Subgroups — #36445082

Order 22784.

530 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022730	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667922	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#10623494	C₂	1	1	yes	maximal, characteristic
5	#26018092	C₂	1	1	yes	maximal, characteristic
6	#26018091	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
9	C₂×#15667922	—	2	2	no	—
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₈₉	—	2	1	no	—
12	#19151505	C₄	1	2	yes	—
13	#19151497	C₄	1	2	yes	—
14	#19151484	—	2	2	no	—
15	C₄×C₄⋊Dic₈₉	—	2	1	no	—
16	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
17	#19151444	—	2	1	no	—
18	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
19	#19151352	—	2	1	no	—
20	Q₈⋊#10623494	C₄	1	4	yes	complemented
21	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	Q₈⋊C₈×C₈₉	C₄	1	4	yes	complemented
25	C₂²×#10623494	—	2	1	no	—
26	C₂²×#10623494	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₈₉	—	2	1	no	—
30	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
34	C₄⋊#10623494	—	2	1	no	—
35	C₄⋊#10623494	C₂×C₄	1	2	yes	—
36	#15667922	—	2	8	no	—
37	C₄×#10623494	—	2	1	no	—
38	C₄×#10623494	C₂×C₄	1	2	yes	—
39	C₈×Dic₈₉	—	4	2	no	—
40	#15667919	—	2	2	no	—
41	#15667919	—	2	2	no	—
42	#15667919	—	2	2	no	—
43	#15667919	—	2	1	no	—
44	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	—	2	1	no	—
46	C₂²×C₈×C₈₉	—	2	1	no	—
47	C₄×Q₈×C₈₉	—	2	1	no	—
48	C₄×Q₈×C₈₉	—	2	1	no	—
49	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	—	2	1	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	—	2	1	no	—
56	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₂²×Dic₈₉	—	2	2	no	—
60	C₂²×Dic₈₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×Dic₈₉	C₂²⋊C₄	1	2	yes	—
62	C₂×#10623494	—	2	4	no	—
63	C₂×#10623494	—	2	2	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	4	1	no	—
66	C₂×#10623494	—	4	1	no	—
67	C₄⋊Dic₈₉	—	2	2	no	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	C₄⋊Dic₈₉	—	4	4	no	—
70	C₄×Dic₈₉	—	2	2	no	—
71	C₄×Dic₈₉	—	2	2	no	—
72	C₄×Dic₈₉	—	2	1	no	—
73	C₄×Dic₈₉	—	2	1	no	—
74	C₄×Dic₈₉	—	2	1	no	—
75	C₄×Dic₈₉	—	2	1	no	—
76	C₄×Dic₈₉	—	2	4	no	—
77	C₄×Dic₈₉	—	4	1	no	—
78	C₄×Dic₈₉	—	4	1	no	—
79	C₂×Q₈×C₈₉	—	2	1	no	—
80	C₂×Q₈×C₈₉	—	2	2	no	—
81	C₂×Q₈×C₈₉	—	2	2	no	—
82	C₂×Q₈×C₈₉	C₄²	1	2	yes	—
83	C₂²×C₄×C₈₉	—	2	1	no	—
84	C₂²×C₄×C₈₉	—	2	1	no	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	—	2	4	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	4	1	no	—
92	C₂×C₈×C₈₉	—	4	1	no	—
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₄⋊C₄×C₈₉	—	2	2	no	—
96	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₄²×C₈₉	—	2	2	no	—
99	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₉	#41	1	1	yes	characteristic
102	C₄²×C₈₉	#41	1	1	yes	characteristic
103	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₈₉	—	2	2	no	—
105	Q₈×C₈₉	—	2	2	no	—
106	Q₈×C₈₉	—	4	2	no	—
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	—	4	2	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#131	1	2	yes	—
119	C₂×C₄×C₈₉	#131	1	2	yes	—
120	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
124	C₂×Dic₈₉	—	2	2	no	—
125	C₂×Dic₈₉	—	2	2	no	—
126	C₂×Dic₈₉	—	2	2	no	—
127	C₂×Dic₈₉	—	2	2	no	—
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	C₂×Dic₈₉	#104	1	4	yes	—
133	C₈×C₈₉	—	2	4	no	—
134	C₈×C₈₉	—	4	2	no	—
135	#10623494	—	2	4	no	—
136	#10623494	—	4	2	no	—
137	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₈₉	#408	1	2	yes	—
139	C₂²×C₈₉	#417	1	1	yes	characteristic
140	C₂²×C₈₉	#419	1	1	yes	characteristic
141	C₂²×C₈₉	C₂×#104	1	2	yes	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	4	2	no	—
145	C₄×C₈₉	—	4	2	no	—
146	C₄×C₈₉	#405	1	1	yes	characteristic
147	C₄×C₈₉	#405	1	1	yes	characteristic
148	C₄×C₈₉	#433	1	2	yes	—
149	C₄×C₈₉	#440	1	1	yes	characteristic
150	C₄×C₈₉	#440	1	1	yes	characteristic
151	C₄×C₈₉	#441	1	1	yes	characteristic
152	C₄×C₈₉	#441	1	1	yes	characteristic
153	Dic₈₉	—	2	4	no	—
154	Dic₈₉	—	4	2	no	—
155	Dic₈₉	—	8	2	no	—
156	#8180	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	#1483	1	2	yes	—
158	C₂×C₈₉	#1485	1	2	yes	—
159	C₂×C₈₉	#1750	1	1	yes	characteristic
160	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
161	C₂×C₈₉	#1763	1	1	yes	characteristic
162	#2295	—	89	1	no	—
163	C₄×C₄⋊C₈	—	89	1	no	—
164	C₄×#97	—	89	1	no	—
165	C₂×Q₈⋊C₈	—	89	1	no	—
166	C₂×Q₈⋊C₈	D₈₉	1	1	yes	characteristic
167	#1279	—	89	1	no	—
168	#1279	—	89	1	no	—
