	C₂×C₄×C₈₉	#95	1	2	yes	—
117	C₂×C₄×C₈₉	#95	1	2	yes	—
118	C₂×C₄×C₈₉	#102	1	2	yes	—
119	C₂×C₄×C₈₉	#102	1	4	yes	—
120	C₂×C₄×C₈₉	#103	1	2	yes	—
121	C₂×C₄×C₈₉	#103	1	4	yes	—
122	C₂×C₄×C₈₉	#106	1	4	yes	—
123	C₂×C₄×C₈₉	#107	1	4	yes	—
124	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₈₉	#137	1	2	yes	—
136	C₂×Dic₈₉	—	4	4	no	—
137	C₂×Dic₈₉	—	4	4	no	—
138	C₂×Dic₈₉	—	4	4	no	—
139	C₈×C₈₉	—	4	4	no	—
140	#10623494	—	4	4	no	—
141	C₂²×C₈₉	—	2	4	no	—
142	C₂²×C₈₉	—	2	4	no	—
143	C₂²×C₈₉	—	2	2	no	—
144	C₂²×C₈₉	—	2	2	no	—
145	C₂²×C₈₉	#340	1	4	yes	—
146	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
147	C₂²×C₈₉	#413	1	2	yes	—
148	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
149	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
150	C₂²×C₈₉	#417	1	2	yes	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	#340	1	4	yes	—
156	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
157	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
158	C₄×C₈₉	#418	1	1	yes	characteristic
159	C₄×C₈₉	#419	1	1	yes	characteristic
160	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
162	C₄×C₈₉	#428	1	2	yes	—
163	Dic₈₉	—	8	4	no	—
164	#8045	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	4	no	—
166	C₂×C₈₉	#1547	1	2	yes	—
167	C₂×C₈₉	#1550	1	2	yes	—
168	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
169	C₂×C₈₉	#1736	1	1	yes	characteristic
170	C₂×C₈₉	#1742	1	1	yes	characteristic
171	C₂×#423	D₈₉	1	1	yes	characteristic
172	C₂²×C₄⋊C₈	—	89	1	no	—
173	#2285	—	89	1	no	—
174	#1296	—	89	4	no	—
175	C₈₉	#8045	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	—	2	1	no	—
177	C₂³×C₈	—	178	1	no	—
178	C₂×#117	—	178	1	no	—
179	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
180	#423	—	2	2	no	—
181	#423	Dic₈₉	1	4	yes	complemented
182	C₂×C₄⋊C₈	—	2	1	no	—
183	C₂×C₄⋊C₈	—	89	4	no	—
184	C₂×C₄⋊C₈	—	89	2	no	—
185	C₂×C₄⋊C₈	—	178	1	no	—
186	C₂×C₄⋊C₈	—	178	2	no	—
187	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
188	#384	—	89	2	no	—
189	#382	—	178	1	no	—
190	#380	—	89	2	no	—
191	#380	—	178	1	no	—
192	C₄×C₄⋊C₄	—	89	2	no	—
193	C₄×C₄⋊C₄	—	178	1	no	—
194	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
195	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	4	2	no	—
199	C₂×M₁₆	—	4	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	356	4	no	—
202	C₂²×C₈	—	178	4	no	—
203	C₂²×C₈	—	178	2	no	—
204	#117	—	356	2	no	—
205	#117	—	356	2	no	—
206	C₂×C₄⋊C₄	—	89	4	no	—
207	C₂×C₄⋊C₄	—	178	2	no	—
208	C₂×C₄⋊C₄	—	178	2	no	—
209	C₂×C₂²⋊C₄	—	178	2	no	—
210	C₂×C₄²	—	178	2	no	—
211	C₂×C₄²	Dic₁₇₈	1	2	yes	—
212	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₈₉	1	2	yes	—
214	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	89	4	no	—
222	C₄⋊C₈	—	178	4	no	—
223	C₄⋊C₈	—	178	2	no	—
224	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
225	#95	—	178	2	no	—
226	#95	—	178	2	no	—
227	C₂⁴	#13083551	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	178	4	no	—
232	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
233	C₂²×C₄	#13083551	1	1	yes	characteristic
234	C₂²×C₄	#13083551	1	1	yes	characteristic
235	C₂²×C₄	#13083551	1	2	yes	—
236	C₂²×C₄	#13083551	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
238	C₂²×C₄	#13083553	1	2	yes	—
