Subgroups — #36707814

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029128	C₂	1	2	yes	maximal, complemented
2	#26022117	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₈₉	C₂	1	2	yes	maximal, complemented
4	D₁₆×Dic₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021964	C₂	1	2	yes	maximal, complemented
7	#26021848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152081	—	2	2	no	—
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	#19151699	—	2	4	no	—
12	Q₁₆×Dic₈₉	C₂²	1	4	yes	—
13	D₈×Dic₈₉	C₂²	1	4	yes	—
14	C₂×#15667946	C₂²	1	2	yes	—
15	#19151599	—	2	2	no	—
16	#19151553	C₂²	1	1	yes	characteristic
17	#19151424	C₂²	1	2	yes	—
18	#19151424	C₂²	1	2	yes	—
19	#19151402	C₂²	1	4	yes	—
20	#19151399	C₂²	1	4	yes	—
21	#19151373	C₂²	1	2	yes	—
22	C₁₆×Dic₈₉	C₂²	1	2	yes	—
23	C₂×#135×C₈₉	C₂²	1	2	yes	—
24	#508×C₈₉	C₄	1	8	yes	complemented
25	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	—
27	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#13083558	D₄	1	1	yes	characteristic
30	Q₈×Dic₈₉	—	2	4	no	—
31	D₄×Dic₈₉	—	2	4	no	—
32	D₄×Dic₈₉	—	4	4	no	—
33	#15667971	—	4	2	no	—
34	#15667971	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	—	2	2	no	—
36	C₂×C₄⋊Dic₈₉	—	4	2	no	—
37	#15667952	—	2	2	no	—
38	C₂×#13083549	C₂³	1	2	yes	—
39	#15667946	—	2	2	no	—
40	#15667946	C₂³	1	4	yes	—
41	#15667936	—	2	4	no	—
42	#15667933	—	2	4	no	—
43	#15667922	D₄	1	2	yes	—
44	C₈×Dic₈₉	C₂³	1	2	yes	—
45	C₂×#41×C₈₉	—	2	2	no	—
46	#135×C₈₉	—	2	4	no	—
47	#135×C₈₉	C₂×C₄	1	8	yes	complemented
48	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₈₉	—	2	2	no	—
50	C₂×D₈×C₈₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	Q₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
53	SD₃₂×C₈₉	C₂×C₄	1	8	yes	complemented
54	D₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₈₉	—	4	2	no	—
59	C₂²×Dic₈₉	—	4	4	no	—
60	#13083558	—	2	2	no	—
61	C₂×#10623494	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₈₉	—	2	4	no	—
63	C₄⋊Dic₈₉	—	4	2	no	—
64	C₄⋊Dic₈₉	—	4	4	no	—
65	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
66	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
67	C₂²⋊Dic₈₉	—	4	4	no	—
68	C₂²⋊Dic₈₉	D₈	1	2	yes	—
69	C₄×Dic₈₉	—	4	4	no	—
70	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
71	#13083549	—	2	2	no	—
72	#41×C₈₉	—	2	8	no	—
73	#41×C₈₉	—	4	4	no	—
74	C₂×Q₈×C₈₉	—	2	2	no	—
75	C₂×D₄×C₈₉	—	2	2	no	—
76	C₂×D₄×C₈₉	—	4	2	no	—
77	C₂²×C₄×C₈₉	—	4	2	no	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₈₉	—	2	2	no	—
80	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
81	SD₁₆×C₈₉	—	2	8	no	—
82	D₈×C₈₉	—	2	2	no	—
83	D₈×C₈₉	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	#41	1	2	yes	—
87	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
89	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
90	C₂³×C₈₉	—	4	2	no	—
91	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₈₉	—	2	4	no	—
93	Q₈×C₈₉	—	4	2	no	—
94	D₄×C₈₉	—	2	4	no	—
95	D₄×C₈₉	—	4	8	no	—
96	D₄×C₈₉	—	4	2	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	—	4	8	no	—
99	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#135	1	2	yes	—
102	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×Dic₈₉	—	4	4	no	—
104	C₂×Dic₈₉	—	16	4	no	—
105	C₂×Dic₈₉	—	16	2	no	—
106	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
107	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
108	C₈×C₈₉	C₄×D₄	1	2	yes	—
109	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
111	#10623494	—	2	2	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	—	4	4	no	—
115	C₂²×C₈₉	—	8	2	no	—
116	C₂²×C₈₉	—	8	2	no	—
117	C₂²×C₈₉	C₄×D₈	1	2	yes	—
118	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₈₉	—	4	4	no	—
120	C₄×C₈₉	C₄×D₈	1	2	yes	—
121	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
123	Dic₈₉	—	2	2	no	—
124	Dic₈₉	—	2	2	no	—
125	Dic₈₉	—	8	4	no	—
126	#33459	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	—	2	2	no	—
128	C₂×C₈₉	—	8	4	no	—
129	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
130	C₂×C₈₉	#3415	1	1	yes	characteristic
131	C₂×C₈₉	#3417	1	1	yes	characteristic
132	C₂×#508	D₈₉	1	1	yes	characteristic
133	#2951	—	89	2	no	—
134	#2175	—	89	4	no	—
135	C₄×Q₃₂	—	89	2	no	—
136	C₄×D₁₆	—	89	2	no	—
137	C₂×#366	—	89	1	no	—
138	#2139	—	89	2	no	—
139	#2108	—	89	1	no	—
