Subgroups — #36447488

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26020412	C₂	1	1	yes	maximal, characteristic
3	#26020258	C₂	1	2	yes	maximal
4	#26018136	C₂	1	2	yes	maximal
5	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₇₈	—	2	2	no	—
8	C₈×Dic₁₇₈	C₄	1	4	yes	complemented
9	C₂×#15667946	—	2	1	no	—
10	C₂×#15667946	—	2	1	no	—
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×#15667941	—	2	1	no	—
13	C₂×#15667937	—	2	2	no	—
14	C₂×C₈×Dic₈₉	—	2	1	no	—
15	#19151468	C₂²	1	2	yes	—
16	#19151361	—	2	2	no	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#335×C₈₉	—	2	2	no	—
20	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
21	C₂²×#10623494	—	2	1	no	—
22	C₄×Dic₁₇₈	—	2	1	no	—
23	C₄×Dic₁₇₈	—	2	1	no	—
24	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₇₈	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
30	C₂×#13083553	—	2	1	no	—
31	C₂×C₄×Dic₈₉	—	2	1	no	—
32	#15667946	—	4	2	no	—
33	#15667946	—	4	2	no	—
34	C₄⋊#10623494	—	2	1	no	—
35	C₄⋊#10623494	C₂×C₄	1	2	yes	—
36	#15667941	—	2	4	no	—
37	#15667937	—	4	4	no	—
38	C₈×Dic₈₉	—	2	4	no	—
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	—	2	2	no	—
44	C₄⋊C₈×C₈₉	—	2	2	no	—
45	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₈₉	D₄	1	2	yes	—
47	C₂×Dic₁₇₈	—	2	1	no	—
48	C₂×Dic₁₇₈	—	2	1	no	—
49	C₂×Dic₁₇₈	—	2	1	no	—
50	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
51	C₂×Dic₁₇₈	D₈	1	2	yes	—
52	C₂×Dic₁₇₈	Q₁₆	1	2	yes	—
53	C₂²×Dic₈₉	—	2	1	no	—
54	C₂²×Dic₈₉	—	2	1	no	—
55	C₂²×Dic₈₉	—	4	2	no	—
56	C₂×#10623494	—	2	2	no	—
57	C₂×#10623494	—	4	1	no	—
58	C₂×#10623494	—	4	1	no	—
59	C₄⋊Dic₈₉	—	2	1	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	—	8	2	no	—
65	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
66	#13083553	—	2	2	no	—
67	#13083553	—	2	2	no	—
68	C₄×Dic₈₉	—	2	2	no	—
69	C₄×Dic₈₉	—	2	2	no	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	2	1	no	—
74	C₂×C₈×C₈₉	—	2	1	no	—
75	C₂×C₈×C₈₉	—	4	2	no	—
76	C₂×C₈×C₈₉	—	4	2	no	—
77	C₂×C₈×C₈₉	—	4	2	no	—
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
81	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
82	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₈₉	#41	1	1	yes	characteristic
86	C₂³×C₈₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
101	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
103	C₂×Dic₈₉	—	2	2	no	—
104	C₂×Dic₈₉	—	2	2	no	—
105	C₂×Dic₈₉	—	2	2	no	—
106	C₂×Dic₈₉	—	2	2	no	—
107	C₂×Dic₈₉	—	4	1	no	—
108	C₂×Dic₈₉	—	4	1	no	—
109	C₂×Dic₈₉	—	8	2	no	—
110	C₂×Dic₈₉	—	8	2	no	—
111	C₂×Dic₈₉	—	8	2	no	—
112	Dic₁₇₈	—	2	2	no	—
113	Dic₁₇₈	—	2	2	no	—
114	Dic₁₇₈	—	2	2	no	—
115	Dic₁₇₈	#107	1	4	yes	complemented
116	C₈×C₈₉	—	2	2	no	—
117	C₈×C₈₉	—	8	2	no	—
118	C₈×C₈₉	#103	1	4	yes	complemented
119	#10623494	—	4	2	no	—
120	C₂²×C₈₉	#380	1	1	yes	characteristic
121	C₂²×C₈₉	#460	1	1	yes	characteristic
