Subgroups — #36712012

Order 22784.

464 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032621	C₂	1	2	yes	maximal
2	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022461	C₂	1	2	yes	maximal, complemented
4	#26022455	C₂	1	2	yes	maximal, complemented
5	#26022417	C₂	1	2	yes	maximal, complemented
6	#26022360	C₂	1	2	yes	maximal, complemented
7	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022040	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic
10	#19152288	—	2	2	no	—
11	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
12	#19152000	C₂²	1	2	yes	—
13	#19151940	—	2	2	no	—
14	#19151929	—	2	2	no	—
15	#19151906	—	2	2	no	—
16	#19151839	C₂²	1	2	yes	—
17	#19151839	C₂²	1	2	yes	—
18	C₄×Dic₃₅₆	C₂²	1	2	yes	—
19	#19151636	C₂²	1	2	yes	—
20	C₂×#15667946	C₂²	1	2	yes	—
21	#19151426	C₂²	1	2	yes	—
22	#19151426	C₂²	1	2	yes	—
23	#19151424	C₂²	1	2	yes	—
24	#19151424	C₂²	1	1	yes	characteristic
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	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
31	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
34	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₃₅₆	C₂³	1	2	yes	—
36	#15668005	—	4	2	no	—
37	#15668004	—	2	2	no	—
38	#15668002	—	4	2	no	—
39	#15667993	—	4	2	no	—
40	#15667991	—	2	2	no	—
41	#15667991	—	2	2	no	—
42	C₄×Dic₁₇₈	—	2	2	no	—
43	#15667971	—	2	2	no	—
44	#15667968	—	4	2	no	—
45	C₂×C₄⋊Dic₈₉	—	2	2	no	—
46	Dic₇₁₂	—	2	2	no	—
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	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
57	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
58	M₃₂×C₈₉	—	2	4	no	—
59	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
64	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₇₈	—	2	2	no	—
66	C₂²×Dic₈₉	—	4	2	no	—
67	Dic₃₅₆	—	2	2	no	—
68	Dic₃₅₆	—	2	2	no	—
69	C₄⋊Dic₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	—	2	2	no	—
72	C₄⋊Dic₈₉	—	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	C₂²⋊Dic₈₉	—	4	2	no	—
79	C₂²⋊Dic₈₉	—	4	2	no	—
80	C₂²⋊Dic₈₉	—	4	2	no	—
81	C₄×Dic₈₉	—	4	2	no	—
82	C₄×Dic₈₉	—	4	2	no	—
83	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
91	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₈₉	—	2	2	no	—
94	C₁₆×C₈₉	—	2	1	no	—
95	C₁₆×C₈₉	#41	1	2	yes	—
96	C₂³×C₈₉	C₂×D₈	1	2	yes	—
97	D₄×C₈₉	—	4	1	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×Dic₈₉	—	4	2	no	—
105	C₂×Dic₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	8	4	no	—
109	Dic₁₇₈	—	4	2	no	—
110	Dic₁₇₈	—	4	2	no	—
111	C₈×C₈₉	—	2	2	no	—
112	C₈×C₈₉	—	2	1	no	—
113	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
114	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂²×C₈₉	—	2	2	no	—
116	C₂²×C₈₉	—	2	2	no	—
117	C₂²×C₈₉	—	2	2	no	—
118	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	—	2	1	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	#547	1	1	yes	characteristic
123	C₄×C₈₉	#578	1	1	yes	characteristic
124	Dic₈₉	—	8	2	no	—
125	Dic₈₉	—	8	2	no	—
126	Dic₈₉	—	8	2	no	—
127	Dic₈₉	—	8	2	no	—
128	#33919	—	89	1	no	maximal, Sylow
129	C₂×C₈₉	—	4	2	no	—
130	C₂×C₈₉	#3334	1	1	yes	characteristic
131	C₂×C₈₉	#3414	1	1	yes	characteristic
132	C₂×C₈₉	#3417	1	1	yes	characteristic
133	#3336	—	89	2	no	—
134	#2253	—	89	1	no	—
135	#2237	—	89	2	no	—
136	#2236	—	89	2	no	—
137	#2229	—	89	2	no	—
138	#2219	—	89	2	no	—
139	C₄×Q₃₂	—	89	1	no	—
140	#2169	D₈₉	1	1	yes	characteristic
141	#2159	—	89	2	no	—
142	C₈₉	#33919	1	1	yes	characteristic, complemented, Sylow
143	#552	—	178	2	no	—
144	C₂×Q₃₂	—	89	1	no	—
145	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
146	#499	—	89	2	no	—
147	#485	—	178	2	no	—
148	#483	—	178	2	no	—
149	#479	—	178	2	no	—
150	#467	—	89	2	no	—
151	#467	—	89	2	no	—
152	C₄×Q₁₆	—	89	2	no	—
153	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
