Subgroups — #36710852

Order 22784.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 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, complemented
2	#26022413	C₂	1	2	yes	maximal, complemented
3	#26022409	C₂	1	2	yes	maximal, complemented
4	#26022363	C₂	1	2	yes	maximal, complemented
5	#26022346	C₂	1	2	yes	maximal, complemented
6	#26022297	C₂	1	2	yes	maximal, complemented
7	#26022289	C₂	1	2	yes	maximal, complemented
8	#26022011	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3176	C₂	1	1	yes	maximal, characteristic
10	#19152210	—	2	1	no	—
11	C₂×#15667956	C₂²	1	2	yes	—
12	C₂×#15667956	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	#19151401	C₂²	1	2	yes	—
27	#19151401	C₂²	1	2	yes	—
28	#19151399	C₂²	1	2	yes	—
29	#19151399	C₂²	1	2	yes	—
30	#540×C₈₉	—	2	1	no	—
31	#499×C₈₉	C₂²	1	1	yes	characteristic
32	#493×C₈₉	C₂²	1	1	yes	characteristic
33	#490×C₈₉	—	2	1	no	—
34	#469×C₈₉	—	2	1	no	—
35	#466×C₈₉	C₂²	1	2	yes	—
36	#442×C₈₉	—	2	1	no	—
37	C₄×D₈×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	#15667956	—	4	2	no	—
48	#15667956	—	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×D₈×C₈₉	—	2	1	no	—
58	C₂×D₈×C₈₉	C₂³	1	2	yes	—
59	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	#127×C₈₉	—	2	1	no	—
62	#125×C₈₉	—	2	1	no	—
63	#123×C₈₉	—	4	2	no	—
64	#121×C₈₉	—	2	2	no	—
65	#121×C₈₉	—	4	2	no	—
66	C₄×D₄×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	#102×C₈₉	—	2	2	no	—
71	#102×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×D₄×C₈₉	—	2	2	no	—
90	C₂×D₄×C₈₉	—	4	2	no	—
91	C₂×D₄×C₈₉	—	4	2	no	—
92	C₂²×C₄×C₈₉	—	4	2	no	—
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	D₈×C₈₉	—	2	2	no	—
95	D₈×C₈₉	—	2	2	no	—
96	D₈×C₈₉	—	4	2	no	—
97	M₁₆×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	—	2	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	—	4	2	no	—
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	C₂³×C₈₉	—	4	2	no	—
113	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
114	D₄×C₈₉	—	4	2	no	—
115	D₄×C₈₉	—	4	2	no	—
116	D₄×C₈₉	—	16	2	no	—
117	D₄×C₈₉	—	16	2	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	16	2	no	—
122	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	Dic₁₇₈	—	4	2	no	—
129	Dic₁₇₈	—	4	2	no	—
130	C₈×C₈₉	—	2	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	8	2	no	—
138	C₂²×C₈₉	—	8	2	no	—
139	C₂²×C₈₉	—	8	2	no	—
140	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	8	2	no	—
145	C₄×C₈₉	#537	1	1	yes	characteristic
146	C₄×C₈₉	#575	1	1	yes	characteristic
147	Dic₈₉	—	8	2	no	—
148	Dic₈₉	—	8	2	no	—
149	#33826	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	—	4	1	no	—
151	C₂×C₈₉	—	16	2	no	—
152	C₂×C₈₉	#3245	1	1	yes	characteristic
153	C₂×C₈₉	#3416	1	2	yes	—
154	#3177	—	89	1	no	—
155	#3176	D₈₉	1	1	yes	characteristic
156	#2229	—	89	2	no	—
157	#2228	—	89	2	no	—
158	#2220	—	89	2	no	—
159	#2217	—	89	2	no	—
160	#2208	—	89	2	no	—
161	#2207	—	89	2	no	—
162	#2151	—	89	1	no	—
163	C₈₉	#33826	1	1	yes	characteristic, complemented, Sylow
164	#541	—	178	1	no	—
165	#540	—	2	1	no	—
166	C₂×SD₃₂	—	89	2	no	—
167	C₂×SD₃₂	—	89	2	no	—
168	C₂×M₃₂	—	89	1	no	—
169	C₂²×C₁₆	—	89	1	no	—
170	#499	—	89	1	no	—
171	#499	C₂×D₈₉	1	1	yes	characteristic
172	#494	—	89	1	no	—
173	#493	C₂×D₈₉	1	1	yes	characteristic
174	#491	—	178	1	no	—
175	#490	—	2	1	no	—
176	#470	—	178	1	no	—
177	#469	—	2	1	no	—
178	#467	—	89	2	no	—
