Subgroups — #36712519

Order 22784.

504 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033241	C₂	1	2	yes	maximal, complemented
2	#26022523	C₂	1	2	yes	maximal, complemented
3	#26022513	C₂	1	1	yes	maximal, characteristic
4	#26022315	C₂	1	2	yes	maximal, complemented
5	#26022308	C₂	1	2	yes	maximal
6	#26022303	C₂	1	2	yes	maximal, complemented
7	#26022118	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152321	—	2	2	no	—
11	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
13	C₂×Dic₇₁₂	C₂²	1	2	yes	—
14	C₂×#15667957	C₂²	1	2	yes	—
15	C₂×#15667957	C₂²	1	2	yes	—
16	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
17	#19151982	C₂²	1	2	yes	—
18	#19151982	C₂²	1	2	yes	—
19	#19151978	C₂²	1	2	yes	—
20	#19151821	—	2	2	no	—
21	#19151821	—	2	2	no	—
22	#19151804	—	2	2	no	—
23	C₄×Dic₃₅₆	C₂²	1	2	yes	—
24	C₄×Dic₃₅₆	C₂²	1	2	yes	—
25	C₄×D₇₁₂	C₂²	1	2	yes	—
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151403	C₂²	1	2	yes	—
28	#19151403	C₂²	1	2	yes	—
29	#19151400	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	—	2	2	no	—
36	C₂×Dic₃₅₆	C₂³	1	2	yes	—
37	C₂×Dic₃₅₆	C₂³	1	2	yes	—
38	C₂×#13083563	—	2	2	no	—
39	C₂×#13083563	—	2	2	no	—
40	C₂×D₇₁₂	C₂³	1	2	yes	—
41	#15668012	—	2	2	no	—
42	#15668007	—	4	2	no	—
43	#15668007	—	4	2	no	—
44	#15668000	—	2	2	no	—
45	#15668000	—	2	2	no	—
46	#15667988	—	4	2	no	—
47	C₄×Dic₁₇₈	—	2	2	no	—
48	C₄×Dic₁₇₈	—	2	2	no	—
49	C₄×D₃₅₆	—	2	2	no	—
50	#15667970	—	4	2	no	—
51	Dic₇₁₂	—	2	2	no	—
52	Dic₇₁₂	—	4	1	no	—
53	Dic₇₁₂	—	4	2	no	—
54	#15667957	—	2	4	no	—
55	#15667957	—	4	2	no	—
56	D₁₄₂₄	—	4	1	no	—
57	#15667946	C₂³	1	2	yes	—
58	#15667937	—	2	2	no	—
59	#15667937	—	2	2	no	—
60	#15667934	—	2	2	no	—
61	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
63	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
66	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	2	yes	—
68	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₇₈	—	2	2	no	—
70	C₂×Dic₁₇₈	—	2	2	no	—
71	C₂×D₃₅₆	—	2	2	no	—
72	C₂×C₄×D₈₉	—	4	2	no	—
73	Dic₃₅₆	—	2	2	no	—
74	Dic₃₅₆	—	2	2	no	—
75	Dic₃₅₆	—	2	2	no	—
76	Dic₃₅₆	—	2	2	no	—
77	Dic₃₅₆	—	8	2	no	—
78	#13083563	—	8	2	no	—
79	#13083563	—	8	2	no	—
80	D₇₁₂	—	2	2	no	—
81	D₇₁₂	—	2	2	no	—
82	C₄⋊Dic₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	—	4	2	no	—
84	C₄⋊Dic₈₉	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	#13083553	—	4	2	no	—
87	#13083551	—	4	2	no	—
88	#13083551	—	4	2	no	—
89	#13083551	—	4	2	no	—
90	#13083551	—	4	2	no	—
91	#13083551	—	4	2	no	—
92	C₄×Dic₈₉	—	4	2	no	—
93	C₄×Dic₈₉	—	4	2	no	—
94	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
99	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
100	C₄²×C₈₉	C₂×D₄	1	2	yes	—
101	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₈₉	—	2	1	no	—
103	C₁₆×C₈₉	—	2	2	no	—
