Subgroups — #36444544

Order 22784.

630 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151655	C₂	1	1	yes	maximal, characteristic
2	#26022659	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal, complemented
4	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667971	—	2	1	no	—
6	#19151655	—	2	2	no	—
7	#19151655	C₄	1	4	yes	complemented
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₂²⋊#10623494	—	2	1	no	—
10	C₂×#15667923	—	2	1	no	—
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×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#431×C₈₉	—	2	2	no	—
19	#431×C₈₉	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
22	C₂×C₄×C₈×C₈₉	—	2	1	no	—
23	C₂²×#10623494	—	2	2	no	—
24	#15667971	—	4	4	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄×Dic₈₉	—	2	2	no	—
30	C₄⋊#10623494	—	2	2	no	—
31	C₄⋊#10623494	C₂×C₄	1	4	yes	—
32	C₂²⋊#10623494	—	2	4	no	—
33	#15667923	—	2	4	no	—
34	#15667919	—	2	2	no	—
35	#15667919	—	2	2	no	—
36	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	D₄	1	2	yes	—
46	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	—	2	2	no	—
49	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₈₉	—	2	4	no	—
51	C₄×C₈×C₈₉	—	2	4	no	—
52	C₂²×Dic₈₉	—	2	4	no	—
53	C₂×#10623494	—	2	4	no	—
54	C₂×#10623494	—	4	4	no	—
55	C₄⋊Dic₈₉	—	4	4	no	—
56	C₄⋊Dic₈₉	—	4	4	no	—
57	C₄⋊Dic₈₉	—	4	4	no	—
58	C₂²⋊Dic₈₉	—	4	4	no	—
59	C₄×Dic₈₉	—	4	4	no	—
60	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	—	2	4	no	—
62	C₂²×C₄×C₈₉	—	2	2	no	—
63	C₂²×C₄×C₈₉	—	2	2	no	—
64	C₂²×C₄×C₈₉	—	4	2	no	—
65	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	—	2	4	no	—
78	C₂×C₈×C₈₉	—	4	2	no	—
79	C₂×C₈×C₈₉	—	4	2	no	—
80	C₄²×C₈₉	C₄²	1	2	yes	—
81	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₂³×C₈₉	—	2	2	no	—
91	C₂³×C₈₉	—	2	2	no	—
92	C₂³×C₈₉	—	2	1	no	—
93	C₂³×C₈₉	—	2	1	no	—
94	C₂³×C₈₉	#95	1	2	yes	—
95	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	4	no	—
102	C₂×C₄×C₈₉	—	2	4	no	—
103	C₂×C₄×C₈₉	—	2	4	no	—
104	C₂×C₄×C₈₉	—	2	4	no	—
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	#95	1	2	yes	—
108	C₂×C₄×C₈₉	#95	1	2	yes	—
109	C₂×C₄×C₈₉	#95	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#135	1	4	yes	—
117	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₈₉	#137	1	2	yes	—
119	C₂×Dic₈₉	—	4	4	no	—
120	C₂×Dic₈₉	—	4	4	no	—
121	C₂×Dic₈₉	—	4	4	no	—
122	C₈×C₈₉	—	4	4	no	—
123	#10623494	—	4	4	no	—
124	C₂²×C₈₉	—	2	2	no	—
125	C₂²×C₈₉	—	2	2	no	—
126	C₂²×C₈₉	—	2	2	no	—
127	C₂²×C₈₉	—	4	2	no	—
128	C₂²×C₈₉	—	4	2	no	—
129	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
130	C₂²×C₈₉	#413	1	2	yes	—
131	C₂²×C₈₉	#416	1	2	yes	—
132	C₂²×C₈₉	#418	1	1	yes	characteristic
133	C₂²×C₈₉	#419	1	1	yes	characteristic
134	C₄×C₈₉	—	2	2	no	—
135	C₄×C₈₉	—	2	2	no	—
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	#416	1	2	yes	—
140	C₄×C₈₉	#417	1	2	yes	—
141	C₄×C₈₉	#427	1	2	yes	—
142	C₄×C₈₉	#428	1	2	yes	—
143	Dic₈₉	—	8	4	no	—
144	#8046	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	4	2	no	—
146	C₂×C₈₉	#1553	1	2	yes	—
147	C₂×C₈₉	#1554	1	2	yes	—
148	C₂×C₈₉	#1735	1	1	yes	characteristic
149	C₂×C₈₉	#1736	1	1	yes	characteristic
150	C₂×C₈₉	#1742	1	1	yes	characteristic
151	C₂×#432	—	89	1	no	—
152	C₂×#431	D₈₉	1	1	yes	characteristic
153	#2285	—	89	1	no	—
154	#1296	—	89	4	no	—
155	C₈₉	#8046	1	1	yes	characteristic, complemented, Sylow
156	C₂×#117	—	178	1	no	—
157	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
158	#432	—	89	4	no	—
159	#432	—	178	2	no	—
160	#431	—	2	2	no	—