179	#466	C₂×D₈₉	1	2	yes	—
180	#442	—	2	1	no	—
181	#441	—	178	1	no	—
182	C₄×Q₁₆	—	89	2	no	—
183	C₄×D₈	C₂×D₈₉	1	2	yes	—
184	#405	C₂×D₈₉	1	1	yes	characteristic
185	#367	—	89	2	no	—
186	#366	—	89	2	no	—
187	C₄⋊C₁₆	—	89	2	no	—
188	C₄⋊C₁₆	—	89	2	no	—
189	#358	—	89	2	no	—
190	#358	—	89	2	no	—
191	#357	—	89	2	no	—
192	#357	—	89	2	no	—
193	C₂×Q₁₆	—	89	2	no	—
194	C₂×Q₁₆	—	178	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	C₂²×D₈₉	1	2	yes	—
197	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	178	1	no	—
200	#127	—	2	1	no	—
201	#126	—	356	2	no	—
202	#125	—	2	1	no	—
203	#125	—	356	2	no	—
204	#125	—	178	1	no	—
205	#123	—	4	2	no	—
206	C₂²⋊Q₈	—	178	2	no	—
207	#121	—	2	2	no	—
208	#121	—	4	2	no	—
209	C₄×Q₈	—	178	2	no	—
210	C₄×D₄	—	2	2	no	—
211	#117	C₂²⋊D₈₉	1	1	yes	characteristic
212	SD₃₂	—	356	2	no	—
213	SD₃₂	—	356	2	no	—
214	M₃₂	—	356	1	no	—
215	C₂×C₁₆	—	89	2	no	—
216	C₂×C₁₆	—	89	2	no	—
217	C₂×C₁₆	—	356	1	no	—
218	#107	—	89	2	no	—
219	#107	C₂²×D₈₉	1	2	yes	—
220	#106	—	2	1	no	—
221	#106	—	178	1	no	—
222	#103	—	178	2	no	—
223	#103	—	178	2	no	—
224	#102	—	2	2	no	—
225	#102	—	2	2	no	—
226	#97	C₂²⋊D₈₉	1	1	yes	characteristic
227	#97	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×Q₈	—	178	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
236	Q₁₆	—	356	2	no	—
237	Q₁₆	—	178	2	no	—
238	Q₁₆	—	178	2	no	—
239	D₈	—	2	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	4	2	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	356	2	no	—
253	C₄⋊C₄	—	356	2	no	—
254	C₄⋊C₄	—	356	2	no	—
255	C₄⋊C₄	—	178	2	no	—
256	C₄⋊C₄	#13083559	1	2	yes	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	356	2	no	—
260	C₂²⋊C₄	#13083559	1	2	yes	—
261	C₄²	—	356	2	no	—
262	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₁₆	—	178	2	no	—
265	C₁₆	—	178	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×#13083559	1	1	yes	characteristic
268	Q₈	—	356	2	no	—
269	Q₈	—	356	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	16	2	no	—
273	D₄	—	16	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	16	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	356	2	no	—
280	C₂×C₄	C₂×#13083559	1	2	yes	—
281	C₂×C₄	C₂×#13083559	1	2	yes	—
282	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	#15668058	1	1	yes	characteristic
288	C₈	#15668059	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×#13083559	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	712	2	no	—
300	C₄	—	712	2	no	—
301	C₄	#19152176	1	1	yes	characteristic
302	C₄	#19152396	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	#26031559	1	1	yes	characteristic
306	C₂	#26033399	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3176	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3176	D₈₉
hypocenter	C₈₉	#33826
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

Complements

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

N	H ≅ G/N	type
1. #26030911	C₂	semidirect
2. #26022413	C₂	semidirect
3. #26022409	C₂	semidirect
4. #26022363	C₂	semidirect
5. #26022346	C₂	semidirect
6. #26022297	C₂	semidirect
7. #26022289	C₂	semidirect
8. #26022011	C₂	semidirect
163. C₈₉	#33826	semidirect

Maximal subgroups

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

subgroup	index
#26030911	2
#26022413	2
#26022409	2
#26022363	2
#26022346	2
#26022297	2
#26022289	2
#26022011	2
C₈₉×#3176	2
#33826	89