Subgroups — #36709653

Order 22784.

520 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030911	C₂	1	1	yes	maximal, characteristic
2	#26029121	C₂	1	1	yes	maximal, characteristic
3	#26022461	C₂	1	2	yes	maximal
4	#26022417	C₂	1	2	yes	maximal, complemented
5	#26022303	C₂	1	2	yes	maximal, complemented
6	#26022188	C₂	1	2	yes	maximal
7	#26022124	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₇₁₂	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2164	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
11	#19152210	—	2	1	no	—
12	#19152075	—	2	1	no	—
13	C₂×Dic₇₁₂	C₂²	1	2	yes	—
14	C₂×Dic₇₁₂	C₂²	1	2	yes	—
15	#19152000	C₂²	1	1	yes	characteristic
16	#19151978	C₂²	1	1	yes	characteristic
17	#19151967	—	2	1	no	—
18	#19151856	—	2	1	no	—
19	#19151839	C₂²	1	2	yes	—
20	#19151706	—	2	1	no	—
21	#19151706	—	2	1	no	—
22	#19151706	—	2	1	no	—
23	C₄×Dic₃₅₆	C₂²	1	2	yes	—
24	C₄×Dic₃₅₆	C₂²	1	2	yes	—
25	C₄×Dic₃₅₆	C₂²	1	2	yes	—
26	#19151636	C₂²	1	1	yes	characteristic
27	#19151608	—	2	1	no	—
28	#19151426	C₂²	1	2	yes	—
29	#19151424	C₂²	1	2	yes	—
30	#19151403	C₂²	1	2	yes	—
31	#19151403	C₂²	1	2	yes	—
32	#19151403	C₂²	1	2	yes	—
33	#19151403	C₂²	1	2	yes	—
34	#405×C₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
36	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
37	#346×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×Dic₁₇₈	—	2	1	no	—
40	C₂×Dic₃₅₆	—	2	4	no	—
41	C₂×Dic₃₅₆	—	2	1	no	—
42	C₂×Dic₃₅₆	—	2	1	no	—
43	C₂×Dic₃₅₆	C₂³	1	2	yes	—
44	C₂×Dic₃₅₆	C₂³	1	2	yes	—
45	#15668012	—	2	1	no	—
46	#15668005	—	4	2	no	—
47	#15668004	—	2	1	no	—
48	#15668002	—	4	2	no	—
49	#15667991	—	2	2	no	—
50	C₄×Dic₁₇₈	—	2	2	no	—
51	C₄×Dic₁₇₈	—	2	2	no	—
52	C₄×Dic₁₇₈	—	2	2	no	—
53	#15667977	—	4	2	no	—
54	#15667971	—	2	1	no	—
55	#15667968	—	4	2	no	—
56	Dic₇₁₂	—	2	4	no	—
57	Dic₇₁₂	—	2	4	no	—
58	#15667946	C₂³	1	2	yes	—
59	#15667946	C₂³	1	2	yes	—
60	#15667944	—	2	1	no	—
61	#15667944	—	2	1	no	—
62	#15667937	—	2	2	no	—
63	#15667937	—	2	2	no	—
64	#15667937	—	2	2	no	—
65	#15667937	—	2	2	no	—
66	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	#117×C₈₉	D₄	1	1	yes	characteristic
69	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
71	#97×C₈₉	D₄	1	1	yes	characteristic
72	#97×C₈₉	D₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₇₈	—	2	4	no	—
76	C₂×Dic₁₇₈	—	2	2	no	—
77	C₂×Dic₁₇₈	—	2	2	no	—
78	C₂×Dic₁₇₈	—	4	2	no	—
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂²×Dic₈₉	—	4	1	no	—
81	Dic₃₅₆	—	2	4	no	—
82	Dic₃₅₆	—	2	2	no	—
83	Dic₃₅₆	—	2	2	no	—
84	Dic₃₅₆	—	2	2	no	—
85	Dic₃₅₆	—	4	2	no	—
86	Dic₃₅₆	#41	1	4	yes	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	#13083553	—	4	2	no	—
91	#13083553	—	4	2	no	—
92	#13083553	—	4	2	no	—
93	#13083553	—	4	2	no	—
94	#13083553	—	4	2	no	—
95	C₂²⋊Dic₈₉	—	4	1	no	—
96	C₂²⋊Dic₈₉	—	4	2	no	—
97	C₄×Dic₈₉	—	4	2	no	—
98	C₄×Dic₈₉	—	4	2	no	—
99	C₄×Dic₈₉	—	4	1	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	M₁₆×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
109	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₈₉	—	2	2	no	—
113	C₁₆×C₈₉	—	2	2	no	—
114	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	—	4	4	no	—
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	4	2	no	—
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	Dic₁₇₈	—	2	4	no	—
127	Dic₁₇₈	—	4	4	no	—
128	Dic₁₇₈	—	4	2	no	—
129	Dic₁₇₈	—	4	2	no	—
130	Dic₁₇₈	—	4	2	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	—	2	1	no	—
134	C₈×C₈₉	C₂×#41	1	2	yes	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	2	2	no	—
137	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	—	2	2	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	#525	1	1	yes	characteristic
142	C₄×C₈₉	#575	1	1	yes	characteristic
143	Dic₈₉	—	4	4	no	—
144	Dic₈₉	—	8	2	no	—
145	Dic₈₉	—	8	2	no	—
146	Dic₈₉	—	8	2	no	—
147	#33687	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	4	1	no	—
149	C₂×C₈₉	#3100	1	1	yes	characteristic
150	C₂×C₈₉	#3417	1	2	yes	—
151	#3177	—	89	1	no	—
152	#2950	—	89	1	no	—
153	#2237	—	89	2	no	—
154	#2229	—	89	2	no	—
155	#2209	—	89	2	no	—
156	#2188	—	89	2	no	—
157	#2176	—	89	2	no	—
158	C₄×Q₃₂	—	89	2	no	—