154	#427	—	89	2	no	—
155	C₂×#107	—	89	2	no	—
156	#367	—	89	2	no	—
157	#367	—	89	2	no	—
158	#366	—	89	1	no	—
159	#366	—	89	2	no	—
160	#366	—	89	1	no	—
161	#366	—	89	1	no	—
162	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
163	#358	—	89	2	no	—
164	#358	—	89	2	no	—
165	#358	—	89	2	no	—
166	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
167	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
168	C₂×Q₁₆	—	89	2	no	—
169	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
170	#126	—	356	2	no	—
171	#125	—	356	2	no	—
172	#125	—	178	2	no	—
173	#123	—	356	2	no	—
174	C₂²⋊Q₈	—	178	2	no	—
175	C₂²⋊Q₈	—	178	2	no	—
176	C₄×Q₈	—	178	2	no	—
177	C₄×D₄	D₃₅₆	1	1	yes	characteristic
178	#117	—	356	2	no	—
179	#117	—	178	2	no	—
180	C₂×C₄⋊C₄	—	178	2	no	—
181	Q₃₂	—	178	2	no	—
182	M₃₂	—	2	4	no	—
183	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
185	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
186	#107	—	89	2	no	—
187	#107	—	89	2	no	—
188	#107	—	89	2	no	—
189	#107	—	178	2	no	—
190	#106	—	178	2	no	—
191	#106	—	178	2	no	—
192	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
193	#103	—	178	2	no	—
194	#103	—	178	2	no	—
195	#103	—	178	2	no	—
196	C₂²⋊C₈	D₃₅₆	1	2	yes	—
197	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₂×Q₈	—	178	2	no	—
199	C₂×D₄	D₇₁₂	1	1	yes	characteristic
200	C₂²×C₄	—	356	2	no	—
201	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
202	Q₁₆	—	178	2	no	—
203	Q₁₆	—	178	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
207	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
208	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
209	C₄⋊C₄	—	356	2	no	—
210	C₄⋊C₄	—	356	2	no	—
211	C₄⋊C₄	—	356	2	no	—
212	C₄⋊C₄	—	356	2	no	—
213	C₄⋊C₄	—	356	2	no	—
214	C₄⋊C₄	—	356	2	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₄⋊C₄	—	178	2	no	—
217	C₄⋊C₄	—	178	2	no	—
218	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
219	C₂²⋊C₄	—	356	2	no	—
220	C₂²⋊C₄	—	356	2	no	—
221	C₂²⋊C₄	—	356	2	no	—
222	C₂²⋊C₄	D₇₁₂	1	2	yes	—
223	C₄²	—	356	2	no	—
224	C₄²	—	356	2	no	—
225	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
226	C₁₆	—	2	2	no	—
227	C₁₆	—	2	1	no	—
228	C₁₆	#13083573	1	2	yes	—
229	C₂³	C₂×D₇₁₂	1	2	yes	—
230	Q₈	—	356	2	no	—
231	Q₈	—	356	2	no	—
232	D₄	—	4	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	356	2	no	—
236	C₂×C₄	—	356	2	no	—
237	C₂×C₄	—	356	2	no	—
238	C₂×C₄	—	356	2	no	—
239	C₂×C₄	—	712	4	no	—
240	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
241	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
242	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	2	no	—
246	C₈	C₂×#13083573	1	1	yes	characteristic
247	C₈	#15668059	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	712	2	no	—
256	C₄	—	712	2	no	—
257	C₄	—	712	2	no	—
258	C₄	—	712	2	no	—
259	C₄	#19152252	1	1	yes	characteristic
260	C₄	#19152415	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	#26032600	1	1	yes	characteristic
263	C₂	#26033388	1	1	yes	characteristic
264	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×#2169	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₄×C₈₉	D₈
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#33919
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

Complements

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

N	H ≅ G/N	type
2. #26022531	C₂	semidirect
3. #26022461	C₂	semidirect
4. #26022455	C₂	semidirect
5. #26022417	C₂	semidirect
6. #26022360	C₂	semidirect
7. C₄×Dic₇₁₂	C₂	semidirect
8. #26022040	C₂	semidirect
142. C₈₉	#33919	semidirect

Maximal subgroups

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

subgroup	index
#26032621	2
#26022531	2
#26022461	2
#26022455	2
#26022417	2
#26022360	2
C₄×Dic₇₁₂	2
#26022040	2
C₈₉×#2169	2
#33919	89