Subgroups — #36709079

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26022406	C₂	1	4	yes	maximal, complemented
3	#26022396	C₂	1	2	yes	maximal, complemented
4	#26022386	C₂	1	2	yes	maximal, complemented
5	#26022041	C₂	1	2	yes	maximal, complemented
6	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021967	C₂	1	2	yes	maximal
8	C₈₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₅₆	—	2	1	no	—
11	C₂×#15667952	C₂²	1	2	yes	—
12	C₄×D₇₁₂	—	2	1	no	—
13	C₄×D₇₁₂	—	2	1	no	—
14	C₄×D₇₁₂	C₂²	1	2	yes	—
15	C₄×D₇₁₂	C₂²	1	2	yes	—
16	C₂×#15667946	C₂²	1	1	yes	characteristic
17	C₂×#15667934	—	2	1	no	—
18	#19151425	C₂²	1	4	yes	complemented
19	#19151423	C₂²	1	4	yes	complemented
20	C₄⋊#13083549	C₂²	1	2	yes	complemented
21	C₄⋊#13083549	C₂²	1	2	yes	complemented
22	#19151398	C₂²	1	4	yes	—
23	#19151398	C₂²	1	4	yes	—
24	#533×C₈₉	—	2	1	no	—
25	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	#499×C₈₉	—	2	1	no	—
28	#499×C₈₉	C₂²	1	2	yes	complemented
29	#498×C₈₉	—	2	1	no	—
30	#427×C₈₉	—	2	1	no	—
31	#427×C₈₉	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×D₃₅₆	—	2	1	no	—
34	C₂²×C₄×D₈₉	—	4	1	no	—
35	C₂×D₇₁₂	—	2	1	no	—
36	C₂×D₇₁₂	—	2	2	no	—
37	C₂×D₇₁₂	Q₈	1	4	yes	complemented
38	C₂×D₇₁₂	C₂³	1	2	yes	—
39	C₄×D₃₅₆	—	2	2	no	—
40	C₄×D₃₅₆	—	2	2	no	—
41	C₄×D₃₅₆	—	4	1	no	—
42	C₄×D₃₅₆	—	4	1	no	—
43	C₂×C₄⋊Dic₈₉	—	2	1	no	—
44	C₂×#13083551	—	4	1	no	—
45	#15667952	—	2	4	no	—
46	C₂×#13083549	C₂³	1	4	yes	—
47	#15667946	—	2	1	no	—
48	#15667946	C₂³	1	2	yes	—
49	#15667934	—	2	2	no	—
50	#15667934	—	2	2	no	—
51	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₉	—	2	1	no	—
54	C₄⋊Q₈×C₈₉	—	2	1	no	—
55	#125×C₈₉	—	2	2	no	—
56	C₂²⋊Q₈×C₈₉	—	4	1	no	—
57	C₂²⋊Q₈×C₈₉	—	4	1	no	—
58	C₄×Q₈×C₈₉	—	4	1	no	—
59	C₄×Q₈×C₈₉	—	4	1	no	—
60	#117×C₈₉	—	2	2	no	—
61	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
62	#107×C₈₉	—	2	1	no	—
63	#107×C₈₉	—	2	1	no	—
64	#107×C₈₉	C₂³	1	2	yes	—
65	#107×C₈₉	C₂³	1	2	yes	—
66	#106×C₈₉	—	2	2	no	—
67	#106×C₈₉	—	2	2	no	—
68	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×D₈₉	—	4	1	no	—
73	C₂×D₃₅₆	—	2	4	no	—
74	C₂×D₃₅₆	—	2	2	no	—
75	C₂×D₃₅₆	—	4	1	no	—
76	C₂×D₃₅₆	—	4	2	no	—
77	C₂²×Dic₈₉	—	4	1	no	—
78	C₂×C₄×D₈₉	—	4	2	no	—
79	C₂×C₄×D₈₉	—	4	2	no	—
80	C₂×C₄×D₈₉	—	8	4	no	—
81	D₇₁₂	—	2	4	no	—
82	D₇₁₂	—	2	2	no	—
83	D₇₁₂	C₂×Q₈	1	4	yes	—
84	C₄⋊Dic₈₉	—	2	2	no	—
85	C₄⋊Dic₈₉	—	4	1	no	—
86	#13083551	—	4	2	no	—
87	#13083551	—	4	2	no	—
88	#13083549	—	2	4	no	—
89	C₂×Q₈×C₈₉	—	4	1	no	—
90	C₂×Q₈×C₈₉	—	4	1	no	—
91	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₈₉	#41	1	2	yes	—
100	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	—	2	2	no	—
103	C₄⋊C₄×C₈₉	—	4	2	no	—
104	C₄⋊C₄×C₈₉	—	4	2	no	—
105	C₄⋊C₄×C₈₉	—	4	1	no	—
106	C₄⋊C₄×C₈₉	—	4	1	no	—
107	C₂²⋊C₄×C₈₉	—	4	2	no	—
108	C₄²×C₈₉	—	4	2	no	—
109	C₄²×C₈₉	D₈	1	1	yes	characteristic
110	C₄²×C₈₉	D₈	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
114	C₂²×D₈₉	—	4	2	no	—
115	C₂²×D₈₉	—	4	4	no	—
116	C₂²×D₈₉	—	8	2	no	—
117	C₂²×D₈₉	—	8	2	no	—
118	Q₈×C₈₉	—	8	2	no	—
