Subgroups — #36709647

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	#26029121	C₂	1	1	yes	maximal, characteristic
2	#26022457	C₂	1	2	yes	maximal
3	#26022413	C₂	1	2	yes	maximal, complemented
4	#26022299	C₂	1	2	yes	maximal, complemented
5	#26022185	C₂	1	2	yes	maximal
6	#26022121	C₂	1	2	yes	maximal, complemented
7	C₄×#15667958	C₂	1	2	yes	maximal, complemented
8	#26022059	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#3177	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
11	#19152075	—	2	1	no	—
12	C₂×#15667958	C₂²	1	2	yes	—
13	C₂×#15667958	C₂²	1	2	yes	—
14	#19151706	—	2	1	no	—
15	#19151706	—	2	1	no	—
16	C₄×Dic₃₅₆	C₂²	1	2	yes	—
17	C₄×Dic₃₅₆	C₂²	1	2	yes	—
18	#19151636	C₂²	1	1	yes	characteristic
19	#19151608	—	2	1	no	—
20	#19151425	C₂²	1	2	yes	—
21	#19151423	C₂²	1	2	yes	—
22	C₄⋊#13083549	C₂²	1	2	yes	—
23	#19151402	C₂²	1	2	yes	—
24	#19151402	C₂²	1	2	yes	—
25	#19151401	C₂²	1	2	yes	—
26	#19151401	C₂²	1	2	yes	—
27	C₂²⋊#13083549	C₂²	1	2	yes	—
28	#19151374	C₂²	1	1	yes	characteristic
29	C₄×#13083549	C₂²	1	1	yes	characteristic
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	C₂²×Dic₁₇₈	—	2	1	no	—
40	C₂×Dic₃₅₆	—	2	4	no	—
41	C₂×Dic₃₅₆	—	2	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	2	yes	—
43	C₄×Dic₁₇₈	—	2	2	no	—
44	C₄×Dic₁₇₈	—	2	2	no	—
45	#15667977	—	4	2	no	—
46	#15667971	—	2	1	no	—
47	#15667968	—	4	2	no	—
48	#15667958	—	2	4	no	—
49	#15667958	—	2	4	no	—
50	C₂×#13083549	C₂³	1	2	yes	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	#15667946	C₂³	1	2	yes	—
53	#15667944	—	2	1	no	—
54	#15667937	—	2	2	no	—
55	#15667937	—	2	2	no	—
56	C₂×Q₁₆×C₈₉	—	2	1	no	—
57	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
58	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₈₉	—	2	1	no	—
61	#126×C₈₉	—	4	2	no	—
62	#125×C₈₉	—	2	1	no	—
63	#125×C₈₉	—	4	2	no	—
64	C₂²⋊Q₈×C₈₉	—	2	2	no	—
65	C₄×Q₈×C₈₉	—	2	2	no	—
66	#117×C₈₉	D₄	1	1	yes	characteristic
67	#107×C₈₉	C₂³	1	2	yes	—
68	#106×C₈₉	—	2	1	no	—
69	#103×C₈₉	—	2	2	no	—
70	#103×C₈₉	—	2	2	no	—
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	—	4	2	no	—
78	C₂×Dic₁₇₈	—	4	1	no	—
79	C₂²×Dic₈₉	—	4	1	no	—
80	Dic₃₅₆	—	2	4	no	—
81	Dic₃₅₆	—	2	2	no	—
82	Dic₃₅₆	#41	1	4	yes	—
83	C₄⋊Dic₈₉	—	2	2	no	—
84	#13083553	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	C₂²⋊Dic₈₉	—	4	1	no	—
87	C₄×Dic₈₉	—	4	2	no	—
88	C₄×Dic₈₉	—	4	2	no	—
89	C₄×Dic₈₉	—	4	1	no	—
90	#13083549	—	2	2	no	—
91	#13083549	—	2	2	no	—
92	C₂×Q₈×C₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₈₉	—	2	2	no	—
95	Q₁₆×C₈₉	—	2	2	no	—
96	Q₁₆×C₈₉	—	4	2	no	—
97	M₁₆×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₄⋊C₄×C₈₉	—	4	2	no	—
105	C₄⋊C₄×C₈₉	—	4	2	no	—
106	C₄⋊C₄×C₈₉	—	4	2	no	—
107	C₄⋊C₄×C₈₉	—	4	2	no	—
108	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
109	C₂²⋊C₄×C₈₉	—	4	2	no	—
110	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
111	C₄²×C₈₉	—	4	2	no	—
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₈₉	—	4	2	no	—
116	Q₈×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	—	4	4	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	Dic₁₇₈	—	2	4	no	—
129	Dic₁₇₈	—	4	4	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	—	8	2	no	—
142	C₄×C₈₉	—	8	2	no	—
143	C₄×C₈₉	#525	1	1	yes	characteristic
144	C₄×C₈₉	#575	1	1	yes	characteristic
