Subgroups — #36709832

Order 22784.

834 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152028	C₂	1	4	yes	maximal, complemented
2	#26030680	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022511	C₂	1	4	yes	maximal, complemented
4	#26022509	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022507	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022504	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022501	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	—	2	1	no	—
11	C₂×#15668034	C₂²	1	2	yes	—
12	#19152148	—	2	1	no	—
13	#19152028	—	2	16	no	—
14	C₂×#15667958	C₂²	1	4	yes	—
15	C₂×#15667956	C₂²	1	4	yes	—
16	C₂×#15667955	C₂²	1	4	yes	—
17	C₂×#15667953	C₂²	1	4	yes	—
18	C₂²×#13083549	C₂²	1	2	yes	—
19	#19151982	—	2	1	no	—
20	#19151982	C₂²	1	2	yes	—
21	#19151978	C₂²	1	1	yes	characteristic
22	#19151975	C₂²	1	1	yes	characteristic
23	#19151971	—	2	1	no	—
24	C₄×#13083549	C₂²	1	1	yes	characteristic
25	C₄×#13083549	C₂²	1	1	yes	characteristic
26	C₄×#13083549	C₂²	1	2	yes	—
27	C₂×#135×C₈₉	—	2	1	no	—
28	C₂×#135×C₈₉	C₂²	1	2	yes	—
29	#532×C₈₉	—	2	1	no	—
30	#495×C₈₉	—	2	1	no	—
31	#495×C₈₉	C₂²	1	2	yes	—
32	#494×C₈₉	C₂²	1	1	yes	characteristic
33	#493×C₈₉	C₂²	1	1	yes	characteristic
34	#492×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#13083574	—	2	2	no	—
37	#15668034	—	2	8	no	—
38	#15668034	—	4	4	no	—
39	C₂×Dic₃₅₆	—	2	1	no	—
40	C₂×Dic₃₅₆	C₂³	1	2	yes	—
41	C₂×#13083563	—	2	2	no	—
42	C₂×#13083563	—	2	2	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	C₂³	1	2	yes	—
45	#15668012	—	2	1	no	—
46	#15668009	—	2	1	no	—
47	#15668000	—	2	2	no	—
48	#15667986	—	4	2	no	—
49	C₄×D₃₅₆	—	4	2	no	—
50	#15667958	—	2	8	no	—
51	#15667956	—	2	8	no	—
52	#15667955	—	2	8	no	—
53	#15667953	—	2	8	no	—
54	C₂×#13083549	D₄	1	8	yes	complemented
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	2	yes	—
57	C₂×#41×C₈₉	—	2	2	no	—
58	#135×C₈₉	—	2	8	no	—
59	#135×C₈₉	—	4	4	no	—
60	C₂×Q₁₆×C₈₉	—	2	1	no	—
61	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₈₉	—	2	2	no	—
63	C₂×SD₁₆×C₈₉	—	2	2	no	—
64	C₂×D₈×C₈₉	—	2	1	no	—
65	C₂×D₈×C₈₉	C₂³	1	2	yes	—
66	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₈₉	—	2	1	no	—
69	#127×C₈₉	—	2	1	no	—
70	#124×C₈₉	—	2	2	no	—
71	#121×C₈₉	—	4	2	no	—
72	C₄×D₄×C₈₉	—	4	2	no	—
73	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
75	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	#13083574	—	4	4	no	—
79	#13083574	—	4	4	no	—
80	C₂×Dic₁₇₈	—	2	2	no	—
81	C₂×D₃₅₆	—	2	2	no	—
82	C₂×D₃₅₆	—	4	2	no	—
83	C₂×C₂²⋊D₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	—	4	2	no	—
85	Dic₃₅₆	—	2	4	no	—
86	Dic₃₅₆	—	4	2	no	—
87	#13083563	—	4	4	no	—
88	#13083563	—	4	4	no	—
89	D₇₁₂	—	2	4	no	—
90	D₇₁₂	—	4	2	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	#13083551	—	4	4	no	—
93	#13083549	C₂×D₄	1	4	yes	—
94	#13083549	C₂×D₄	1	4	yes	—
95	#41×C₈₉	—	4	4	no	—
96	#41×C₈₉	—	4	4	no	—
97	C₂×Q₈×C₈₉	—	2	2	no	—
98	C₂×D₄×C₈₉	—	2	2	no	—
99	C₂×D₄×C₈₉	—	4	2	no	—
100	C₂×D₄×C₈₉	—	4	2	no	—
101	C₂²×C₄×C₈₉	—	4	2	no	—
102	C₂²×C₄×C₈₉	D₈	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₈₉	—	2	4	no	—
105	Q₁₆×C₈₉	—	4	2	no	—
106	SD₁₆×C₈₉	—	4	4	no	—
107	SD₁₆×C₈₉	—	4	4	no	—
108	D₈×C₈₉	—	2	4	no	—
109	D₈×C₈₉	—	4	2	no	—