119	Q₈×C₈₉	—	8	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#137	1	2	yes	—
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	—	4	2	no	—
133	C₂×Dic₈₉	—	8	2	no	—
134	D₃₅₆	—	2	4	no	—
135	D₃₅₆	—	4	2	no	—
136	D₃₅₆	—	4	4	no	—
137	C₄×D₈₉	—	8	4	no	—
138	C₄×D₈₉	—	8	4	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
142	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	#474	1	2	yes	—
147	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₈₉	—	4	4	no	—
149	C₂×D₈₉	—	8	8	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₂×D₈₉	—	8	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	—	8	2	no	—
157	C₄×C₈₉	#474	1	2	yes	—
158	C₄×C₈₉	#509	1	2	yes	—
159	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
161	Dic₈₉	—	8	2	no	—
162	#33627	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	2	2	no	—
164	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
165	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
166	C₂×C₈₉	#3414	1	1	yes	characteristic
167	D₈₉	—	8	4	no	—
168	#3160	D₈₉	1	1	yes	characteristic, complemented
169	C₂×C₄×D₈	—	89	1	no	—
170	#2228	—	89	4	no	—
171	#2226	—	89	2	no	—
172	#2224	—	89	2	no	—
173	#2159	—	89	2	no	—
174	#2150	—	89	1	no	—
175	#2140	—	89	2	no	—
176	C₈₉	#33627	1	1	yes	characteristic, complemented, Sylow
177	C₂²×D₈	—	89	1	no	—
178	#533	—	2	1	no	—
179	C₂×C₄×D₄	—	178	1	no	—
180	C₂×M₃₂	—	89	2	no	—
181	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
182	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
183	#499	—	2	1	no	—
184	#499	C₂×D₈₉	1	2	yes	complemented
185	#498	—	2	1	no	—
186	C₄×D₈	—	89	2	no	—
187	C₄×D₈	—	89	2	no	—
188	C₄×D₈	—	178	1	no	—
189	C₄×D₈	—	178	1	no	—
190	#427	—	2	1	no	—
191	#427	C₂×D₈₉	1	2	yes	—
192	C₂×#107	—	89	1	no	—
193	C₂×#102	—	178	1	no	—
194	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
195	#367	—	89	4	no	—
196	#366	—	89	4	no	—
197	C₄⋊C₁₆	—	89	2	no	—
198	C₄⋊C₁₆	—	89	2	no	—
199	#357	—	89	4	no	—
200	#357	—	89	4	no	—
201	C₂²×D₄	—	178	1	no	—
202	C₂³×C₄	—	356	1	no	—
203	C₂×D₈	—	89	4	no	—
204	C₂×D₈	—	89	2	no	—
205	C₂×D₈	—	178	1	no	—
206	C₂×D₈	—	178	2	no	—
207	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	#125	—	2	2	no	—
212	C₂²⋊Q₈	—	4	1	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	C₄×Q₈	—	4	1	no	—
215	C₄×Q₈	—	4	1	no	—
216	C₄×D₄	—	356	1	no	—
217	C₄×D₄	—	356	1	no	—
218	C₄×D₄	—	178	2	no	—
219	C₄×D₄	—	178	2	no	—
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	178	1	no	—
222	C₂×C₂²⋊C₄	—	356	1	no	—
223	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
224	M₃₂	—	178	4	no	—
225	C₂×C₁₆	—	89	4	no	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	—	89	2	no	—
229	#107	—	178	1	no	—
230	#107	C₂²×D₈₉	1	2	yes	—
231	#107	C₂²×D₈₉	1	2	yes	—
232	#106	—	2	2	no	—
233	#106	—	2	2	no	—
234	#102	—	178	2	no	—
235	#102	—	178	2	no	—
236	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₂⁴	—	356	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	356	1	no	—
244	C₂×D₄	—	356	2	no	—
245	C₂×D₄	—	178	4	no	—
246	C₂×D₄	—	178	2	no	—
247	C₂²×C₄	—	356	1	no	—
248	C₂²×C₄	—	356	2	no	—
