Subgroups — #36709068

Order 22784.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668034	C₂	1	1	yes	maximal, characteristic
2	#26022413	C₂	1	2	yes	maximal, complemented
3	#26022406	C₂	1	2	yes	maximal, complemented
4	#26022401	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022396	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022391	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022386	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022034	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021957	C₂	1	2	yes	maximal
11	C₈₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₂²	1	1	yes	characteristic
13	C₄×#13083574	—	2	1	no	—
14	C₂²×#13083549	C₂²	1	2	yes	—
15	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
16	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
17	C₄×#13083563	—	2	1	no	—
18	C₄×#13083563	—	2	1	no	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
21	#19151636	C₂²	1	1	yes	characteristic
22	#19151595	—	2	1	no	—
23	#19151425	C₂²	1	2	yes	—
24	#19151425	C₂²	1	2	yes	—
25	#19151423	C₂²	1	2	yes	—
26	#19151423	C₂²	1	2	yes	—
27	C₄⋊#13083549	C₂²	1	1	yes	characteristic
28	C₄⋊#13083549	C₂²	1	2	yes	—
29	C₄⋊#13083549	C₂²	1	1	yes	characteristic
30	#19151401	C₂²	1	2	yes	—
31	#19151401	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	—
33	#19151398	C₂²	1	2	yes	—
34	#533×C₈₉	—	2	1	no	—
35	#500×C₈₉	C₂²	1	1	yes	characteristic
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	#499×C₈₉	—	2	1	no	—
38	#499×C₈₉	C₂²	1	2	yes	—
39	#498×C₈₉	—	2	1	no	—
40	#427×C₈₉	—	2	1	no	—
41	#427×C₈₉	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083574	—	2	1	no	—
44	#15668034	—	2	2	no	—
45	#15668034	Q₈	1	4	yes	complemented
46	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
47	C₂×#13083563	—	2	1	no	—
48	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
49	C₄×Dic₁₇₈	—	2	1	no	—
50	C₄×Dic₁₇₈	—	2	1	no	—
51	C₄×D₃₅₆	—	2	1	no	—
52	C₄×D₃₅₆	—	2	1	no	—
53	C₄×C₂²⋊D₈₉	—	4	1	no	—
54	C₄×C₂²⋊D₈₉	—	4	1	no	—
55	#15667971	—	2	1	no	—
56	#15667969	—	4	1	no	—
57	C₄²×D₈₉	—	4	1	no	—
58	C₂×#13083549	—	2	4	no	—
59	C₂×#13083549	C₂³	1	2	yes	—
60	C₂×#13083549	C₂³	1	2	yes	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	1	yes	characteristic
63	#15667944	—	2	1	no	—
64	#15667937	—	2	1	no	—
65	#15667937	—	2	1	no	—
66	#15667934	—	2	1	no	—
67	#15667934	—	2	1	no	—
68	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	#125×C₈₉	—	2	2	no	—
73	C₂²⋊Q₈×C₈₉	—	4	1	no	—
74	C₂²⋊Q₈×C₈₉	—	4	1	no	—
75	C₄×Q₈×C₈₉	—	4	1	no	—
76	C₄×Q₈×C₈₉	—	4	1	no	—
77	#117×C₈₉	—	2	2	no	—
78	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
79	#107×C₈₉	—	2	1	no	—
80	#107×C₈₉	—	2	1	no	—
81	#107×C₈₉	C₂³	1	2	yes	—
82	#107×C₈₉	C₂³	1	2	yes	—
83	#106×C₈₉	—	2	2	no	—
84	#106×C₈₉	—	2	2	no	—
85	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	#13083574	—	2	4	no	—
90	#13083574	—	4	2	no	—
91	C₂×Dic₁₇₈	—	2	1	no	—
92	C₂×D₃₅₆	—	2	1	no	—
93	C₂×C₂²⋊D₈₉	—	4	1	no	—
94	C₂×C₄×D₈₉	—	4	1	no	—
95	C₂×C₄×D₈₉	—	4	1	no	—
96	C₂×C₄×D₈₉	—	4	1	no	—
97	Dic₃₅₆	—	2	1	no	—
98	Dic₃₅₆	C₂×Q₈	1	2	yes	—
99	#13083563	—	2	4	no	—
100	D₇₁₂	—	2	1	no	—
101	D₇₁₂	C₂×Q₈	1	2	yes	—
102	C₄⋊Dic₈₉	—	2	1	no	—
103	C₄⋊Dic₈₉	—	2	1	no	—
104	#13083553	—	4	1	no	—
105	#13083553	—	4	1	no	—
106	C₂²⋊Dic₈₉	—	4	1	no	—
107	#13083551	—	4	1	no	—
108	#13083551	—	4	1	no	—
109	C₄×Dic₈₉	—	4	1	no	—
110	C₄×Dic₈₉	—	4	1	no	—
111	C₄×Dic₈₉	—	4	1	no	—
112	#13083549	—	2	2	no	—
113	#13083549	—	2	2	no	—
114	C₂×Q₈×C₈₉	—	4	1	no	—
115	C₂×Q₈×C₈₉	—	4	1	no	—
116	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
124	C₂×C₈×C₈₉	#41	1	2	yes	—