239	C₂²×C₄	#13083553	1	2	yes	—
240	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
241	C₂²×C₄	#13083559	1	2	yes	—
242	C₂²×C₄	#13083561	1	2	yes	—
243	C₂²×C₄	#13083562	1	2	yes	—
244	C₂²×C₄	#13083564	1	2	yes	—
245	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
248	M₁₆	—	4	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	356	8	no	—
253	C₂×C₈	—	356	4	no	—
254	C₂×C₈	—	178	4	no	—
255	C₄⋊C₄	—	356	4	no	—
256	C₄⋊C₄	—	356	2	no	—
257	C₄⋊C₄	—	356	2	no	—
258	C₄⋊C₄	—	356	2	no	—
259	C₄⋊C₄	—	356	2	no	—
260	C₂²⋊C₄	—	356	4	no	—
261	C₄²	—	356	4	no	—
262	C₄²	C₄×Dic₈₉	1	2	yes	—
263	C₄²	#13083551	1	2	yes	—
264	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
265	C₄²	#13083553	1	2	yes	—
266	C₄²	#13083553	1	2	yes	—
267	C₄²	C₄⋊Dic₈₉	1	2	yes	—
268	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
269	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
270	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
271	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	#15667919	1	2	yes	—
275	C₂³	#15667932	1	2	yes	—
276	C₂³	#15667935	1	2	yes	—
277	C₂³	C₂×#13083551	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	356	4	no	—
287	C₂×C₄	—	356	4	no	—
288	C₂×C₄	—	356	4	no	—
289	C₂×C₄	#15667919	1	2	yes	—
290	C₂×C₄	#15667919	1	2	yes	—
291	C₂×C₄	#15667919	1	2	yes	—
292	C₂×C₄	#15667932	1	2	yes	—
293	C₂×C₄	#15667933	1	4	yes	—
294	C₂×C₄	#15667935	1	2	yes	—
295	C₂×C₄	#15667936	1	4	yes	—
296	C₂×C₄	#15667943	1	4	yes	—
297	C₂×C₄	#15667945	1	4	yes	—
298	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
299	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
301	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
302	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
304	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
306	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
307	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
308	C₂×C₄	#15668040	1	2	yes	—
309	C₂×C₄	#15668045	1	2	yes	—
310	C₈	—	4	4	no	—
311	C₈	—	356	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	#19151360	1	4	yes	—
317	C₂²	C₂×#15667919	1	1	yes	characteristic
318	C₂²	#19151580	1	2	yes	—
319	C₂²	C₂×#15667932	1	1	yes	characteristic
320	C₂²	C₂×#15667935	1	1	yes	characteristic
321	C₂²	#19151600	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	712	4	no	—
327	C₄	#19151360	1	4	yes	—
328	C₄	C₂×#15667933	1	1	yes	characteristic
329	C₄	C₂×#15667936	1	1	yes	characteristic
330	C₄	#19151604	1	1	yes	characteristic
331	C₄	#19151608	1	1	yes	characteristic
332	C₄	C₂×#15667943	1	1	yes	characteristic
333	C₄	C₂×#15667945	1	1	yes	characteristic
334	C₄	#19151640	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	#26018930	1	2	yes	—
337	C₂	#26018943	1	2	yes	—
338	C₂	C₂×#19151360	1	1	yes	characteristic
339	C₂	#26019972	1	1	yes	characteristic
340	C₂	#26019990	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#423×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#423	D₈₉
hypocenter	C₈₉	#8045
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #26018136	C₂	semidirect
9. C₂×C₄⋊#10623494	C₄	semidirect
20. #423×C₈₉	C₄	semidirect
175. C₈₉	#8045	semidirect
181. #423	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#10623494	2
#26022659	2
#26018136	2
C₂×#423×C₈₉	2
#8045	89