Subgroups — #36445141

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667939	C₂	1	1	yes	maximal, characteristic
2	#26019997	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018634	C₂	1	1	yes	maximal, characteristic
4	C₈×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018088	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19151651	C₂²	1	1	yes	characteristic
9	C₂×#15667939	C₄	1	2	yes	—
10	C₄×#13083558	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
13	#19151357	C₄	1	2	yes	—
14	#19151352	—	2	2	no	—
15	Q₈⋊#10623494	C₄	1	2	yes	complemented
16	D₄⋊#10623494	C₄	1	2	yes	complemented
17	#19151320	—	2	1	no	—
18	C₈×#10623494	—	2	1	no	—
19	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₈₉	C₄	1	2	yes	complemented
21	C₈×D₄×C₈₉	—	2	2	no	—
22	C₈×D₄×C₈₉	C₄	1	2	yes	complemented
23	#431×C₈₉	—	2	1	no	—
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	—	2	1	no	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#13083558	C₂×C₄	1	2	yes	—
28	C₂²×#10623494	—	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	C₄⋊#10623494	C₂×C₄	1	2	yes	—
33	#15667939	C₈	1	8	yes	complemented
34	C₂²⋊#10623494	—	2	1	no	—
35	#15667923	—	2	1	no	—
36	C₄×#10623494	—	2	1	no	—
37	C₄×#10623494	—	2	1	no	—
38	C₄×#10623494	—	2	1	no	—
39	C₄×#10623494	C₂×C₄	1	2	yes	—
40	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	—	2	2	no	—
42	C₂²×C₈×C₈₉	D₄	1	2	yes	—
43	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×D₄×C₈₉	—	2	1	no	—
46	C₄×D₄×C₈₉	—	2	1	no	—
47	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
49	#117×C₈₉	—	2	1	no	—
50	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	—	2	1	no	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	—	2	2	no	—
54	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₈₉	—	2	1	no	—
56	C₂²⋊C₈×C₈₉	—	2	2	no	—
57	C₄×C₈×C₈₉	—	2	2	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂²×Dic₈₉	—	2	2	no	—
63	#13083558	—	2	2	no	—
64	#13083558	C₂×C₈	1	4	yes	complemented
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	4	2	no	—
69	C₄×Dic₈₉	—	2	2	no	—
70	C₄×Dic₈₉	—	4	2	no	—
71	C₄×Dic₈₉	—	4	2	no	—
72	C₄×Dic₈₉	C₂×C₈	1	4	yes	complemented
73	#41×C₈₉	—	2	2	no	—
74	#41×C₈₉	C₂×C₈	1	4	yes	—
75	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
76	C₂×D₄×C₈₉	—	2	1	no	—
77	C₂×D₄×C₈₉	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	—	2	1	no	—
79	C₂²×C₄×C₈₉	—	2	1	no	—
80	C₂²×C₄×C₈₉	—	2	1	no	—
81	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₉	—	2	2	no	—
85	C₂×C₈×C₈₉	—	4	2	no	—
86	C₂×C₈×C₈₉	—	4	4	no	—
87	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
93	C₂²⋊C₄×C₈₉	—	2	1	no	—
94	C₂²⋊C₄×C₈₉	—	2	1	no	—
95	C₂²⋊C₄×C₈₉	—	2	1	no	—
96	C₄²×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	—	2	1	no	—
98	C₄²×C₈₉	—	2	1	no	—
99	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₈₉	#41	1	1	yes	characteristic
103	C₂³×C₈₉	—	2	1	no	—
104	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₈₉	—	2	1	no	—
106	Q₈×C₈₉	C₄×C₈	1	2	yes	—
107	D₄×C₈₉	—	2	4	no	—
108	D₄×C₈₉	—	2	1	no	—
109	D₄×C₈₉	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₈₉	#131	1	2	yes	—
126	C₂×Dic₈₉	—	2	4	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	C₈×C₈₉	—	2	2	no	—
130	C₈×C₈₉	—	2	2	no	—
131	C₈×C₈₉	—	4	2	no	—
132	#10623494	—	2	4	no	—
133	#10623494	—	4	2	no	—
134	#10623494	—	4	2	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	2	no	—
138	C₂²×C₈₉	—	4	1	no	—
139	C₂²×C₈₉	—	4	1	no	—
140	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₈₉	C₈×D₄	1	2	yes	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	4	1	no	—
146	C₄×C₈₉	—	4	1	no	—
147	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
148	C₄×C₈₉	#405	1	1	yes	characteristic
149	C₄×C₈₉	#421	1	2	yes	—
150	C₄×C₈₉	C₈×D₄	1	2	yes	—
151	C₄×C₈₉	#444	1	2	yes	—
152	Dic₈₉	—	4	4	no	—
153	#8191	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	2	2	no	—
155	C₂×C₈₉	—	4	2	no	—
156	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₈₉	#1759	1	1	yes	characteristic
158	C₂×C₈₉	#1765	1	1	yes	characteristic
159	C₈×#41	D₈₉	1	1	yes	characteristic
160	C₄×#104	—	89	1	no	—
161	#1744	—	89	1	no	—
162	#1476	—	89	1	no	—
163	C₈×M₁₆	—	89	1	no	—
164	#1278	—	89	1	no	—
