Subgroups — #36710856

Order 22784.

464 conjugacy classes of subgroups.

The subgroup lattice has 296 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	#26022416	C₂	1	2	yes	maximal, complemented
3	#26022413	C₂	1	2	yes	maximal, complemented
4	#26022358	C₂	1	2	yes	maximal
5	#26022341	C₂	1	2	yes	maximal
6	#26022302	C₂	1	2	yes	maximal, complemented
7	#26022299	C₂	1	2	yes	maximal, complemented
8	#26022011	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#3177	C₂	1	1	yes	maximal, characteristic
10	#19152210	—	2	1	no	—
11	C₂×#15667958	C₂²	1	2	yes	—
12	C₂×#15667958	C₂²	1	2	yes	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
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	C₄×Dic₃₅₆	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151423	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	2	yes	—
25	C₄⋊#13083549	C₂²	1	2	yes	—
26	#19151402	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151401	C₂²	1	2	yes	—
29	#19151401	C₂²	1	2	yes	—
30	#541×C₈₉	—	2	1	no	—
31	#499×C₈₉	C₂²	1	1	yes	characteristic
32	#494×C₈₉	C₂²	1	1	yes	characteristic
33	#491×C₈₉	—	2	1	no	—
34	#470×C₈₉	—	2	1	no	—
35	#467×C₈₉	C₂²	1	2	yes	—
36	#441×C₈₉	—	2	1	no	—
37	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₅₆	—	2	1	no	—
40	C₂×Dic₃₅₆	C₂³	1	2	yes	—
41	#15668012	—	2	1	no	—
42	#15668005	—	4	2	no	—
43	#15668004	—	2	1	no	—
44	#15668002	—	4	2	no	—
45	#15667991	—	2	2	no	—
46	C₄×Dic₁₇₈	—	2	2	no	—
47	#15667958	—	4	2	no	—
48	#15667958	—	4	2	no	—
49	#15667952	—	4	1	no	—
50	C₂×#13083549	—	4	1	no	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	C₂×#13083549	C₂³	1	2	yes	—
53	#15667946	C₂³	1	2	yes	—
54	#15667944	—	2	1	no	—
55	#15667937	—	2	2	no	—
56	#15667937	—	2	2	no	—
57	C₂×Q₁₆×C₈₉	—	2	1	no	—
58	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
59	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₈₉	—	2	1	no	—
62	#126×C₈₉	—	4	2	no	—
63	#125×C₈₉	—	2	1	no	—
64	#125×C₈₉	—	4	2	no	—
65	C₂²⋊Q₈×C₈₉	—	2	2	no	—
66	C₄×Q₈×C₈₉	—	2	2	no	—
67	#117×C₈₉	D₄	1	1	yes	characteristic
68	#107×C₈₉	C₂³	1	2	yes	—
69	#106×C₈₉	—	2	1	no	—
70	#103×C₈₉	—	2	2	no	—
71	#103×C₈₉	—	2	2	no	—
72	#97×C₈₉	D₄	1	1	yes	characteristic
73	#97×C₈₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₇₈	—	2	2	no	—
77	Dic₃₅₆	—	2	2	no	—
78	Dic₃₅₆	—	2	2	no	—
79	Dic₃₅₆	—	4	2	no	—
80	C₄⋊Dic₈₉	—	2	2	no	—
81	C₄⋊Dic₈₉	—	4	2	no	—
82	#13083553	—	4	2	no	—
83	#13083553	—	4	2	no	—
84	#13083553	—	4	2	no	—
85	C₂²⋊Dic₈₉	—	4	2	no	—
86	C₄×Dic₈₉	—	4	2	no	—
87	#13083549	—	2	2	no	—
88	#13083549	—	2	2	no	—
89	C₂×Q₈×C₈₉	—	2	2	no	—
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₈₉	—	2	2	no	—
92	Q₁₆×C₈₉	—	2	2	no	—
93	Q₁₆×C₈₉	—	4	2	no	—
94	M₁₆×C₈₉	—	4	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	—	4	2	no	—
103	C₄⋊C₄×C₈₉	—	4	2	no	—
104	C₄⋊C₄×C₈₉	—	4	2	no	—
105	C₄⋊C₄×C₈₉	—	4	2	no	—
106	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
107	C₂²⋊C₄×C₈₉	—	4	2	no	—
108	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
109	C₄²×C₈₉	—	4	2	no	—
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₈₉	—	4	2	no	—
114	Q₈×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₈₉	—	4	2	no	—
124	C₂×Dic₈₉	—	4	2	no	—
125	Dic₁₇₈	—	4	2	no	—
126	Dic₁₇₈	—	4	2	no	—
127	C₈×C₈₉	—	2	1	no	—
128	C₈×C₈₉	—	2	1	no	—
129	C₈×C₈₉	—	2	1	no	—
130	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
131	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	1	no	—
134	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₈₉	—	2	2	no	—
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	8	2	no	—
139	C₄×C₈₉	—	8	2	no	—
140	C₄×C₈₉	#537	1	1	yes	characteristic
141	C₄×C₈₉	#575	1	1	yes	characteristic
142	Dic₈₉	—	8	2	no	—
143	Dic₈₉	—	8	2	no	—
144	#33827	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	4	1	no	—
146	C₂×C₈₉	#3245	1	1	yes	characteristic
147	C₂×C₈₉	#3417	1	2	yes	—