145	Dic₈₉	—	4	4	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	D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
160	C₈₉	#33687	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₁₆	—	89	1	no	—
162	#541	—	2	1	no	—
163	#519	—	178	1	no	—
164	C₂×Q₃₂	—	89	2	no	—
165	C₂×Q₃₂	—	89	2	no	—
166	#499	C₂×D₈₉	1	1	yes	characteristic
167	#494	C₂×D₈₉	1	1	yes	characteristic
168	#491	—	2	1	no	—
169	#470	—	2	1	no	—
170	#467	C₂×D₈₉	1	2	yes	—
171	#441	—	2	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	C₂×D₈₉	1	2	yes	—
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	—	89	2	no	—
183	#358	—	89	2	no	—
184	#358	—	89	2	no	—
185	#358	—	89	2	no	—
186	#358	—	89	2	no	—
187	C₂²⋊C₁₆	—	89	2	no	—
188	#346	—	89	1	no	—
189	C₄×C₁₆	—	89	1	no	—
190	C₂²×Q₈	—	178	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	89	2	no	—
193	C₂×Q₁₆	—	178	4	no	—
194	C₂×Q₁₆	—	178	1	no	—
195	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
196	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	1	no	—
199	#126	—	4	2	no	—
200	#125	—	2	1	no	—
201	#125	—	4	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	356	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	C₂²⋊D₈₉	1	1	yes	characteristic
210	Q₃₂	—	178	4	no	—
211	Q₃₂	—	178	4	no	—
212	C₂×C₁₆	—	89	2	no	—
213	C₂×C₁₆	—	89	2	no	—
214	#107	—	89	2	no	—
215	#107	C₂²×D₈₉	1	2	yes	—
216	#106	—	2	1	no	—
217	#106	—	178	1	no	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	#103	—	178	2	no	—
221	#103	—	178	2	no	—
222	#97	C₂²⋊D₈₉	1	1	yes	characteristic
223	#97	C₂²⋊D₈₉	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	—	2	2	no	—
227	C₂×Q₈	—	356	2	no	—
228	C₂×Q₈	—	356	1	no	—
229	C₂×Q₈	—	178	4	no	—
230	C₂×Q₈	—	178	2	no	—
231	C₂²×C₄	—	356	1	no	—
232	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	2	no	—
236	Q₁₆	—	89	4	no	—
237	Q₁₆	—	178	4	no	—
238	Q₁₆	—	178	2	no	—
239	M₁₆	—	4	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	178	2	no	—
253	C₄⋊C₄	#13083559	1	2	yes	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	356	1	no	—
256	C₂²⋊C₄	#13083559	1	2	yes	—
257	C₄²	—	4	2	no	—
258	C₄²	—	356	2	no	—
259	C₄²	—	356	2	no	—
260	C₄²	—	356	1	no	—
261	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
263	C₁₆	—	178	2	no	—
264	C₁₆	—	178	2	no	—
265	C₂³	C₂×#13083559	1	1	yes	characteristic
266	Q₈	—	4	2	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	356	4	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	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	356	4	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	C₂×#13083559	1	2	yes	—
279	C₂×C₄	C₂×#13083559	1	2	yes	—
280	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₂²⋊D₈₉	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×#13083559	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	356	4	no	—
295	C₄	—	712	2	no	—
296	C₄	#19152105	1	1	yes	characteristic
297	C₄	#19152396	1	1	yes	characteristic
298	C₂	—	4	1	no	—
299	C₂	#26030248	1	1	yes	characteristic
300	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	#33687
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
3. #26022413	C₂	semidirect
4. #26022299	C₂	semidirect
6. #26022121	C₂	semidirect
7. C₄×#15667958	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
#26029121	2
#26022457	2
#26022413	2
#26022299	2
#26022185	2
#26022121	2
C₄×#15667958	2
#26022059	2
C₈₉×#3177	2
#33687	89