Subgroups — #36707571

Order 22784.

528 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028674	C₂	1	1	yes	maximal, characteristic
2	#26022066	C₂	1	4	yes	maximal, complemented
3	#26022062	C₂	1	4	yes	maximal, complemented
4	C₂×#19151418	C₂	1	2	yes	maximal, complemented
5	#26021872	C₂	1	2	yes	maximal
6	#26021857	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668020	C₄	1	1	yes	characteristic
9	C₂×#15668018	C₂²	1	1	yes	characteristic
10	#19152066	C₄	1	1	yes	characteristic
11	C₂×#15667952	C₂²	1	2	yes	—
12	#19151555	C₂²	1	2	yes	—
13	#19151554	C₄	1	1	yes	characteristic
14	#19151552	C₄	1	2	yes	—
15	#19151550	C₂²	1	2	yes	—
16	M₁₆×Dic₈₉	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×D₈₉	C₄	1	2	yes	—
19	#19151418	C₂²	1	4	yes	—
20	#19151418	C₂²	1	4	yes	—
21	#19151379	C₂²	1	8	yes	—
22	#19151376	C₂²	1	4	yes	—
23	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
24	#405×C₈₉	C₂²	1	1	yes	characteristic
25	#347×C₈₉	C₂²	1	4	yes	—
26	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
27	#15668020	—	2	4	no	—
28	#15668018	—	2	2	no	—
29	#15668018	—	2	2	no	—
30	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083557	C₂×C₄	1	2	yes	—
32	C₂×#13083557	C₂³	1	2	yes	—
33	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
36	#15667977	—	2	1	no	—
37	#15667974	—	2	1	no	—
38	#15667972	—	2	2	no	—
39	#15667969	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×D₈₉	C₂×C₄	1	2	yes	—
42	#15667952	—	2	2	no	—
43	#15667952	C₂³	1	4	yes	—
44	C₂×#13083549	—	2	2	no	—
45	#15667923	C₂×C₄	1	2	yes	—
46	#15667922	C₂×C₄	1	2	yes	—
47	C₄×#10623494	C₂×C₄	1	2	yes	—
48	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
50	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
52	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₈₉	—	2	2	no	—
57	M₃₂×C₈₉	C₂³	1	4	yes	—
58	C₂×C₁₆×C₈₉	—	2	2	no	—
59	#97×C₈₉	C₂³	1	2	yes	—
60	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
61	#13083574	—	2	2	no	—
62	#13083574	—	2	2	no	—
63	C₂×Dic₁₇₈	M₁₆	1	1	yes	characteristic
64	C₂×D₃₅₆	M₁₆	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	M₁₆	1	2	yes	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	2	no	—
68	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
69	#13083558	—	2	2	no	—
70	#13083557	—	2	4	no	—
71	#13083557	—	2	4	no	—
72	C₂×#10623494	M₁₆	1	4	yes	—
73	C₂×#10623494	C₂²×C₄	1	2	yes	—
74	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
75	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
76	C₈×D₈₉	—	2	2	no	—
77	C₈×D₈₉	—	2	2	no	—
78	C₈×D₈₉	—	2	4	no	—
79	#13083553	—	2	2	no	—
80	#13083551	—	2	2	no	—
81	C₄×Dic₈₉	—	2	1	no	—
82	C₄×Dic₈₉	—	2	2	no	—
83	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
85	#13083549	—	2	4	no	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	M₁₆×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	#41	1	2	yes	—
