Subgroups — #36708088

Order 22784.

460 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029181	C₂	1	1	yes	maximal, characteristic
2	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022159	C₂	1	2	yes	maximal, complemented
5	C₁₆×Dic₁₇₈	C₂	1	2	yes	maximal, complemented
6	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022058	C₂	1	2	yes	maximal
8	#26022057	C₂	1	2	yes	maximal
9	#26022050	C₂	1	1	yes	maximal, characteristic
10	#26022049	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152152	C₄	1	1	yes	characteristic
13	#19151729	C₄	1	1	yes	characteristic
14	#19151727	C₄	1	2	yes	—
15	#19151725	C₄	1	1	yes	characteristic
16	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₇₈	C₂²	1	2	yes	—
18	#19151623	C₄	1	2	yes	—
19	#19151620	C₄	1	1	yes	characteristic
20	#19151552	C₂²	1	2	yes	—
21	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊#13083549	C₂²	1	2	yes	—
23	C₄⋊#13083549	C₂²	1	1	yes	characteristic
24	#19151416	C₂²	1	2	yes	—
25	#19151416	C₂²	1	1	yes	characteristic
26	#19151416	C₂²	1	2	yes	—
27	#19151416	C₂²	1	1	yes	characteristic
28	#19151377	C₂²	1	1	yes	characteristic
29	#19151377	C₂²	1	1	yes	characteristic
30	#19151377	C₂²	1	2	yes	—
31	#19151374	C₂²	1	2	yes	—
32	#19151373	C₂²	1	2	yes	—
33	C₄×#13083549	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₈₉	C₂²	1	2	yes	—
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
39	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
44	#15668011	C₈	1	1	yes	characteristic
45	#15668010	C₈	1	1	yes	characteristic
46	#15668002	C₈	1	2	yes	—
47	#15667992	C₈	1	1	yes	characteristic
48	#15667990	C₈	1	2	yes	—
49	#15667988	C₈	1	1	yes	characteristic
50	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
52	#15667969	C₂×C₄	1	2	yes	—
53	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×#13083549	C₂³	1	1	yes	characteristic
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#10623494	C₂×C₄	1	2	yes	—
59	#15667941	C₂×C₄	1	2	yes	—
60	#15667941	C₂×C₄	1	2	yes	—
61	#15667941	C₂×C₄	1	2	yes	—
62	#15667923	C₂×C₄	1	2	yes	—
63	#15667922	C₂×C₄	1	2	yes	—
64	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
67	C₈×Dic₈₉	C₂³	1	2	yes	—
68	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	2	yes	—
76	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
77	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
81	#13083557	—	4	2	no	—
82	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
83	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
84	C₂×#10623494	C₂²×C₄	1	2	yes	—
85	C₈×D₈₉	—	2	1	no	—
86	C₈×D₈₉	—	2	1	no	—
87	C₈×D₈₉	—	4	1	no	—
88	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
90	#13083553	C₂×C₈	1	2	yes	—
91	#13083553	C₂×C₈	1	2	yes	—
92	#13083553	C₂×C₈	1	2	yes	—
93	#13083553	C₂×C₈	1	2	yes	—
94	#13083553	C₂×C₈	1	2	yes	—
95	#13083551	C₂×C₈	1	2	yes	—
96	#13083551	C₂×C₈	1	2	yes	—
97	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
100	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
101	#13083549	—	2	1	no	—
102	#13083549	—	2	1	no	—
103	#13083549	—	2	2	no	—
104	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