148	#3177	—	89	1	no	—
149	#3177	D₈₉	1	1	yes	characteristic
150	#2229	—	89	2	no	—
151	#2229	—	89	2	no	—
152	#2219	—	89	2	no	—
153	#2216	—	89	2	no	—
154	#2209	—	89	2	no	—
155	#2209	—	89	2	no	—
156	#2151	—	89	1	no	—
157	C₈₉	#33827	1	1	yes	characteristic, complemented, Sylow
158	#541	—	2	1	no	—
159	#541	—	178	1	no	—
160	C₂×Q₃₂	—	89	2	no	—
161	C₂×Q₃₂	—	89	2	no	—
162	C₂×M₃₂	—	89	1	no	—
163	C₂²×C₁₆	—	89	1	no	—
164	#499	—	89	1	no	—
165	#499	C₂×D₈₉	1	1	yes	characteristic
166	#494	—	89	1	no	—
167	#494	C₂×D₈₉	1	1	yes	characteristic
168	#491	—	2	1	no	—
169	#491	—	178	1	no	—
170	#470	—	2	1	no	—
171	#470	—	178	1	no	—
172	#467	—	89	2	no	—
173	#467	C₂×D₈₉	1	2	yes	—
174	#441	—	2	1	no	—
175	#441	—	178	1	no	—
176	C₄×Q₁₆	—	89	2	no	—
177	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
178	#405	C₂×D₈₉	1	1	yes	characteristic
179	#367	—	89	2	no	—
180	#366	—	89	2	no	—
181	C₄⋊C₁₆	—	89	2	no	—
182	C₄⋊C₁₆	—	89	2	no	—
183	#358	—	89	2	no	—
184	#358	—	89	2	no	—
185	#358	—	89	2	no	—
186	#358	—	89	2	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	89	2	no	—
189	C₂×Q₁₆	—	178	1	no	—
190	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
191	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	178	1	no	—
195	#126	—	4	2	no	—
196	#126	—	356	2	no	—
197	#125	—	2	1	no	—
198	#125	—	4	2	no	—
199	#125	—	356	2	no	—
200	#125	—	178	1	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	178	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	178	2	no	—
205	#117	C₂²⋊D₈₉	1	1	yes	characteristic
206	Q₃₂	—	356	2	no	—
207	Q₃₂	—	356	2	no	—
208	M₃₂	—	356	1	no	—
209	C₂×C₁₆	—	89	2	no	—
210	C₂×C₁₆	—	89	2	no	—
211	C₂×C₁₆	—	356	1	no	—
212	#107	—	89	2	no	—
213	#107	C₂²×D₈₉	1	2	yes	—
214	#106	—	2	1	no	—
215	#106	—	178	1	no	—
216	#103	—	2	2	no	—
217	#103	—	2	2	no	—
218	#103	—	178	2	no	—
219	#103	—	178	2	no	—
220	#97	C₂²⋊D₈₉	1	1	yes	characteristic
221	#97	C₂²⋊D₈₉	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	178	2	no	—
226	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	4	2	no	—
230	Q₁₆	—	356	2	no	—
231	Q₁₆	—	178	2	no	—
232	Q₁₆	—	178	2	no	—
233	M₁₆	—	4	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
239	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
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	—	178	2	no	—
250	C₄⋊C₄	#13083559	1	2	yes	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	356	2	no	—
253	C₂²⋊C₄	#13083559	1	2	yes	—
254	C₄²	—	4	2	no	—
255	C₄²	—	356	2	no	—
256	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₁₆	—	178	2	no	—
259	C₁₆	—	178	2	no	—
260	C₂³	C₂×#13083559	1	1	yes	characteristic
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	356	2	no	—
264	Q₈	—	356	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	356	2	no	—
270	C₂×C₄	—	356	2	no	—
271	C₂×C₄	C₂×#13083559	1	2	yes	—
272	C₂×C₄	C₂×#13083559	1	2	yes	—
273	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	#15668058	1	1	yes	characteristic
279	C₈	#15668059	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₂²×#13083559	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	2	no	—
288	C₄	—	712	2	no	—
289	C₄	—	712	2	no	—
290	C₄	#19152176	1	1	yes	characteristic
291	C₄	#19152396	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	#26031559	1	1	yes	characteristic
294	C₂	#26033403	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3177	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	#3177	D₈₉
hypocenter	C₈₉	#33827
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #26022416	C₂	semidirect
3. #26022413	C₂	semidirect
6. #26022302	C₂	semidirect
7. #26022299	C₂	semidirect
157. C₈₉	#33827	semidirect

Maximal subgroups

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

subgroup	index
#26030911	2
#26022416	2
#26022413	2
#26022358	2
#26022341	2
#26022302	2
#26022299	2
#26022011	2
C₈₉×#3177	2
#33827	89