122	C₂²×C₈₉	#462	1	1	yes	characteristic
123	C₂²×C₈₉	#466	1	1	yes	characteristic
124	C₂²×C₈₉	#467	1	1	yes	characteristic
125	C₂²×C₈₉	#472	1	1	yes	characteristic
126	C₂²×C₈₉	#473	1	1	yes	characteristic
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
130	C₄×C₈₉	#417	1	1	yes	characteristic
131	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
132	C₄×C₈₉	#427	1	1	yes	characteristic
133	C₄×C₈₉	#466	1	1	yes	characteristic
134	C₄×C₈₉	#467	1	1	yes	characteristic
135	C₄×C₈₉	#475	1	1	yes	characteristic
136	C₄×C₈₉	#477	1	1	yes	characteristic
137	Dic₈₉	—	2	4	no	—
138	Dic₈₉	—	4	2	no	—
139	Dic₈₉	—	16	2	no	—
140	#8645	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	#1668	1	1	yes	characteristic
142	C₂×C₈₉	#1670	1	1	yes	characteristic
143	C₂×C₈₉	#1674	1	1	yes	characteristic
144	C₂×C₈₉	#1676	1	1	yes	characteristic
145	C₂×C₈₉	#1803	1	1	yes	characteristic
146	C₂×C₈₉	#1808	1	1	yes	characteristic
147	C₂×C₈₉	#1816	1	1	yes	characteristic
148	C₂×C₈×Q₈	—	89	1	no	—
149	#1845	—	89	1	no	—
150	#1803	—	89	2	no	—
151	C₂×#335	D₈₉	1	1	yes	characteristic
152	#1296	—	89	2	no	—
153	C₈₉	#8645	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₄×Q₈	—	89	1	no	—
155	C₈×Q₈	—	89	4	no	—
156	C₈×Q₈	—	178	2	no	—
157	C₂×#107	—	178	1	no	—
158	C₂×#107	—	178	1	no	—
159	C₂×C₄⋊C₈	—	89	1	no	—
160	C₂×C₄⋊C₈	—	178	1	no	—
161	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
162	C₂×#103	—	178	2	no	—
163	C₂×C₄×C₈	—	178	1	no	—
164	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
165	#384	—	89	2	no	—
166	#340	—	178	2	no	—
167	#335	—	2	2	no	—
168	C₂²×Q₈	—	89	1	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	178	1	no	—
171	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
172	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
173	C₄×Q₈	—	89	2	no	—
174	C₄×Q₈	—	89	2	no	—
175	C₄×Q₈	—	178	1	no	—
176	C₄×Q₈	—	178	1	no	—
177	C₂×C₄⋊C₄	—	89	1	no	—
178	C₂×C₄⋊C₄	—	178	2	no	—
179	C₂×C₄⋊C₄	—	178	2	no	—
180	C₂×C₄⋊C₄	—	178	1	no	—
181	C₂×C₄⋊C₄	—	178	2	no	—
182	C₂×C₄²	—	178	1	no	—
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	#107	—	356	2	no	—
185	#107	—	356	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	89	2	no	—
189	C₄⋊C₈	—	178	4	no	—
190	C₄⋊C₈	—	178	1	no	—
191	#103	—	356	4	no	—
192	C₄×C₈	—	178	4	no	—
193	C₄×C₈	C₄×D₈₉	1	2	yes	—
194	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
195	C₂×Q₈	—	89	2	no	—
196	C₂×Q₈	—	89	2	no	—
197	C₂×Q₈	—	89	2	no	—
198	C₂×Q₈	—	178	1	no	—
199	C₂×Q₈	—	178	1	no	—
200	C₂×Q₈	—	178	1	no	—
201	C₂²×C₄	—	356	2	no	—
202	C₂²×C₄	—	178	1	no	—
203	C₂²×C₄	—	178	1	no	—
204	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
206	C₂²×C₄	#13083575	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	356	1	no	—
213	C₂×C₈	—	356	1	no	—
214	C₂×C₈	—	178	2	no	—
215	C₂×C₈	#13083551	1	2	yes	—
216	C₂×C₈	#13083551	1	2	yes	—
217	C₂×C₈	#13083563	1	2	yes	—