125	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄⋊C₄×C₈₉	—	4	2	no	—
129	C₄⋊C₄×C₈₉	—	4	2	no	—
130	C₄⋊C₄×C₈₉	—	4	1	no	—
131	C₄⋊C₄×C₈₉	—	4	1	no	—
132	C₂²⋊C₄×C₈₉	—	4	2	no	—
133	C₄²×C₈₉	—	4	2	no	—
134	C₄²×C₈₉	D₈	1	1	yes	characteristic
135	C₄²×C₈₉	D₈	1	1	yes	characteristic
136	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₈₉	—	4	1	no	—
140	Q₈×C₈₉	—	8	2	no	—
141	Q₈×C₈₉	—	8	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₈₉	#137	1	2	yes	—
153	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂²⋊D₈₉	—	4	4	no	—
155	C₂×Dic₈₉	—	4	1	no	—
156	C₂×Dic₈₉	—	4	1	no	—
157	C₂×Dic₈₉	—	4	1	no	—
158	D₃₅₆	—	2	2	no	—
159	D₃₅₆	—	4	1	no	—
160	C₄×D₈₉	—	4	4	no	—
161	C₄×D₈₉	—	8	2	no	—
162	C₄×D₈₉	—	8	2	no	—
163	Dic₁₇₈	—	2	2	no	—
164	Dic₁₇₈	—	4	1	no	—
165	C₈×C₈₉	—	2	1	no	—
166	C₈×C₈₉	—	2	1	no	—
167	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
168	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	#474	1	2	yes	—
173	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₈₉	—	4	2	no	—
175	C₂×D₈₉	—	8	1	no	—
176	C₂×D₈₉	—	8	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	8	2	no	—
181	C₄×C₈₉	—	8	2	no	—
182	C₄×C₈₉	#474	1	2	yes	—
183	C₄×C₈₉	#508	1	2	yes	—
184	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
185	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
186	Dic₈₉	—	4	2	no	—
187	Dic₈₉	—	8	1	no	—
188	Dic₈₉	—	8	1	no	—
189	#33626	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	—	2	2	no	—
191	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
192	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
193	C₂×C₈₉	#3414	1	1	yes	characteristic
194	D₈₉	—	8	2	no	—
195	#3160	D₈₉	1	1	yes	characteristic, complemented
196	C₄×#135	—	89	1	no	—
197	#2229	—	89	2	no	—
198	#2228	—	89	2	no	—
199	#2227	—	89	1	no	—
200	#2226	—	89	1	no	—
201	#2225	—	89	1	no	—
202	#2224	—	89	1	no	—
203	#2158	—	89	2	no	—
204	C₂×C₄⋊C₁₆	—	89	1	no	—
205	#2138	—	89	2	no	—
206	C₈₉	#33626	1	1	yes	characteristic, complemented, Sylow
207	C₂×#135	—	89	1	no	—
208	#533	—	2	1	no	—
209	C₄×#41	—	178	1	no	—
210	C₂²×C₁₆	—	89	2	no	—
211	#500	C₂×D₈₉	1	1	yes	characteristic
212	#500	C₂×D₈₉	1	1	yes	characteristic
213	#499	—	2	1	no	—
214	#499	C₂×D₈₉	1	2	yes	—
215	#498	—	2	1	no	—
216	C₄×Q₁₆	—	89	1	no	—
217	C₄×Q₁₆	—	89	1	no	—
218	C₄×SD₁₆	—	178	1	no	—
219	C₄×SD₁₆	—	178	1	no	—
220	C₄×D₈	—	89	1	no	—
221	C₄×D₈	—	89	1	no	—
222	#427	—	2	1	no	—
223	#427	—	89	1	no	—
224	#427	C₂×D₈₉	1	2	yes	—
225	#416	—	178	1	no	—
226	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
227	#367	—	89	2	no	—
228	#367	—	89	2	no	—
229	#366	—	89	2	no	—
230	#366	—	89	2	no	—
231	C₄⋊C₁₆	—	89	1	no	—
232	C₄⋊C₁₆	—	89	1	no	—
233	C₄⋊C₁₆	—	89	2	no	—
234	#358	—	89	2	no	—
235	#358	—	89	2	no	—
236	#357	—	89	2	no	—
237	#357	—	89	2	no	—
238	C₂×#41	—	178	1	no	—
239	#135	—	89	4	no	—
240	#135	—	178	2	no	—
241	C₂×Q₁₆	—	89	1	no	—
242	C₂×SD₁₆	—	178	1	no	—
243	C₂×D₈	—	89	1	no	—
244	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	#125	—	2	2	no	—
249	C₂²⋊Q₈	—	4	1	no	—
250	C₂²⋊Q₈	—	4	1	no	—
251	C₄×Q₈	—	4	1	no	—
252	C₄×Q₈	—	4	1	no	—
253	C₄×Q₈	—	178	1	no	—
254	C₄×Q₈	—	178	1	no	—
255	C₄×D₄	—	356	1	no	—
256	C₄×D₄	—	356	1	no	—
257	C₄×D₄	—	178	1	no	—
258	C₄×D₄	—	178	1	no	—
259	#117	—	2	2	no	—
260	#117	—	356	1	no	—
261	#117	—	178	1	no	—
262	C₂×C₄²	—	356	1	no	—
263	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂×C₁₆	—	89	2	no	—
265	C₂×C₁₆	—	89	2	no	—
266	C₂×C₁₆	—	178	4	no	—
267	#107	—	2	1	no	—
268	#107	—	2	1	no	—
269	#107	—	89	1	no	—
270	#107	—	89	1	no	—
271	#107	C₂²×D₈₉	1	2	yes	—
272	#107	C₂²×D₈₉	1	2	yes	—
273	#106	—	2	2	no	—
274	#106	—	2	2	no	—
275	#106	—	178	1	no	—
276	#103	—	178	1	no	—