169	C₈₉	#8180	1	1	yes	characteristic, complemented, Sylow
170	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	89	1	no	—
172	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
173	C₂×#97	—	178	2	no	—
174	C₂×C₄×C₈	—	89	1	no	—
175	C₂×C₄×C₈	—	178	1	no	—
176	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
177	#391	—	89	2	no	—
178	#389	—	89	2	no	—
179	#387	—	178	2	no	—
180	C₄×C₄⋊C₄	—	89	1	no	—
181	C₄×C₄⋊C₄	—	178	1	no	—
182	#376	—	178	1	no	—
183	C₄³	—	89	1	no	—
184	#336	—	178	1	no	—
185	Q₈⋊C₈	—	89	4	no	—
186	Q₈⋊C₈	Dic₈₉	1	4	yes	complemented
187	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	178	1	no	—
191	C₂²×C₈	—	178	1	no	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	—	2	1	no	—
194	C₄×Q₈	C₄×D₈₉	1	2	yes	—
195	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	89	2	no	—
198	C₂×C₄⋊C₄	—	178	2	no	—
199	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	89	2	no	—
202	C₂×C₄²	—	89	2	no	—
203	C₂×C₄²	—	89	1	no	—
204	C₂×C₄²	—	89	1	no	—
205	C₂×C₄²	—	178	1	no	—
206	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	89	2	no	—
209	C₄⋊C₈	—	178	1	no	—
210	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
211	#97	—	178	8	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	89	2	no	—
214	C₄×C₈	—	356	2	no	—
215	C₄×C₈	—	178	1	no	—
216	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
217	#95	—	178	2	no	—
218	#95	—	178	2	no	—
219	#95	—	178	2	no	—
220	#95	—	178	1	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	C₄×Dic₈₉	1	2	yes	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	89	2	no	—
228	C₂²×C₄	—	89	2	no	—
229	C₂²×C₄	—	178	2	no	—
230	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	356	1	no	—
239	C₂×C₈	—	356	1	no	—
240	C₂×C₈	—	178	2	no	—
241	C₂×C₈	—	178	4	no	—
242	C₂×C₈	—	178	2	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	356	4	no	—
247	C₄⋊C₄	—	178	2	no	—
248	C₄⋊C₄	—	178	2	no	—
249	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	356	1	no	—
253	C₄²	—	356	1	no	—
254	C₄²	—	178	4	no	—
255	C₄²	—	178	2	no	—
256	C₄²	—	178	2	no	—
257	C₄²	—	178	1	no	—
258	C₄²	—	178	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	—	178	1	no	—
261	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
262	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
263	C₄²	#13083573	1	1	yes	characteristic
264	C₄²	#13083573	1	1	yes	characteristic
265	C₂³	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	—	2	2	no	—
268	Q₈	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	89	4	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	178	2	no	—
279	C₂×C₄	—	178	2	no	—
280	C₂×C₄	—	178	2	no	—
281	C₂×C₄	—	178	2	no	—
282	C₂×C₄	—	178	2	no	—
283	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
284	C₂×C₄	#15667968	1	1	yes	characteristic
285	C₂×C₄	#15667972	1	1	yes	characteristic
286	C₂×C₄	#15667972	1	1	yes	characteristic
287	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
288	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
289	C₂×C₄	#15668018	1	2	yes	—
290	C₂×C₄	#15668020	1	2	yes	—
291	C₂×C₄	#15668039	1	1	yes	characteristic
292	C₂×C₄	#15668041	1	1	yes	characteristic
293	C₂×C₄	#15668044	1	1	yes	characteristic
294	C₂×C₄	#15668044	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	4	2	no	—
297	C₈	—	356	2	no	—
298	C₈	—	178	4	no	—
299	C₂²	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
300	C₂²	#19151563	1	2	yes	—
301	C₂²	#19151598	1	1	yes	characteristic
302	C₂²	#19151605	1	1	yes	characteristic
303	C₂²	#104×D₈₉	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	356	2	no	—
309	C₄	—	178	4	no	—
310	C₄	—	712	2	no	—
311	C₄	#19151553	1	1	yes	characteristic
312	C₄	#19151556	1	1	yes	characteristic
313	C₄	#19151656	1	2	yes	—
314	C₄	#19151699	1	1	yes	characteristic
315	C₄	#19151701	1	1	yes	characteristic
316	C₄	#19151702	1	1	yes	characteristic
317	C₄	#19151705	1	1	yes	characteristic
318	C₂	#26018664	1	2	yes	—
319	C₂	#26018671	1	2	yes	—
320	C₂	#26020020	1	1	yes	characteristic
321	C₂	#104×Dic₈₉	1	1	yes	characteristic
322	C₂	#26020084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×Q₈⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×Q₈⋊C₈	D₈₉
hypocenter	C₈₉	#8180
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
20. Q₈⋊#10623494	C₄	semidirect
24. Q₈⋊C₈×C₈₉	C₄	semidirect
169. C₈₉	#8180	semidirect
186. Q₈⋊C₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022730	2
C₄⋊C₈×Dic₈₉	2
C₄×#15667922	2
C₂×Q₈⋊#10623494	2
#26018092	2
#26018091	2
C₂×Q₈⋊C₈×C₈₉	2
#8180	89