110	C₂×C₈×C₈₉	D₈	1	4	yes	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	—	4	2	no	—
117	C₂²⋊C₄×C₈₉	—	4	4	no	—
118	C₄²×C₈₉	D₈	1	2	yes	—
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₈₉	—	4	2	no	—
122	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₈₉	—	4	2	no	—
124	Q₈×C₈₉	—	4	2	no	—
125	Q₈×C₈₉	—	4	2	no	—
126	D₄×C₈₉	—	4	2	no	—
127	D₄×C₈₉	—	4	2	no	—
128	D₄×C₈₉	—	8	4	no	—
129	D₄×C₈₉	—	8	4	no	—
130	C₂×C₄×C₈₉	—	2	4	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	8	4	no	—
133	C₂×C₄×C₈₉	#127	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₈₉	—	8	4	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	D₃₅₆	—	4	2	no	—
142	D₃₅₆	—	4	2	no	—
143	D₃₅₆	—	8	4	no	—
144	C₄×D₈₉	—	8	4	no	—
145	Dic₁₇₈	—	4	2	no	—
146	Dic₁₇₈	—	4	2	no	—
147	C₈×C₈₉	#127	1	2	yes	—
148	C₈×C₈₉	C₂×D₈	1	2	yes	—
149	C₈×C₈₉	C₂×D₈	1	2	yes	—
150	C₈×C₈₉	C₂²×D₄	1	2	yes	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	—	8	2	no	—
153	C₂²×C₈₉	—	8	4	no	—
154	C₂²×C₈₉	#493	1	2	yes	—
155	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₈₉	—	8	2	no	—
157	C₂×D₈₉	—	8	4	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	#493	1	2	yes	—
161	C₄×C₈₉	#508	1	4	yes	complemented
162	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
163	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
164	Dic₈₉	—	8	2	no	—
165	#33713	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	2	2	no	—
167	C₂×C₈₉	—	16	2	no	—
168	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#508	1	2	yes	—
170	D₈₉	—	16	2	no	—
171	C₂×#508	—	89	4	no	—
172	#3147	—	89	1	no	—
173	#3147	D₈₉	1	1	yes	characteristic, complemented
174	#2249	—	89	4	no	—
175	#2248	—	89	1	no	—
176	#2248	—	89	1	no	—
177	#2247	—	89	1	no	—
178	#2246	—	89	1	no	—
179	C₂×C₄×C₁₆	—	89	1	no	—
180	C₈₉	#33713	1	1	yes	characteristic, complemented, Sylow
181	C₂×#135	—	2	1	no	—
182	C₂×#135	—	89	2	no	—
183	C₂×#135	—	178	1	no	—
184	C₂×#135	C₂×D₈₉	1	2	yes	—
185	#532	—	2	1	no	—
186	#532	—	178	1	no	—
187	#508	—	178	16	no	—
188	C₂×Q₃₂	—	89	4	no	—
189	C₂×SD₃₂	—	89	4	no	—
190	C₂×SD₃₂	—	89	4	no	—
191	C₂×D₁₆	—	89	4	no	—
192	C₂²×C₁₆	—	89	2	no	—
193	#495	—	2	1	no	—
194	#495	—	89	2	no	—
195	#495	—	178	1	no	—
196	#495	C₂×D₈₉	1	2	yes	—
197	#494	—	89	1	no	—
198	#494	C₂×D₈₉	1	1	yes	characteristic
199	#493	—	89	1	no	—
200	#493	C₂×D₈₉	1	1	yes	characteristic
201	#492	—	2	1	no	—
202	#492	—	178	1	no	—
203	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
204	C₄×C₁₆	—	89	1	no	—
205	C₄×C₁₆	—	89	1	no	—
206	C₄×C₁₆	—	89	2	no	—
207	C₂×#41	—	2	2	no	—
208	C₂×#41	—	178	2	no	—
209	#135	—	2	8	no	—
210	#135	—	4	4	no	—
211	#135	—	356	4	no	—
212	#135	—	178	8	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	89	2	no	—
215	C₂×Q₁₆	—	178	1	no	—
216	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	178	2	no	—
220	C₂×SD₁₆	—	178	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	89	2	no	—
223	C₂×D₈	—	178	1	no	—
224	C₂×D₈	C₂²×D₈₉	1	2	yes	—
225	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	178	1	no	—
229	#127	—	2	1	no	—
230	#127	—	178	1	no	—
231	#124	—	2	2	no	—
232	#124	—	178	2	no	—
233	#121	—	4	2	no	—
234	#121	—	356	2	no	—
235	C₄×D₄	—	4	2	no	—
236	C₄×D₄	—	356	2	no	—
237	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