109	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
110	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
111	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₁₆×C₈₉	—	2	2	no	—
113	C₁₆×C₈₉	—	2	1	no	—
114	C₁₆×C₈₉	—	2	1	no	—
115	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
116	Q₈×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
125	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
126	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
127	C₄×D₈₉	—	2	1	no	—
128	C₄×D₈₉	—	2	1	no	—
129	C₄×D₈₉	—	2	2	no	—
130	C₄×D₈₉	—	2	2	no	—
131	Dic₁₇₈	—	2	2	no	—
132	Dic₁₇₈	—	2	4	no	—
133	C₈×C₈₉	—	2	1	no	—
134	C₈×C₈₉	—	2	2	no	—
135	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
136	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
137	#10623494	—	2	1	no	—
138	#10623494	—	2	1	no	—
139	#10623494	—	2	2	no	—
140	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
141	C₂×D₈₉	—	2	1	no	—
142	C₂×D₈₉	—	2	1	no	—
143	C₂×D₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	#504	1	2	yes	—
148	C₄×C₈₉	#519	1	1	yes	characteristic
149	C₄×C₈₉	#568	1	1	yes	characteristic
150	Dic₈₉	—	2	2	no	—
151	Dic₈₉	—	2	2	no	—
152	Dic₈₉	—	2	1	no	—
153	Dic₈₉	—	2	1	no	—
154	Dic₈₉	#504	1	2	yes	—
155	#33500	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	#2982	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
158	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
159	D₈₉	—	4	1	no	—
160	#2960	—	89	1	no	—
161	#2183	—	89	1	no	—
162	#2183	—	89	2	no	—
163	#2183	—	89	1	no	—
164	Q₈×C₁₆	—	89	1	no	—
165	Q₈×C₁₆	—	89	2	no	—
166	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
167	#2164	—	89	2	no	—
168	#2164	—	89	2	no	—
169	#2162	—	89	1	no	—
170	#2162	—	89	1	no	—
171	C₈₉	#33500	1	1	yes	characteristic, complemented, Sylow
172	#533	—	89	1	no	—
173	#446	—	89	1	no	—
174	#446	—	89	1	no	—
175	#446	—	89	2	no	—
176	C₈×Q₈	—	89	2	no	—
177	C₈×Q₈	—	89	1	no	—
178	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
179	#424	—	89	2	no	—
180	#423	—	89	1	no	—
181	#405	—	89	2	no	—
182	C₂×C₄×C₈	—	89	1	no	—
183	C₄⋊C₁₆	—	89	1	no	—
184	C₄⋊C₁₆	—	89	1	no	—
185	C₄⋊C₁₆	—	89	2	no	—
186	C₄⋊C₁₆	—	89	2	no	—
187	C₄⋊C₁₆	—	89	1	no	—
188	C₄⋊C₁₆	—	89	2	no	—
189	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
191	C₂²⋊C₁₆	—	89	1	no	—
192	C₂²⋊C₁₆	—	89	2	no	—
193	C₂²⋊C₁₆	—	89	1	no	—
194	#346	—	89	2	no	—
195	#346	—	89	2	no	—
196	C₄×C₁₆	—	89	2	no	—
197	C₄×C₁₆	—	89	2	no	—
198	C₄×C₁₆	—	89	1	no	—
199	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
200	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
201	C₂×M₁₆	—	89	2	no	—
202	C₂²×C₈	—	89	1	no	—
203	C₂²×C₈	—	89	1	no	—
204	C₄⋊Q₈	—	89	1	no	—
205	C₄⋊Q₈	—	89	1	no	—
206	#125	—	89	2	no	—
207	C₂²⋊Q₈	—	89	1	no	—
208	C₂²⋊Q₈	—	89	2	no	—
209	C₂²⋊Q₈	—	89	1	no	—
210	C₄×Q₈	—	89	2	no	—
211	C₄×Q₈	—	89	1	no	—
212	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
213	#117	—	89	2	no	—
214	C₂×C₄²	—	89	1	no	—
215	C₂×C₁₆	—	89	1	no	—
216	C₂×C₁₆	—	89	1	no	—
217	C₂×C₁₆	—	89	2	no	—
218	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
219	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
221	C₄⋊C₈	—	89	2	no	—
222	C₄⋊C₈	—	89	2	no	—