104	C₁₆×C₈₉	#41	1	2	yes	—
105	C₂²×D₈₉	—	4	2	no	—
106	Q₈×C₈₉	—	4	1	no	—
107	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	4	2	no	—
115	D₃₅₆	—	4	2	no	—
116	D₃₅₆	—	4	2	no	—
117	C₄×D₈₉	—	8	4	no	—
118	Dic₁₇₈	—	4	2	no	—
119	Dic₁₇₈	—	4	2	no	—
120	Dic₁₇₈	—	4	2	no	—
121	Dic₁₇₈	—	4	2	no	—
122	C₈×C₈₉	—	2	1	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
125	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
127	C₂×D₈₉	—	8	2	no	—
128	C₂×D₈₉	—	8	2	no	—
129	C₂×D₈₉	—	8	2	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	2	2	no	—
134	C₄×C₈₉	#550	1	1	yes	characteristic
135	C₄×C₈₉	#576	1	1	yes	characteristic
136	Dic₈₉	—	8	2	no	—
137	Dic₈₉	—	8	2	no	—
138	Dic₈₉	—	8	2	no	—
139	#33969	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	#3391	1	1	yes	characteristic
141	C₂×C₈₉	#3414	1	1	yes	characteristic
142	C₂×C₈₉	#3417	1	1	yes	characteristic
143	D₈₉	—	16	2	no	—
144	#3393	—	89	2	no	—
145	#2251	—	89	2	no	—
146	#2249	—	89	1	no	—
147	#2211	—	89	2	no	—
148	#2210	—	89	2	no	—
149	#2209	—	89	2	no	—
150	#2183	D₈₉	1	1	yes	characteristic, complemented
151	#2175	—	89	2	no	—
152	C₄×Q₃₂	—	89	1	no	—
153	C₈₉	#33969	1	1	yes	characteristic, complemented, Sylow
154	#556	—	178	2	no	—
155	C₂×Q₃₂	—	89	1	no	—
156	C₂×Q₃₂	—	89	2	no	—
157	C₂×Q₃₂	—	89	1	no	—
158	C₂×SD₃₂	—	89	2	no	—
159	C₂×SD₃₂	—	89	2	no	—
160	C₂×D₁₆	—	89	1	no	—
161	#495	—	89	2	no	—
162	#495	—	89	2	no	—
163	#494	—	89	2	no	—
164	#464	—	178	2	no	—
165	#464	—	178	2	no	—
166	#461	—	178	2	no	—
167	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
168	C₄×Q₁₆	—	89	2	no	—
169	C₄×Q₁₆	—	89	2	no	—
170	C₄×D₈	—	89	2	no	—
171	#366	—	89	1	no	—
172	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
173	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
174	#358	—	89	2	no	—
175	#358	—	89	2	no	—
176	#357	—	89	2	no	—
177	#346	C₂×D₈₉	1	2	yes	—
178	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
179	C₂×Q₁₆	—	89	2	no	—
180	C₂×Q₁₆	—	89	2	no	—
181	C₂×Q₁₆	—	178	2	no	—
182	C₂×SD₁₆	—	178	2	no	—
183	C₂×SD₁₆	—	178	2	no	—
184	C₂×D₈	—	89	2	no	—
185	C₄⋊Q₈	—	178	2	no	—
186	#126	—	356	2	no	—
187	#126	—	356	2	no	—
188	#124	—	178	2	no	—
189	#124	—	178	2	no	—
190	C₂²⋊Q₈	—	356	2	no	—
191	C₄×Q₈	—	178	2	no	—
192	C₄×Q₈	—	178	2	no	—
193	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
194	C₄×D₄	—	178	2	no	—
195	#117	—	356	2	no	—
196	Q₃₂	—	356	1	no	—
197	Q₃₂	—	356	2	no	—
198	Q₃₂	—	178	2	no	—
199	SD₃₂	—	356	2	no	—
200	SD₃₂	—	178	4	no	—
201	D₁₆	—	356	1	no	—
202	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
203	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
204	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
205	#107	—	89	2	no	—
206	C₄⋊C₈	D₃₅₆	1	2	yes	—
207	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