277	#103	—	178	1	no	—
278	#102	—	178	1	no	—
279	#102	—	178	1	no	—
280	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
281	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
284	#41	—	356	2	no	—
285	#41	—	178	4	no	—
286	C₂×Q₈	—	4	1	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	178	1	no	—
289	C₂×D₄	—	356	1	no	—
290	C₂×D₄	—	178	1	no	—
291	C₂²×C₄	—	356	1	no	—
292	C₂²×C₄	—	356	1	no	—
293	C₂²×C₄	—	356	1	no	—
294	C₂²×C₄	#13083559	1	1	yes	characteristic
295	C₂²×C₄	#13083559	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
297	Q₁₆	—	89	2	no	—
298	Q₁₆	—	178	1	no	—
299	SD₁₆	—	178	4	no	—
300	D₈	—	89	2	no	—
301	D₈	—	178	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
304	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
306	C₂×C₈	Q₈×D₈₉	1	2	yes	—
307	C₂×C₈	#13083575	1	2	yes	—
308	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	356	1	no	—
316	C₄⋊C₄	—	356	1	no	—
317	C₄⋊C₄	—	178	1	no	—
318	C₄⋊C₄	—	178	1	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	356	1	no	—
321	C₂²⋊C₄	—	356	1	no	—
322	C₂²⋊C₄	—	356	1	no	—
323	C₄²	—	4	2	no	—
324	C₄²	—	356	1	no	—
325	C₄²	—	356	1	no	—
326	C₄²	—	356	1	no	—
327	C₄²	#13083559	1	1	yes	characteristic
328	C₄²	#13083559	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
330	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₁₆	—	178	2	no	—
332	C₁₆	—	178	2	no	—
333	C₂³	—	356	1	no	—
334	C₂³	C₂×#13083559	1	1	yes	characteristic
335	Q₈	—	8	2	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	356	1	no	—
338	Q₈	—	178	2	no	—
339	D₄	—	356	4	no	—
340	D₄	—	356	1	no	—
341	D₄	—	178	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	356	1	no	—
347	C₂×C₄	—	356	4	no	—
348	C₂×C₄	—	356	1	no	—
349	C₂×C₄	—	356	1	no	—
350	C₂×C₄	—	712	2	no	—
351	C₂×C₄	—	712	2	no	—
352	C₂×C₄	#15667988	1	2	yes	—
353	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
354	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
355	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
356	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
357	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
358	C₂×C₄	#15668043	1	2	yes	—
359	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	#15667988	1	2	yes	—
363	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
364	C₈	C₂×#13083575	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	356	2	no	—
368	C₂²	—	712	1	no	—
369	C₂²	—	712	1	no	—
370	C₂²	#19151875	1	2	yes	—
371	C₂²	C₂²×#13083559	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	356	2	no	—
378	C₄	—	712	1	no	—
379	C₄	—	712	1	no	—
380	C₄	#19151875	1	2	yes	—
381	C₄	#19152028	1	2	yes	—
382	C₄	C₂×#15667988	1	1	yes	characteristic
383	C₄	C₂×#15668043	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	712	2	no	—
386	C₂	C₂×#19151875	1	1	yes	characteristic
387	C₂	C₂×#19152028	1	1	yes	characteristic
388	C₂	#26033386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3160	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15668034	C₂²
hypercenter	#3160	D₈₉
hypocenter	C₈₉	#33626
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #26022413	C₂	semidirect
3. #26022406	C₂	semidirect
4. #26022401	C₂	semidirect
5. #26022396	C₂	semidirect
6. #26022391	C₂	semidirect
7. #26022386	C₂	semidirect
8. #26022034	C₂	semidirect
9. C₂×C₄⋊#13083549	C₂	semidirect
11. C₈₉×#3160	C₂	semidirect
45. #15668034	Q₈	semidirect
195. #3160	D₈₉	semidirect
206. C₈₉	#33626	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668034	2
#26022413	2
#26022406	2
#26022401	2
#26022396	2
#26022391	2
#26022386	2
#26022034	2
C₂×C₄⋊#13083549	2
#26021957	2
C₈₉×#3160	2
#33626	89