238	Q₃₂	—	178	8	no	—
239	SD₃₂	—	178	8	no	—
240	SD₃₂	—	178	8	no	—
241	D₁₆	—	178	8	no	—
242	C₂×C₁₆	—	89	8	no	—
243	C₂×C₁₆	—	89	2	no	—
244	C₂×C₁₆	—	89	2	no	—
245	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
246	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	#41	—	4	4	no	—
250	#41	—	4	4	no	—
251	#41	—	356	4	no	—
252	#41	—	356	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	178	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	356	2	no	—
259	C₂×D₄	—	356	2	no	—
260	C₂×D₄	—	178	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	356	2	no	—
263	C₂²×C₄	#13083559	1	2	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	4	2	no	—
267	Q₁₆	—	356	2	no	—
268	Q₁₆	—	178	4	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	4	4	no	—
271	SD₁₆	—	356	4	no	—
272	SD₁₆	—	356	4	no	—
273	D₈	—	2	4	no	—
274	D₈	—	4	2	no	—
275	D₈	—	356	2	no	—
276	D₈	—	178	4	no	—
277	C₂×C₈	#13083559	1	4	yes	—
278	C₂×C₈	D₄×D₈₉	1	4	yes	—
279	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
282	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	356	2	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	356	4	no	—
287	C₄²	#13083559	1	2	yes	—
288	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₁₆	—	89	4	no	—
291	C₁₆	—	89	4	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	356	2	no	—
294	C₂³	C₂×#13083559	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	356	2	no	—
298	Q₈	—	356	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	356	2	no	—
304	D₄	—	356	2	no	—
305	D₄	—	712	4	no	—
306	D₄	—	712	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	712	4	no	—
312	C₂×C₄	#15668008	1	2	yes	—
313	C₂×C₄	C₂×#13083559	1	2	yes	—
314	C₂×C₄	C₂×#13083559	1	2	yes	—
315	C₂×C₄	C₂×#13083559	1	2	yes	—
316	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
318	C₈	#15668008	1	2	yes	—
319	C₈	C₂×#13083559	1	2	yes	—
320	C₈	C₂×#13083559	1	2	yes	—
321	C₈	C₂×D₄×D₈₉	1	2	yes	—
322	C₂²	—	2	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	4	no	—
325	C₂²	—	712	2	no	—
326	C₂²	—	712	4	no	—
327	C₂²	#19151974	1	2	yes	—
328	C₂²	C₂²×#13083559	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	712	2	no	—
332	C₄	#19151974	1	2	yes	—
333	C₄	#19152028	1	4	yes	complemented
334	C₄	C₂×#15668008	1	1	yes	characteristic
335	C₄	C₂²×#13083559	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	—	1424	2	no	—
339	C₂	C₂×#19151974	1	1	yes	characteristic
340	C₂	C₂×#19152028	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3147	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3147	D₈₉
hypocenter	C₈₉	#33713
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#19152028	C₂	semidirect
2. #26030680	C₂	semidirect
3. #26022511	C₂	semidirect
4. #26022509	C₂	semidirect
5. #26022507	C₂	semidirect
6. #26022504	C₂	semidirect
7. #26022501	C₂	semidirect
8. C₂×C₄×#13083549	C₂	semidirect
9. C₈₉×#3147	C₂	semidirect
54. C₂×#13083549	D₄	semidirect
161. C₄×C₈₉	#508	semidirect
173. #3147	D₈₉	semidirect
180. C₈₉	#33713	semidirect
333. C₄	#19152028	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152028	2
#26030680	2
#26022511	2
#26022509	2
#26022507	2
#26022504	2
#26022501	2
C₂×C₄×#13083549	2
C₈₉×#3147	2
#33713	89