140	C₈₉	#33459	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	C₂×D₈₉	1	2	yes	—
142	#520	—	178	2	no	—
143	#508	Dic₈₉	1	8	yes	complemented
144	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
145	C₂×SD₃₂	C₂×D₈₉	1	2	yes	—
146	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
147	C₂×M₃₂	—	89	1	no	—
148	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
149	#440	—	178	4	no	—
150	C₄×Q₁₆	—	89	4	no	—
151	C₄×D₈	—	89	4	no	—
152	C₂×#107	—	89	2	no	—
153	#417	—	178	2	no	—
154	#405	—	89	1	no	—
155	#366	—	89	2	no	—
156	#366	—	89	2	no	—
157	#358	—	89	4	no	—
158	#357	—	89	4	no	—
159	#346	—	89	2	no	—
160	C₄×C₁₆	—	89	2	no	—
161	C₂×#41	—	2	2	no	—
162	#135	—	2	4	no	—
163	#135	C₂×Dic₈₉	1	8	yes	complemented
164	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×D₈	C₂²×D₈₉	1	2	yes	—
167	C₂×M₁₆	—	89	1	no	—
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	C₄×Q₈	—	178	4	no	—
170	C₄×D₄	—	356	4	no	—
171	C₄×D₄	—	178	4	no	—
172	#117	—	89	1	no	—
173	#117	—	356	2	no	—
174	C₂×C₄⋊C₄	—	356	2	no	—
175	C₂×C₄⋊C₄	—	178	2	no	—
176	Q₃₂	C₂×Dic₈₉	1	4	yes	complemented
177	SD₃₂	C₂×Dic₈₉	1	8	yes	complemented
178	D₁₆	C₂×Dic₈₉	1	4	yes	complemented
179	M₃₂	—	178	2	no	—
180	C₂×C₁₆	—	89	2	no	—
181	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	complemented
182	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
183	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
184	#107	—	89	4	no	—
185	#107	—	178	2	no	—
186	#103	—	178	4	no	—
187	#102	—	178	4	no	—
188	#97	—	89	2	no	—
189	C₄×C₈	—	89	2	no	—
190	#41	—	2	8	no	—
191	#41	—	4	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	356	4	no	—
197	C₂²×C₄	—	356	2	no	—
198	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	C₂²×Dic₈₉	1	4	yes	—
201	SD₁₆	—	2	8	no	—
202	D₈	—	2	2	no	—
203	D₈	C₂²×Dic₈₉	1	4	yes	—
204	M₁₆	—	178	2	no	—
205	C₂×C₈	—	89	2	no	—
206	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
207	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
208	C₂×C₈	#13083573	1	2	yes	—
209	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
210	C₄⋊C₄	—	89	1	no	—
211	C₄⋊C₄	—	89	1	no	—
212	C₄⋊C₄	—	356	4	no	—
213	C₄⋊C₄	—	356	2	no	—
214	C₄⋊C₄	—	178	4	no	—
215	C₂²⋊C₄	—	89	2	no	—
216	C₂²⋊C₄	—	356	4	no	—
217	C₄²	—	89	2	no	—
218	C₄²	—	356	4	no	—
219	C₁₆	—	178	2	no	—
220	C₁₆	C₂²×Dic₈₉	1	2	yes	—
221	C₁₆	C₂²×Dic₈₉	1	2	yes	—
222	C₂³	—	4	2	no	—
223	C₂³	D₈×D₈₉	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	89	2	no	—
232	C₂×C₄	—	89	2	no	—
233	C₂×C₄	—	356	4	no	—
234	C₂×C₄	—	1424	4	no	—
235	C₂×C₄	—	1424	2	no	—
236	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
237	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
238	C₂×C₄	#15668031	1	2	yes	—
239	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
240	C₈	—	178	2	no	—
241	C₈	D₄×Dic₈₉	1	2	yes	—
242	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
243	C₈	C₂×#13083573	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	D₈×Dic₈₉	1	2	yes	—
250	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
251	C₄	—	4	4	no	—
252	C₄	—	178	2	no	—
253	C₄	—	178	2	no	—
254	C₄	—	712	4	no	—
255	C₄	D₈×Dic₈₉	1	2	yes	—
256	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
257	C₄	C₂×#15668031	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	8	4	no	—
260	C₂	C₂×D₈×Dic₈₉	1	1	yes	characteristic
261	C₂	#26033390	1	1	yes	characteristic
262	C₂	#26033402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₂×#508×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#508	D₈₉
hypocenter	C₈₉	#33459
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #26029128	C₂	semidirect
2. #26022117	C₂	semidirect
3. Q₃₂×Dic₈₉	C₂	semidirect
4. D₁₆×Dic₈₉	C₂	semidirect
5. C₂×#19151424	C₂	semidirect
6. #26021964	C₂	semidirect
7. #26021848	C₂	semidirect
24. #508×C₈₉	C₄	semidirect
47. #135×C₈₉	C₂×C₄	semidirect
52. Q₃₂×C₈₉	C₂×C₄	semidirect
53. SD₃₂×C₈₉	C₂×C₄	semidirect
54. D₁₆×C₈₉	C₂×C₄	semidirect
55. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
140. C₈₉	#33459	semidirect
143. #508	Dic₈₉	semidirect
163. #135	C₂×Dic₈₉	semidirect
176. Q₃₂	C₂×Dic₈₉	semidirect
177. SD₃₂	C₂×Dic₈₉	semidirect
178. D₁₆	C₂×Dic₈₉	semidirect
181. C₂×C₁₆	C₂×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029128	2
#26022117	2
Q₃₂×Dic₈₉	2
D₁₆×Dic₈₉	2
C₂×#19151424	2
#26021964	2
#26021848	2
C₂×#508×C₈₉	2
#33459	89