161	#431	Dic₈₉	1	4	yes	complemented
162	C₂×C₄⋊C₈	—	89	2	no	—
163	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
164	C₂×C₂²⋊C₈	—	2	1	no	—
165	C₂×C₂²⋊C₈	—	178	1	no	—
166	C₂×#97	—	178	1	no	—
167	C₂×C₄×C₈	—	2	1	no	—
168	#384	—	89	2	no	—
169	#382	—	178	1	no	—
170	#380	—	89	2	no	—
171	#380	—	178	1	no	—
172	C₄×C₄⋊C₄	—	89	2	no	—
173	C₄×C₄⋊C₄	—	178	1	no	—
174	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
175	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	178	2	no	—
179	#117	—	356	4	no	—
180	C₂×C₄⋊C₄	—	89	4	no	—
181	C₂×C₄⋊C₄	—	178	2	no	—
182	C₂×C₄⋊C₄	—	178	2	no	—
183	C₂×C₂²⋊C₄	—	178	2	no	—
184	C₂×C₄²	—	178	2	no	—
185	C₂×C₄²	Dic₁₇₈	1	2	yes	—
186	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₈₉	1	2	yes	—
188	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	89	4	no	—
195	C₄⋊C₈	—	178	2	no	—
196	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
197	C₂²⋊C₈	—	2	4	no	—
198	C₂²⋊C₈	—	178	4	no	—
199	#97	—	178	4	no	—
200	C₄×C₈	—	2	4	no	—
201	#95	—	178	2	no	—
202	#95	—	178	2	no	—
203	C₂⁴	#13083551	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	178	4	no	—
209	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
210	C₂²×C₄	#13083551	1	1	yes	characteristic
211	C₂²×C₄	#13083551	1	1	yes	characteristic
212	C₂²×C₄	#13083551	1	1	yes	characteristic
213	C₂²×C₄	#13083551	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
215	C₂²×C₄	#13083553	1	2	yes	—
216	C₂²×C₄	#13083553	1	2	yes	—
217	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	356	4	no	—
225	C₂×C₈	—	178	4	no	—
226	C₄⋊C₄	—	356	4	no	—
227	C₄⋊C₄	—	356	4	no	—
228	C₄⋊C₄	—	356	4	no	—
229	C₂²⋊C₄	—	356	4	no	—
230	C₄²	—	356	4	no	—
231	C₄²	C₄×Dic₈₉	1	2	yes	—
232	C₄²	#13083551	1	2	yes	—
233	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
234	C₄²	#13083553	1	2	yes	—
235	C₄²	#13083553	1	2	yes	—
236	C₄²	C₄⋊Dic₈₉	1	2	yes	—
237	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
240	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	#15667919	1	2	yes	—
246	C₂³	C₂×#13083551	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	356	4	no	—
259	C₂×C₄	—	356	4	no	—
260	C₂×C₄	—	356	4	no	—
261	C₂×C₄	#15667919	1	2	yes	—
262	C₂×C₄	#15667919	1	2	yes	—
263	C₂×C₄	#15667919	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
267	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
268	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
270	C₂×C₄	#15668034	1	4	yes	—
271	C₂×C₄	#15668037	1	2	yes	—
272	C₂×C₄	#15668042	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	356	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₂×#15667919	1	1	yes	characteristic
281	C₂²	#19151580	1	2	yes	—
282	C₂²	#19151595	1	2	yes	—
283	C₂²	#19151602	1	1	yes	characteristic
284	C₂²	#19151606	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	712	4	no	—
291	C₄	#19151595	1	2	yes	—
292	C₄	#19151599	1	2	yes	—
293	C₄	#19151636	1	2	yes	—
294	C₄	#19151641	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	#26018955	1	2	yes	—
297	C₂	#26018960	1	2	yes	—
298	C₂	#26019969	1	1	yes	characteristic
299	C₂	#26019973	1	1	yes	characteristic
300	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×#431×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#431	D₈₉
hypocenter	C₈₉	#8046
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #26018136	C₂	semidirect
7. #19151655	C₄	semidirect
19. #431×C₈₉	C₄	semidirect
155. C₈₉	#8046	semidirect
161. #431	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151655	2
#26022659	2
#26018136	2
C₂×#431×C₈₉	2
#8046	89