159	#2164	D₈₉	1	1	yes	characteristic
160	C₈₉	#33687	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₁₆	—	89	1	no	—
162	#541	—	178	1	no	—
163	#519	—	178	1	no	—
164	C₂×Q₃₂	—	89	2	no	—
165	C₂×Q₃₂	—	89	2	no	—
166	#499	—	89	1	no	—
167	#494	—	89	1	no	—
168	#491	—	178	1	no	—
169	#470	—	178	1	no	—
170	#467	—	89	2	no	—
171	#441	—	178	1	no	—
172	#441	—	178	1	no	—
173	#441	—	178	1	no	—
174	C₄×Q₁₆	—	89	2	no	—
175	C₄×Q₁₆	—	89	2	no	—
176	C₄×Q₁₆	—	89	2	no	—
177	#427	—	89	1	no	—
178	#419	—	178	1	no	—
179	#405	C₂×D₈₉	1	1	yes	characteristic
180	#367	—	89	2	no	—
181	#366	—	89	2	no	—
182	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
183	#358	—	89	2	no	—
184	#358	—	89	2	no	—
185	#358	—	89	2	no	—
186	#358	—	89	2	no	—
187	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
188	#346	C₂×D₈₉	1	1	yes	characteristic
189	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₂²×Q₈	—	178	1	no	—
191	C₂×Q₁₆	—	89	2	no	—
192	C₂×Q₁₆	—	89	2	no	—
193	C₂×Q₁₆	—	178	4	no	—
194	C₂×Q₁₆	—	178	1	no	—
195	C₂×Q₁₆	—	178	1	no	—
196	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	178	1	no	—
199	#126	—	356	2	no	—
200	#125	—	356	2	no	—
201	#125	—	178	1	no	—
202	C₂²⋊Q₈	—	178	2	no	—
203	C₄×Q₈	—	356	2	no	—
204	C₄×Q₈	—	178	2	no	—
205	C₄×Q₈	—	178	2	no	—
206	C₄×Q₈	—	178	2	no	—
207	#117	—	356	2	no	—
208	#117	—	178	1	no	—
209	#117	D₃₅₆	1	1	yes	characteristic
210	Q₃₂	—	178	4	no	—
211	Q₃₂	—	178	4	no	—
212	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
213	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
214	#107	—	89	2	no	—
215	#107	—	89	2	no	—
216	#106	—	178	1	no	—
217	#106	—	178	1	no	—
218	#103	—	178	2	no	—
219	#103	—	178	2	no	—
220	#103	—	178	2	no	—
221	#103	—	178	2	no	—
222	#97	D₃₅₆	1	1	yes	characteristic
223	#97	D₃₅₆	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
226	C₂×Q₈	—	356	2	no	—
227	C₂×Q₈	—	356	1	no	—
228	C₂×Q₈	—	178	4	no	—
229	C₂×Q₈	—	178	2	no	—
230	C₂×Q₈	—	178	2	no	—
231	C₂²×C₄	—	356	1	no	—
232	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
233	Q₁₆	—	89	4	no	—
234	Q₁₆	—	356	2	no	—
235	Q₁₆	—	178	4	no	—
236	Q₁₆	—	178	2	no	—
237	Q₁₆	—	178	2	no	—
238	Q₁₆	—	178	2	no	—
239	M₁₆	—	4	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
242	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
243	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	—	356	2	no	—
246	C₄⋊C₄	—	356	2	no	—
247	C₄⋊C₄	—	356	2	no	—
248	C₄⋊C₄	—	356	2	no	—
249	C₄⋊C₄	—	356	2	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	178	2	no	—
252	C₄⋊C₄	—	178	2	no	—
253	C₄⋊C₄	D₇₁₂	1	2	yes	—
254	C₂²⋊C₄	—	356	1	no	—
255	C₂²⋊C₄	—	356	2	no	—
256	C₂²⋊C₄	D₇₁₂	1	2	yes	—
257	C₄²	—	356	2	no	—
258	C₄²	—	356	2	no	—
259	C₄²	—	356	1	no	—
260	C₄²	—	356	2	no	—
261	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
262	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
266	Q₈	—	356	4	no	—
267	Q₈	—	356	2	no	—
268	Q₈	—	356	2	no	—
269	Q₈	—	356	2	no	—
270	Q₈	—	178	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	356	4	no	—
274	C₂×C₄	—	356	2	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
279	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
280	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	#41×D₈₉	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	356	4	no	—
293	C₄	—	712	2	no	—
294	C₄	—	712	2	no	—
295	C₄	—	712	2	no	—
296	C₄	#19152110	1	1	yes	characteristic
297	C₄	#19152398	1	1	yes	characteristic
298	C₂	—	4	1	no	—
299	C₂	#26030253	1	1	yes	characteristic
300	C₂	#26033405	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2164	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	#2164	D₈₉
hypocenter	C₈₉	#33687
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2164

Complements

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

N	H ≅ G/N	type
4. #26022417	C₂	semidirect
5. #26022303	C₂	semidirect
7. #26022124	C₂	semidirect
8. C₄×Dic₇₁₂	C₂	semidirect
160. C₈₉	#33687	semidirect

Maximal subgroups

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

subgroup	index
#26030911	2
#26029121	2
#26022461	2
#26022417	2
#26022303	2
#26022188	2
#26022124	2
C₄×Dic₇₁₂	2
C₈₉×#2164	2
#33687	89