165	#1276	—	89	1	no	—
166	C₈₉	#8191	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
168	C₈×Q₈	Dic₈₉	1	2	yes	complemented
169	C₈×D₄	—	2	2	no	—
170	C₈×D₄	Dic₈₉	1	2	yes	complemented
171	#431	—	2	1	no	—
172	#431	—	89	1	no	—
173	#431	C₂×D₈₉	1	1	yes	characteristic
174	C₂×#104	—	89	2	no	—
175	C₄×M₁₆	—	89	1	no	—
176	C₂×C₄×C₈	—	2	1	no	—
177	C₂×C₄×C₈	—	89	1	no	—
178	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
179	C₄³	—	89	1	no	—
180	#339	—	89	2	no	—
181	#336	—	178	2	no	—
182	Q₈⋊C₈	—	89	2	no	—
183	D₄⋊C₈	—	89	2	no	—
184	#322	—	178	1	no	—
185	C₈²	—	178	1	no	—
186	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
187	C₂×M₁₆	—	89	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	178	1	no	—
190	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
191	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
192	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
196	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
197	#117	—	2	1	no	—
198	#117	C₄×D₈₉	1	1	yes	characteristic
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	89	2	no	—
201	C₂×C₄²	—	178	1	no	—
202	C₂×C₄²	—	178	1	no	—
203	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	89	2	no	—
206	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
207	#104	—	89	8	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	2	2	no	—
210	C₂²⋊C₈	—	178	1	no	—
211	#97	—	178	1	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	89	2	no	—
215	C₄×C₈	—	178	1	no	—
216	C₄×C₈	—	178	1	no	—
217	C₄×C₈	—	178	1	no	—
218	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
219	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
220	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
221	#41	—	2	2	no	—
222	#41	C₈×D₈₉	1	4	yes	—
223	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	C₄×Dic₈₉	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	178	2	no	—
230	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
232	C₂²×C₄	#13083574	1	1	yes	characteristic
233	M₁₆	—	89	4	no	—
234	M₁₆	—	178	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	—	356	2	no	—
239	C₂×C₈	—	178	2	no	—
240	C₂×C₈	—	178	2	no	—
241	C₂×C₈	—	178	2	no	—
242	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
243	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
247	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	89	4	no	—
255	C₄²	—	356	2	no	—
256	C₄²	—	356	2	no	—
257	C₄²	—	178	2	no	—
258	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₄²	#13083574	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₈×Dic₈₉	1	2	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	1	no	—
268	D₄	C₈×Dic₈₉	1	2	yes	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	178	4	no	—
280	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
282	C₂×C₄	#15667971	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
287	C₂×C₄	#15668018	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	356	2	no	—
292	C₈	—	356	2	no	—
293	C₈	—	178	4	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
300	C₂²	C₈×C₂²⋊D₈₉	1	2	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	356	4	no	—
307	C₄	C₂×C₈×Dic₈₉	1	1	yes	characteristic
308	C₄	#19151553	1	1	yes	characteristic
309	C₄	#19151615	1	2	yes	—
310	C₄	C₈×C₂²⋊D₈₉	1	2	yes	—
311	C₄	#19151718	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	C₈×C₂²⋊Dic₈₉	1	1	yes	characteristic
315	C₂	#26020062	1	1	yes	characteristic
316	C₂	#26020094	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×#41×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×#41	D₈₉
hypocenter	C₈₉	#8191
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #26019997	C₂	semidirect
4. C₈×#13083558	C₂	semidirect
5. #26018088	C₂	semidirect
6. #26018084	C₂	semidirect
15. Q₈⋊#10623494	C₄	semidirect
16. D₄⋊#10623494	C₄	semidirect
20. C₈×Q₈×C₈₉	C₄	semidirect
22. C₈×D₄×C₈₉	C₄	semidirect
33. #15667939	C₈	semidirect
64. #13083558	C₂×C₈	semidirect
72. C₄×Dic₈₉	C₂×C₈	semidirect
166. C₈₉	#8191	semidirect
168. C₈×Q₈	Dic₈₉	semidirect
170. C₈×D₄	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667939	2
#26019997	2
#26018634	2
C₈×#13083558	2
#26018088	2
#26018084	2
C₈×#41×C₈₉	2
#8191	89