Subgroups — #36447478

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#26020409	C₂	1	1	yes	maximal, characteristic
3	#26020258	C₂	1	2	yes	maximal
4	#26018136	C₂	1	2	yes	maximal
5	C₂×#334×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×#15667944	—	2	1	no	—
10	C₂×#15667944	—	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	#334×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	#15667944	—	4	2	no	—
33	#15667944	—	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	2	no	—
49	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
50	C₂×Dic₁₇₈	SD₁₆	1	4	yes	—
51	C₂²×Dic₈₉	—	2	1	no	—
52	C₂²×Dic₈₉	—	2	1	no	—
53	C₂²×Dic₈₉	—	4	2	no	—
54	C₂×#10623494	—	2	2	no	—
55	C₂×#10623494	—	4	1	no	—
56	C₂×#10623494	—	4	1	no	—
57	C₄⋊Dic₈₉	—	2	1	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₄⋊Dic₈₉	—	8	2	no	—
63	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
64	#13083553	—	2	2	no	—
65	#13083553	—	2	2	no	—
66	C₄×Dic₈₉	—	2	2	no	—
67	C₄×Dic₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
71	C₂×C₈×C₈₉	—	2	1	no	—
72	C₂×C₈×C₈₉	—	2	1	no	—
73	C₂×C₈×C₈₉	—	4	2	no	—
74	C₂×C₈×C₈₉	—	4	2	no	—
75	C₂×C₈×C₈₉	—	4	2	no	—
76	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
79	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
80	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₈₉	#41	1	1	yes	characteristic
84	C₂³×C₈₉	#121	1	1	yes	characteristic
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#135	1	2	yes	—
96	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
98	C₂×Dic₈₉	—	2	2	no	—
99	C₂×Dic₈₉	—	2	4	no	—
100	C₂×Dic₈₉	—	2	2	no	—
101	C₂×Dic₈₉	—	4	2	no	—
102	C₂×Dic₈₉	—	8	2	no	—
103	C₂×Dic₈₉	—	8	2	no	—
104	C₂×Dic₈₉	—	8	2	no	—
105	Dic₁₇₈	—	2	2	no	—
106	Dic₁₇₈	—	2	4	no	—
107	Dic₁₇₈	#106	1	4	yes	complemented
108	C₈×C₈₉	—	2	2	no	—
109	C₈×C₈₉	—	8	2	no	—
110	C₈×C₈₉	#103	1	4	yes	complemented
111	#10623494	—	4	2	no	—
112	C₂²×C₈₉	#380	1	1	yes	characteristic
113	C₂²×C₈₉	#460	1	1	yes	characteristic
114	C₂²×C₈₉	#462	1	1	yes	characteristic
115	C₂²×C₈₉	#465	1	2	yes	—
116	C₂²×C₈₉	#471	1	2	yes	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
120	C₄×C₈₉	#417	1	1	yes	characteristic
121	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
122	C₄×C₈₉	#427	1	1	yes	characteristic
123	C₄×C₈₉	#465	1	2	yes	—
124	C₄×C₈₉	#475	1	1	yes	characteristic
125	C₄×C₈₉	#477	1	1	yes	characteristic
126	Dic₈₉	—	2	4	no	—
127	Dic₈₉	—	4	2	no	—
128	Dic₈₉	—	16	2	no	—
129	#8643	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	#1666	1	2	yes	—
131	C₂×C₈₉	#1672	1	2	yes	—
132	C₂×C₈₉	#1803	1	1	yes	characteristic
133	C₂×C₈₉	#1807	1	1	yes	characteristic
134	C₂×C₈₉	#1815	1	1	yes	characteristic
135	C₂×C₈×Q₈	—	89	1	no	—
136	#1844	—	89	1	no	—
137	#1803	—	89	2	no	—
138	C₂×#334	D₈₉	1	1	yes	characteristic
139	#1296	—	89	2	no	—
140	C₈₉	#8643	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄×Q₈	—	89	1	no	—
142	C₈×Q₈	—	89	4	no	—
143	C₈×Q₈	—	178	2	no	—
144	C₂×#106	—	178	1	no	—
145	C₂×#106	—	178	1	no	—
146	C₂×C₄⋊C₈	—	89	1	no	—
147	C₂×C₄⋊C₈	—	178	1	no	—
148	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
149	C₂×#103	—	178	2	no	—
150	C₂×C₄×C₈	—	178	1	no	—
151	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
152	#384	—	89	2	no	—
153	#340	—	178	2	no	—
154	#334	—	2	2	no	—
155	C₂²×Q₈	—	89	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	178	1	no	—
158	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
159	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