223	C₄⋊C₈	—	89	2	no	—
224	C₄⋊C₈	—	89	1	no	—
225	C₄⋊C₈	—	89	2	no	—
226	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
227	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
228	#97	—	89	2	no	—
229	#97	—	89	2	no	—
230	C₄×C₈	—	89	1	no	—
231	C₄×C₈	—	89	1	no	—
232	C₄×C₈	—	89	2	no	—
233	C₄×C₈	—	89	1	no	—
234	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₈₉	1	2	yes	—
236	C₂×Q₈	—	89	2	no	—
237	C₂×Q₈	—	89	1	no	—
238	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
239	C₂²×C₄	—	89	1	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	89	1	no	—
242	M₁₆	—	356	2	no	—
243	C₂×C₈	—	89	1	no	—
244	C₂×C₈	—	89	1	no	—
245	C₂×C₈	—	89	2	no	—
246	C₂×C₈	—	356	1	no	—
247	C₂×C₈	—	178	1	no	—
248	C₂×C₈	—	178	1	no	—
249	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
251	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
252	C₄⋊C₄	—	89	2	no	—
253	C₄⋊C₄	—	89	2	no	—
254	C₄⋊C₄	—	89	2	no	—
255	C₄⋊C₄	—	89	2	no	—
256	C₄⋊C₄	—	89	2	no	—
257	C₄⋊C₄	—	89	2	no	—
258	C₄⋊C₄	—	89	1	no	—
259	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
260	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	89	2	no	—
262	C₂²⋊C₄	—	89	2	no	—
263	C₄²	—	89	1	no	—
264	C₄²	—	89	1	no	—
265	C₄²	—	89	1	no	—
266	C₄²	—	89	2	no	—
267	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₄²	C₂×C₄×D₈₉	1	2	yes	—
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	—	178	1	no	—
273	C₁₆	—	178	1	no	—
274	C₁₆	—	178	2	no	—
275	C₂³	—	89	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	178	4	no	—
278	Q₈	—	178	2	no	—
279	C₂×C₄	—	89	1	no	—
280	C₂×C₄	—	89	1	no	—
281	C₂×C₄	—	89	1	no	—
282	C₂×C₄	—	89	2	no	—
283	C₂×C₄	—	89	2	no	—
284	C₂×C₄	—	178	2	no	—
285	C₂×C₄	—	178	2	no	—
286	C₂×C₄	—	178	1	no	—
287	C₂×C₄	—	178	1	no	—
288	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
290	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	2	no	—
295	C₈	—	178	1	no	—
296	C₈	—	178	1	no	—
297	C₈	—	178	2	no	—
298	C₈	#15668059	1	1	yes	characteristic
299	C₈	#15668060	1	1	yes	characteristic
300	C₂²	—	178	1	no	—
301	C₂²	—	178	1	no	—
302	C₂²	—	178	1	no	—
303	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	89	2	no	—
308	C₄	—	178	1	no	—
309	C₄	—	178	2	no	—
310	C₄	—	178	1	no	—
311	C₄	—	178	2	no	—
312	C₄	#19152014	1	2	yes	—
313	C₄	#19152074	1	1	yes	characteristic
314	C₄	#19152363	1	1	yes	characteristic
315	C₂	—	356	1	no	—
316	C₂	#26029331	1	1	yes	characteristic
317	C₂	#504×D₈₉	1	1	yes	characteristic
318	C₂	C₂×#19152014	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	Q₈×C₁₆×C₈₉	C₂
prime radical	C₂²×D₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#33500
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022161	C₂	semidirect
3. #26022160	C₂	semidirect
4. #26022159	C₂	semidirect
5. C₁₆×Dic₁₇₈	C₂	semidirect
6. #26022154	C₂	semidirect
11. Q₈×C₁₆×C₈₉	C₂	semidirect
166. Q₈×C₁₆	D₈₉	semidirect
171. C₈₉	#33500	semidirect

Maximal subgroups

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

subgroup	index
#26029181	2
#26022161	2
#26022160	2
#26022159	2
C₁₆×Dic₁₇₈	2
#26022154	2
#26022058	2
#26022057	2
#26022050	2
#26022049	2
Q₈×C₁₆×C₈₉	2
#33500	89