218	C₂×C₈	Dic₃₅₆	1	2	yes	—
219	C₄⋊C₄	—	89	2	no	—
220	C₄⋊C₄	—	356	2	no	—
221	C₄⋊C₄	—	356	2	no	—
222	C₄⋊C₄	—	356	2	no	—
223	C₄⋊C₄	—	356	2	no	—
224	C₄⋊C₄	—	178	2	no	—
225	C₄⋊C₄	—	178	1	no	—
226	C₄⋊C₄	—	178	2	no	—
227	C₄⋊C₄	—	712	2	no	—
228	C₄²	—	178	2	no	—
229	C₄²	—	178	2	no	—
230	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
232	C₄²	Q₈×D₈₉	1	1	yes	characteristic
233	C₄²	#13083573	1	1	yes	characteristic
234	C₂³	#15667987	1	1	yes	characteristic
235	Q₈	—	89	4	no	—
236	Q₈	—	178	2	no	—
237	Q₈	—	178	2	no	—
238	Q₈	—	178	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	356	1	no	—
242	C₂×C₄	—	356	1	no	—
243	C₂×C₄	—	178	2	no	—
244	C₂×C₄	—	178	2	no	—
245	C₂×C₄	—	178	2	no	—
246	C₂×C₄	—	178	2	no	—
247	C₂×C₄	—	712	2	no	—
248	C₂×C₄	—	712	2	no	—
249	C₂×C₄	—	712	2	no	—
250	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
251	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
252	C₂×C₄	#15667970	1	1	yes	characteristic
253	C₂×C₄	#15667983	1	1	yes	characteristic
254	C₂×C₄	#15667988	1	1	yes	characteristic
255	C₂×C₄	#15667992	1	1	yes	characteristic
256	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
257	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
258	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
260	C₂×C₄	#15668031	1	1	yes	characteristic
261	C₂×C₄	#15668035	1	1	yes	characteristic
262	C₂×C₄	#15668040	1	1	yes	characteristic
263	C₂×C₄	#15668045	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	8	2	no	—
266	C₈	—	356	2	no	—
267	C₈	#15667937	1	4	yes	complemented
268	C₂²	#19151455	1	1	yes	characteristic
269	C₂²	#19151799	1	1	yes	characteristic
270	C₂²	#19151809	1	1	yes	characteristic
271	C₂²	#19151835	1	1	yes	characteristic
272	C₂²	#19151840	1	1	yes	characteristic
273	C₂²	#19151869	1	1	yes	characteristic
274	C₂²	#19151874	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	356	2	no	—
278	C₄	—	1424	2	no	—
279	C₄	—	178	4	no	—
280	C₄	C₂×#15667937	1	1	yes	characteristic
281	C₄	#19151600	1	1	yes	characteristic
282	C₄	#107×D₈₉	1	1	yes	characteristic
283	C₄	#19151635	1	1	yes	characteristic
284	C₄	#19151831	1	1	yes	characteristic
285	C₄	#19151836	1	1	yes	characteristic
286	C₄	#19151884	1	1	yes	characteristic
287	C₄	#19151894	1	1	yes	characteristic
288	C₂	#26019613	1	1	yes	characteristic
289	C₂	#26019623	1	1	yes	characteristic
290	C₂	#26019647	1	1	yes	characteristic
291	C₂	#26019657	1	1	yes	characteristic
292	C₂	#26020258	1	1	yes	characteristic
293	C₂	#26020278	1	1	yes	characteristic
294	C₂	#26020313	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	#8645
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₇₈	C₄	semidirect
115. Dic₁₇₈	#107	semidirect
118. C₈×C₈₉	#103	semidirect
153. C₈₉	#8645	semidirect
267. C₈	#15667937	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₇₈	2
#26020412	2
#26020258	2
#26018136	2
C₂×#335×C₈₉	2
#8645	89