160	C₄×Q₈	—	89	2	no	—
161	C₄×Q₈	—	89	2	no	—
162	C₄×Q₈	—	178	1	no	—
163	C₄×Q₈	—	178	1	no	—
164	C₂×C₄⋊C₄	—	89	1	no	—
165	C₂×C₄⋊C₄	—	178	2	no	—
166	C₂×C₄⋊C₄	—	178	2	no	—
167	C₂×C₄⋊C₄	—	178	1	no	—
168	C₂×C₄⋊C₄	—	178	2	no	—
169	C₂×C₄²	—	178	1	no	—
170	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
171	#106	—	356	2	no	—
172	#106	—	356	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	89	2	no	—
176	C₄⋊C₈	—	178	4	no	—
177	C₄⋊C₈	—	178	1	no	—
178	#103	—	356	4	no	—
179	C₄×C₈	—	178	4	no	—
180	C₄×C₈	C₄×D₈₉	1	2	yes	—
181	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
182	C₂×Q₈	—	89	4	no	—
183	C₂×Q₈	—	89	2	no	—
184	C₂×Q₈	—	178	1	no	—
185	C₂×Q₈	—	178	2	no	—
186	C₂²×C₄	—	356	2	no	—
187	C₂²×C₄	—	178	1	no	—
188	C₂²×C₄	—	178	1	no	—
189	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
191	C₂²×C₄	#13083575	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	356	1	no	—
198	C₂×C₈	—	356	1	no	—
199	C₂×C₈	—	178	2	no	—
200	C₂×C₈	#13083551	1	2	yes	—
201	C₂×C₈	#13083551	1	2	yes	—
202	C₂×C₈	#13083563	1	2	yes	—
203	C₂×C₈	Dic₃₅₆	1	2	yes	—
204	C₄⋊C₄	—	89	2	no	—
205	C₄⋊C₄	—	356	2	no	—
206	C₄⋊C₄	—	356	2	no	—
207	C₄⋊C₄	—	356	2	no	—
208	C₄⋊C₄	—	356	2	no	—
209	C₄⋊C₄	—	178	2	no	—
210	C₄⋊C₄	—	178	1	no	—
211	C₄⋊C₄	—	178	2	no	—
212	C₄⋊C₄	—	712	2	no	—
213	C₄²	—	178	2	no	—
214	C₄²	—	178	2	no	—
215	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
217	C₄²	Q₈×D₈₉	1	1	yes	characteristic
218	C₄²	#13083573	1	1	yes	characteristic
219	C₂³	#15667987	1	1	yes	characteristic
220	Q₈	—	89	4	no	—
221	Q₈	—	178	2	no	—
222	Q₈	—	178	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	356	2	no	—
225	C₂×C₄	—	178	2	no	—
226	C₂×C₄	—	178	2	no	—
227	C₂×C₄	—	178	4	no	—
228	C₂×C₄	—	712	2	no	—
229	C₂×C₄	—	712	2	no	—
230	C₂×C₄	—	712	2	no	—
231	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₄	#15667970	1	1	yes	characteristic
234	C₂×C₄	#15667983	1	1	yes	characteristic
235	C₂×C₄	#15667988	1	1	yes	characteristic
236	C₂×C₄	#15667992	1	1	yes	characteristic
237	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
238	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
239	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
240	C₂×C₄	#15668033	1	2	yes	—
241	C₂×C₄	#15668040	1	1	yes	characteristic
242	C₂×C₄	#15668045	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	356	2	no	—
246	C₈	#15667937	1	4	yes	complemented
247	C₂²	#19151455	1	1	yes	characteristic
248	C₂²	#19151799	1	1	yes	characteristic
249	C₂²	#19151809	1	1	yes	characteristic
250	C₂²	#19151828	1	2	yes	—
251	C₂²	#19151862	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	356	2	no	—
255	C₄	—	1424	2	no	—
256	C₄	—	178	4	no	—
257	C₄	C₂×#15667937	1	1	yes	characteristic
258	C₄	#19151600	1	1	yes	characteristic
259	C₄	#106×D₈₉	1	1	yes	characteristic
260	C₄	#19151633	1	1	yes	characteristic
261	C₄	#19151824	1	2	yes	—
262	C₄	#19151884	1	1	yes	characteristic
263	C₄	#19151894	1	1	yes	characteristic
264	C₂	#26019601	1	2	yes	—
265	C₂	#26019635	1	2	yes	—
266	C₂	#26020258	1	1	yes	characteristic
267	C₂	#26020274	1	1	yes	characteristic
268	C₂	#26020308	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	#8643
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₇₈	C₄	semidirect
107. Dic₁₇₈	#106	semidirect
110. C₈×C₈₉	#103	semidirect
140. C₈₉	#8643	semidirect
246. 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
#26020409	2
#26020258	2
#26018136	2
C₂×#334×C₈₉	2
#8643	89