91	C₂×C₈×C₈₉	#41	1	2	yes	—
92	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₂²⋊C₄×C₈₉	—	2	1	no	—
95	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
96	C₁₆×C₈₉	—	2	4	no	—
97	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₈₉	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	#117	1	2	yes	—
101	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
104	C₂²⋊D₈₉	—	2	4	no	—
105	C₂×Dic₈₉	—	2	1	no	—
106	C₂×Dic₈₉	—	2	1	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	C₂×M₁₆	1	2	yes	—
109	D₃₅₆	—	2	2	no	—
110	C₄×D₈₉	—	2	2	no	—
111	C₄×D₈₉	—	2	2	no	—
112	C₄×D₈₉	—	4	4	no	—
113	Dic₁₇₈	—	2	2	no	—
114	C₈×C₈₉	—	2	2	no	—
115	C₈×C₈₉	#117	1	1	yes	characteristic
116	C₈×C₈₉	#117	1	1	yes	characteristic
117	C₈×C₈₉	C₂×#41	1	2	yes	—
118	#10623494	—	2	2	no	—
119	#10623494	C₂×M₁₆	1	2	yes	—
120	#10623494	C₂×M₁₆	1	2	yes	—
121	C₂²×C₈₉	—	4	1	no	—
122	C₂²×C₈₉	#432	1	2	yes	—
123	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×D₈₉	—	2	2	no	—
125	C₂×D₈₉	—	4	2	no	—
126	C₄×C₈₉	—	4	2	no	—
127	C₄×C₈₉	#432	1	2	yes	—
128	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
129	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
130	Dic₈₉	—	2	2	no	—
131	Dic₈₉	—	2	2	no	—
132	Dic₈₉	—	4	1	no	—
133	#33428	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	—	2	1	no	—
135	C₂×C₈₉	—	2	2	no	—
136	C₂×C₈₉	C₂×#432	1	1	yes	characteristic
137	D₈₉	—	4	2	no	—
138	C₄×#131	—	89	1	no	—
139	#2166	—	89	4	no	—
140	#2165	—	89	4	no	—
141	C₂×#364	—	89	2	no	—
142	#2115	—	89	2	no	—
143	#2110	—	89	1	no	—
144	#2110	D₈₉	1	1	yes	characteristic, complemented
145	C₈₉	#33428	1	1	yes	characteristic, complemented, Sylow
146	C₂×#131	—	89	1	no	—
147	C₂×#131	—	89	1	no	—
148	C₄×#41	—	89	1	no	—
149	C₂×M₃₂	—	89	2	no	—
150	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
151	#405	—	89	1	no	—
152	#405	—	89	2	no	—
153	#405	—	89	2	no	—
154	#405	C₂×D₈₉	1	1	yes	characteristic
155	C₄×M₁₆	—	89	1	no	—
156	C₄×M₁₆	—	89	2	no	—
157	C₂×C₄×C₈	—	89	1	no	—
158	C₂×C₄×C₈	—	89	2	no	—
159	#364	—	89	4	no	—
160	#364	—	89	4	no	—
161	#349	—	89	8	no	—
162	#347	—	89	4	no	—
163	#347	C₂×D₈₉	1	4	yes	—
164	C₂×#41	—	89	1	no	—
165	#131	—	178	4	no	—
166	#131	—	178	2	no	—
167	#131	—	178	2	no	—
168	C₂×M₁₆	—	89	1	no	—
169	C₂×M₁₆	—	89	2	no	—
170	C₂×M₁₆	—	89	2	no	—
171	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
172	C₂²×C₈	—	89	1	no	—
173	C₂²×C₈	—	89	2	no	—
174	C₂²×C₈	—	89	2	no	—
175	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
176	C₄×Q₈	—	178	1	no	—
177	C₄×D₄	—	178	2	no	—
178	C₄×D₄	—	178	1	no	—
179	#117	—	89	2	no	—
180	#117	C₄×D₈₉	1	1	yes	characteristic
181	C₂×C₄²	—	89	1	no	—
182	C₂×C₄²	—	89	2	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	89	4	no	—
185	M₃₂	—	178	2	no	—
186	M₃₂	C₂²×D₈₉	1	4	yes	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	178	2	no	—