208	#103	—	178	2	no	—
209	#103	—	178	2	no	—
210	#102	—	178	2	no	—
211	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₈₉	1	2	yes	—
213	C₂×Q₈	—	178	2	no	—
214	C₂×Q₈	—	178	2	no	—
215	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
216	C₂×D₄	—	178	2	no	—
217	C₂²×C₄	—	356	2	no	—
218	Q₁₆	—	178	2	no	—
219	Q₁₆	—	178	2	no	—
220	Q₁₆	—	178	2	no	—
221	Q₁₆	—	178	2	no	—
222	Q₁₆	—	712	2	no	—
223	SD₁₆	—	712	2	no	—
224	SD₁₆	—	712	2	no	—
225	D₈	—	178	2	no	—
226	D₈	—	178	2	no	—
227	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
228	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
230	C₄⋊C₄	—	356	2	no	—
231	C₄⋊C₄	—	356	2	no	—
232	C₄⋊C₄	—	356	2	no	—
233	C₄⋊C₄	—	356	2	no	—
234	C₄⋊C₄	—	178	2	no	—
235	C₄⋊C₄	D₇₁₂	1	2	yes	—
236	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
237	C₂²⋊C₄	—	356	2	no	—
238	C₂²⋊C₄	—	356	2	no	—
239	C₂²⋊C₄	—	356	2	no	—
240	C₂²⋊C₄	—	356	2	no	—
241	C₂²⋊C₄	—	356	2	no	—
242	C₄²	—	356	2	no	—
243	C₄²	—	356	2	no	—
244	C₄²	C₂×D₃₅₆	1	2	yes	—
245	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	#13083575	1	2	yes	—
249	C₂³	—	356	2	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	356	2	no	—
252	Q₈	—	356	2	no	—
253	Q₈	—	356	2	no	—
254	Q₈	—	356	2	no	—
255	D₄	—	356	2	no	—
256	D₄	—	356	2	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	356	2	no	—
260	C₂×C₄	—	712	4	no	—
261	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
262	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
263	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
264	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
265	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	C₂×#13083575	1	1	yes	characteristic
269	C₈	#15668058	1	1	yes	characteristic
270	C₂²	—	712	2	no	—
271	C₂²	—	712	2	no	—
272	C₂²	—	712	2	no	—
273	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	712	2	no	—
279	C₄	—	712	2	no	—
280	C₄	—	712	2	no	—
281	C₄	#19152270	1	1	yes	characteristic
282	C₄	#19152403	1	1	yes	characteristic
283	C₂	—	1424	2	no	—
284	C₂	#26033227	1	1	yes	characteristic
285	C₂	#26033388	1	1	yes	characteristic
286	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	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₄₂₄	C₂²
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33969
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
1. #26033241	C₂	semidirect
2. #26022523	C₂	semidirect
4. #26022315	C₂	semidirect
6. #26022303	C₂	semidirect
7. #26022118	C₂	semidirect
8. C₄×Dic₇₁₂	C₂	semidirect
9. C₈₉×#2183	C₂	semidirect
150. #2183	D₈₉	semidirect
153. C₈₉	#33969	semidirect

Maximal subgroups

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

subgroup	index
#26033241	2
#26022523	2
#26022513	2
#26022315	2
#26022308	2
#26022303	2
#26022118	2
C₄×Dic₇₁₂	2
C₈₉×#2183	2
#33969	89