249	C₂²×C₄	—	356	2	no	—
250	C₂²×C₄	—	712	4	no	—
251	C₂²×C₄	#13083559	1	1	yes	characteristic
252	C₂²×C₄	#13083559	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
254	D₈	—	89	4	no	—
255	D₈	—	178	4	no	—
256	D₈	—	178	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₂×C₈	Q₈×D₈₉	1	2	yes	—
262	C₂×C₈	#13083575	1	2	yes	—
263	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	356	1	no	—
271	C₄⋊C₄	—	178	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	356	2	no	—
274	C₂²⋊C₄	—	356	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	#13083559	1	1	yes	characteristic
277	C₄²	#13083559	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
280	C₁₆	—	178	4	no	—
281	C₂³	—	356	4	no	—
282	C₂³	—	356	2	no	—
283	C₂³	—	712	2	no	—
284	C₂³	—	712	2	no	—
285	C₂³	C₂×#13083559	1	1	yes	characteristic
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	356	4	no	—
289	D₄	—	356	2	no	—
290	D₄	—	178	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	356	2	no	—
296	C₂×C₄	—	712	4	no	—
297	C₂×C₄	—	712	2	no	—
298	C₂×C₄	—	712	4	no	—
299	C₂×C₄	#15667988	1	2	yes	—
300	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
301	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
302	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
303	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
304	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
305	C₂×C₄	#15668043	1	2	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	#15667988	1	2	yes	—
310	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
311	C₈	C₂×#13083575	1	1	yes	characteristic
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	356	4	no	—
315	C₂²	—	712	8	no	—
316	C₂²	—	712	2	no	—
317	C₂²	—	712	2	no	—
318	C₂²	#19151875	1	2	yes	—
319	C₂²	C₂²×#13083559	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	712	2	no	—
326	C₄	#19151875	1	2	yes	—
327	C₄	#19152033	1	2	yes	—
328	C₄	C₂×#15667988	1	1	yes	characteristic
329	C₄	C₂×#15668043	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	712	4	no	—
332	C₂	C₂×#19151875	1	1	yes	characteristic
333	C₂	C₂×#19152033	1	1	yes	characteristic
334	C₂	#26033386	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3160	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₇₁₂	C₂²
hypercenter	#3160	D₈₉
hypocenter	C₈₉	#33627
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #26022406	C₂	semidirect
3. #26022396	C₂	semidirect
4. #26022386	C₂	semidirect
5. #26022041	C₂	semidirect
6. #26022008	C₂	semidirect
8. C₈₉×#3160	C₂	semidirect
18. #19151425	C₂²	semidirect
19. #19151423	C₂²	semidirect
20. C₄⋊#13083549	C₂²	semidirect
21. C₄⋊#13083549	C₂²	semidirect
25. #500×C₈₉	C₂²	semidirect
26. #500×C₈₉	C₂²	semidirect
28. #499×C₈₉	C₂²	semidirect
37. C₂×D₇₁₂	Q₈	semidirect
168. #3160	D₈₉	semidirect
176. C₈₉	#33627	semidirect
181. #500	C₂×D₈₉	semidirect
182. #500	C₂×D₈₉	semidirect
184. #499	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₁₂	2
#26022406	2
#26022396	2
#26022386	2
#26022041	2
#26022008	2
#26021967	2
C₈₉×#3160	2
#33627	89