189	#97	—	89	2	no	—
190	#97	—	89	2	no	—
191	#97	C₂²×D₈₉	1	2	yes	—
192	C₄×C₈	—	89	1	no	—
193	C₄×C₈	—	89	2	no	—
194	C₄×C₈	—	89	1	no	—
195	C₄×C₈	—	89	1	no	—
196	C₄×C₈	—	89	1	no	—
197	C₄×C₈	—	89	2	no	—
198	C₄×C₈	C₄×D₈₉	1	2	yes	—
199	#41	—	178	2	no	—
200	#41	—	178	2	no	—
201	C₂×Q₈	—	89	1	no	—
202	C₂×D₄	—	89	2	no	—
203	C₂×D₄	—	89	1	no	—
204	C₂²×C₄	—	89	2	no	—
205	C₂²×C₄	—	178	2	no	—
206	C₂²×C₄	—	178	1	no	—
207	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
208	M₁₆	—	2	2	no	—
209	M₁₆	—	178	4	no	—
210	M₁₆	—	178	2	no	—
211	M₁₆	—	178	4	no	—
212	C₂×C₈	—	89	4	no	—
213	C₂×C₈	—	89	1	no	—
214	C₂×C₈	—	89	1	no	—
215	C₂×C₈	—	89	2	no	—
216	C₂×C₈	—	178	4	no	—
217	C₂×C₈	—	178	2	no	—
218	C₂×C₈	—	178	2	no	—
219	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
221	C₂×C₈	#13083574	1	2	yes	—
222	C₂×C₈	#13083574	1	2	yes	—
223	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	178	2	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	178	2	no	—
228	C₄²	—	89	1	no	—
229	C₄²	—	89	1	no	—
230	C₄²	—	178	2	no	—
231	C₄²	—	178	1	no	—
232	C₄²	C₂×C₄×D₈₉	1	2	yes	—
233	C₁₆	—	2	4	no	—
234	C₁₆	—	178	4	no	—
235	C₂³	—	89	2	no	—
236	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
237	Q₈	—	178	2	no	—
238	D₄	—	178	4	no	—
239	D₄	—	178	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	89	2	no	—
242	C₂×C₄	—	356	4	no	—
243	C₂×C₄	—	356	2	no	—
244	C₂×C₄	—	178	1	no	—
245	C₂×C₄	—	178	1	no	—
246	C₂×C₄	—	178	2	no	—
247	C₂×C₄	—	178	2	no	—
248	C₂×C₄	#15667969	1	2	yes	—
249	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	89	2	no	—
254	C₈	—	89	2	no	—
255	C₈	—	178	2	no	—
256	C₈	#15667969	1	1	yes	characteristic
257	C₈	#15667969	1	1	yes	characteristic
258	C₈	C₂×#13083574	1	2	yes	—
259	C₂²	—	4	1	no	—
260	C₂²	—	356	2	no	—
261	C₂²	—	178	2	no	—
262	C₂²	#19151653	1	2	yes	—
263	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
264	C₄	—	4	2	no	—
265	C₄	—	356	1	no	—
266	C₄	—	178	2	no	—
267	C₄	—	178	2	no	—
268	C₄	#19151653	1	2	yes	—
269	C₄	C₂×#15667969	1	1	yes	characteristic
270	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	356	2	no	—
274	C₂	C₂×#19151653	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667969
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2110	C₂
socle	C₂×C₈₉	C₂×#432
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2110	D₈₉
hypocenter	C₈₉	#33428
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. #26022066	C₂	semidirect
3. #26022062	C₂	semidirect
4. C₂×#19151418	C₂	semidirect
6. #26021857	C₂	semidirect
7. C₈₉×#2110	C₂	semidirect
144. #2110	D₈₉	semidirect
145. C₈₉	#33428	semidirect

Maximal subgroups

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

subgroup	index
#26028674	2
#26022066	2
#26022062	2
C₂×#19151418	2
#26021872	2
#26021857	2
C₈₉×#2110	2
#33428	89