Subgroups — #36712407

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033144	C₂	1	2	yes	maximal
2	#26022538	C₂	1	2	yes	maximal
3	#26022531	C₂	1	1	yes	maximal, characteristic
4	#26022417	C₂	1	2	yes	maximal
5	#26022417	C₂	1	2	yes	maximal
6	#26022405	C₂	1	2	yes	maximal
7	#26022124	C₂	1	2	yes	maximal
8	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic
10	#19152326	—	2	2	no	—
11	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
12	#19152003	C₂²	1	2	yes	—
13	#19152000	C₂²	1	2	yes	—
14	#19152000	C₂²	1	2	yes	—
15	#19151906	—	2	2	no	—
16	#19151906	—	2	2	no	—
17	#19151884	—	2	2	no	—
18	C₄×Dic₃₅₆	C₂²	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	C₄×Dic₃₅₆	C₂²	1	2	yes	—
21	#19151426	C₂²	1	2	yes	—
22	#19151426	C₂²	1	2	yes	—
23	#19151424	C₂²	1	1	yes	characteristic
24	#19151424	C₂²	1	2	yes	—
25	#19151424	C₂²	1	1	yes	characteristic
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151403	C₂²	1	2	yes	—
28	#19151403	C₂²	1	2	yes	—
29	#19151403	C₂²	1	2	yes	—
30	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
32	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	#346×C₈₉	C₂²	1	2	yes	—
34	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₃₅₆	C₂³	1	2	yes	—
36	#15668012	—	2	2	no	—
37	#15668010	—	4	2	no	—
38	#15668004	—	2	2	no	—
39	#15668004	—	2	2	no	—
40	#15668002	—	4	2	no	—
41	#15668002	—	4	2	no	—
42	C₄×Dic₁₇₈	—	2	2	no	—
43	C₄×Dic₁₇₈	—	2	2	no	—
44	C₄×Dic₁₇₈	—	2	2	no	—
45	#15667977	—	4	2	no	—
46	Dic₇₁₂	Q₈	1	4	yes	complemented
47	#15667946	—	2	2	no	—
48	#15667946	C₂³	1	2	yes	—
49	#15667946	C₂³	1	2	yes	—
50	#15667946	C₂³	1	2	yes	—
51	#15667944	—	2	2	no	—
52	#15667944	—	2	2	no	—
53	#15667937	—	2	2	no	—
54	#15667937	—	2	2	no	—
55	#15667937	—	2	2	no	—
56	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
59	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
61	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂³	1	2	yes	—
64	C₂×Dic₁₇₈	—	2	2	no	—
65	C₂×Dic₁₇₈	—	4	2	no	—
66	Dic₃₅₆	—	2	2	no	—
67	Dic₃₅₆	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	C₄⋊Dic₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	—	4	2	no	—
72	#13083553	—	4	2	no	—
73	#13083553	—	4	2	no	—
74	#13083553	—	4	2	no	—
75	#13083553	—	4	2	no	—
76	#13083553	—	4	2	no	—
77	#13083553	—	4	2	no	—
78	#13083553	—	4	2	no	—
79	C₄×Dic₈₉	—	4	2	no	—
80	C₄×Dic₈₉	—	4	2	no	—
81	C₄×Dic₈₉	—	4	2	no	—
82	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
87	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂×D₄	1	2	yes	—
90	C₁₆×C₈₉	—	2	1	no	—
91	C₁₆×C₈₉	—	2	2	no	—
92	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
93	Q₈×C₈₉	—	4	1	no	—
94	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	2	no	—
101	C₂×Dic₈₉	—	4	2	no	—
102	C₂×Dic₈₉	—	4	2	no	—
103	Dic₁₇₈	—	2	4	no	—
104	Dic₁₇₈	—	4	2	no	—
105	Dic₁₇₈	—	8	4	no	—
106	C₈×C₈₉	—	2	1	no	—
107	C₈×C₈₉	—	2	2	no	—
108	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
109	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
111	C₄×C₈₉	—	2	2	no	—
112	C₄×C₈₉	—	2	1	no	—
113	C₄×C₈₉	—	2	1	no	—
114	C₄×C₈₉	—	2	2	no	—
115	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
116	C₄×C₈₉	#578	1	1	yes	characteristic
117	Dic₈₉	—	4	4	no	—
118	Dic₈₉	—	8	2	no	—
119	Dic₈₉	—	8	2	no	—
120	Dic₈₉	—	8	2	no	—
121	#33957	—	89	1	no	maximal, Sylow
122	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
123	C₂×C₈₉	#3414	1	1	yes	characteristic
124	C₂×C₈₉	#3417	1	1	yes	characteristic
125	#3383	—	89	2	no	—
126	#2255	—	89	2	no	—
127	#2253	—	89	1	no	—
128	#2229	—	89	2	no	—
129	#2229	—	89	2	no	—
130	#2227	—	89	2	no	—
131	#2183	D₈₉	1	1	yes	characteristic
132	#2176	—	89	2	no	—
133	C₄×Q₃₂	—	89	1	no	—
134	C₈₉	#33957	1	1	yes	characteristic, complemented, Sylow
135	#557	—	178	2	no	—
136	C₂×Q₃₂	—	89	1	no	—
137	#500	—	89	2	no	—
138	#499	—	89	2	no	—
139	#499	—	89	2	no	—
140	#479	—	178	2	no	—
141	#479	—	178	2	no	—
142	#475	—	178	2	no	—
143	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
144	C₄×Q₁₆	—	89	2	no	—
145	C₄×Q₁₆	—	89	2	no	—
146	C₄×Q₁₆	—	89	2	no	—
147	#367	—	89	2	no	—
148	#367	—	89	2	no	—
149	#366	—	89	1	no	—
150	#366	—	89	1	no	—
151	#366	—	89	2	no	—
152	#366	—	89	1	no	—
153	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
154	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
155	#358	—	89	2	no	—
156	#358	—	89	2	no	—
157	#358	—	89	2	no	—
158	#346	C₂×D₈₉	1	2	yes	—
159	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
160	C₂×Q₁₆	—	89	2	no	—
161	C₄⋊Q₈	—	356	2	no	—
162	C₄⋊Q₈	—	178	2	no	—
163	#125	—	356	2	no	—
164	#125	—	356	2	no	—
165	#125	—	178	2	no	—
166	#125	—	178	2	no	—
167	C₄×Q₈	—	356	2	no	—
168	C₄×Q₈	—	178	2	no	—
169	C₄×Q₈	—	178	2	no	—
170	C₄×Q₈	—	178	2	no	—
171	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
172	Q₃₂	—	89	4	no	—
173	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
174	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
175	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
176	#107	—	89	2	no	—
177	#107	—	89	2	no	—
178	#107	—	89	2	no	—
179	#107	—	178	2	no	—
180	#106	—	178	2	no	—
181	#106	—	178	2	no	—
182	C₄⋊C₈	D₃₅₆	1	2	yes	—
183	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
184	#103	—	178	2	no	—
185	#103	—	178	2	no	—
186	#103	—	178	2	no	—
187	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₈₉	1	2	yes	—
189	C₂×Q₈	—	356	2	no	—
190	C₂×Q₈	—	178	2	no	—
191	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
192	Q₁₆	—	89	4	no	—
193	Q₁₆	—	178	2	no	—
194	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
195	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
196	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
197	C₄⋊C₄	—	356	2	no	—
198	C₄⋊C₄	—	356	2	no	—
199	C₄⋊C₄	—	356	2	no	—
200	C₄⋊C₄	—	356	2	no	—
201	C₄⋊C₄	—	356	2	no	—
202	C₄⋊C₄	—	356	2	no	—
203	C₄⋊C₄	—	356	2	no	—
204	C₄⋊C₄	—	356	2	no	—
205	C₄⋊C₄	—	178	2	no	—
206	C₄⋊C₄	—	178	2	no	—
207	C₄⋊C₄	—	178	2	no	—
208	C₄⋊C₄	D₇₁₂	1	2	yes	—
209	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
210	C₄²	—	356	2	no	—
211	C₄²	—	356	2	no	—
212	C₄²	—	356	2	no	—
213	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
214	C₄²	C₂×D₃₅₆	1	2	yes	—
215	C₁₆	—	2	1	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	Q₈×D₈₉	1	2	yes	—
218	Q₈	—	4	1	no	—
219	Q₈	—	356	2	no	—
220	Q₈	—	178	4	no	—
221	Q₈	—	712	4	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	356	2	no	—
224	C₂×C₄	—	356	2	no	—
225	C₂×C₄	—	356	2	no	—
226	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
227	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
228	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
229	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
234	C₈	#15668059	1	1	yes	characteristic
235	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	356	4	no	—
241	C₄	—	712	2	no	—
242	C₄	—	712	2	no	—
243	C₄	—	712	2	no	—
244	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
245	C₄	#19152415	1	1	yes	characteristic
246	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
247	C₂	#26033388	1	1	yes	characteristic
248	C₂	#26033405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2183	C₂
socle, prime radical	C₂²×C₈₉	C₂²×D₈
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33957
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
46. Dic₇₁₂	Q₈	semidirect
134. C₈₉	#33957	semidirect

Maximal subgroups

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

subgroup	index
#26033144	2
#26022538	2
#26022531	2
#26022417	2
#26022405	2
#26022124	2
C₄×Dic₇₁₂	2
C₈₉×#2183	2
#33957	89