Subgroups — #36709074

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	#26030766	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022416	C₂	1	2	yes	maximal, complemented
3	#26022409	C₂	1	2	yes	maximal, complemented
4	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022394	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022389	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	#26021959	C₂	1	2	yes	maximal
11	C₄×#135×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19152155	—	2	1	no	—
13	C₂²×#13083549	C₂²	1	2	yes	—
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152000	—	2	1	no	—
17	#19152000	C₂²	1	2	yes	—
18	#19151996	—	2	1	no	—
19	#19151636	—	2	1	no	—
20	#19151636	C₂²	1	2	yes	—
21	#19151425	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151423	C₂²	1	2	yes	—
24	#19151423	C₂²	1	2	yes	—
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	2	yes	—
27	C₄⋊#13083549	C₂²	1	1	yes	characteristic
28	#19151402	C₂²	1	2	yes	—
29	#19151402	C₂²	1	2	yes	—
30	#19151399	C₂²	1	2	yes	—
31	#19151399	C₂²	1	2	yes	—
32	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
33	C₄×#41×C₈₉	—	2	1	no	—
34	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₈₉	—	2	1	no	—
37	C₄×SD₁₆×C₈₉	—	2	1	no	—
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
40	#427×C₈₉	C₂²	1	1	yes	characteristic
41	#416×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
43	#15668012	—	2	1	no	—
44	#15668012	—	2	1	no	—
45	#15668004	—	2	2	no	—
46	#15667991	—	4	1	no	—
47	#15667991	—	4	1	no	—
48	C₄×Dic₁₇₈	—	4	1	no	—
49	C₄×Dic₁₇₈	—	4	1	no	—
50	#15667971	—	2	2	no	—
51	C₂×#13083549	—	2	4	no	—
52	C₂×#13083549	C₂³	1	2	yes	—
53	C₂×#13083549	C₂³	1	2	yes	—
54	#15667946	—	2	1	no	—
55	#15667946	—	2	1	no	—
56	#15667946	C₂³	1	2	yes	—
57	#15667946	C₂³	1	2	yes	—
58	#15667944	—	2	2	no	—
59	#15667944	—	2	2	no	—
60	C₂×#41×C₈₉	—	2	1	no	—
61	#135×C₈₉	—	2	2	no	—
62	#135×C₈₉	Q₈	1	4	yes	complemented
63	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×SD₁₆×C₈₉	—	2	1	no	—
65	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
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	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×D₄×C₈₉	—	2	1	no	—
71	C₄×D₄×C₈₉	—	2	1	no	—
72	C₄×D₄×C₈₉	—	4	1	no	—
73	C₄×D₄×C₈₉	—	4	1	no	—
74	#117×C₈₉	—	2	1	no	—
75	#117×C₈₉	—	4	1	no	—
76	C₂×C₄²×C₈₉	—	4	1	no	—
77	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#106×C₈₉	—	2	1	no	—
81	#103×C₈₉	—	2	1	no	—
82	#103×C₈₉	—	2	1	no	—
83	#102×C₈₉	—	2	1	no	—
84	#102×C₈₉	—	2	1	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	C₂×Dic₁₇₈	—	4	1	no	—
90	C₂×Dic₁₇₈	—	4	1	no	—
91	C₄⋊Dic₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	C₄⋊Dic₈₉	—	4	1	no	—
95	#13083553	—	4	2	no	—
96	#13083553	—	4	2	no	—
97	C₂²⋊Dic₈₉	—	4	2	no	—
98	C₄×Dic₈₉	—	4	2	no	—
99	#13083549	—	2	2	no	—
100	#13083549	—	2	2	no	—
101	#41×C₈₉	—	2	4	no	—
102	#41×C₈₉	—	4	2	no	—
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×D₄×C₈₉	—	2	1	no	—
105	C₂×D₄×C₈₉	—	4	1	no	—
106	C₂²×C₄×C₈₉	—	4	1	no	—
107	C₂²×C₄×C₈₉	—	4	1	no	—
108	C₂²×C₄×C₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	Q₁₆×C₈₉	—	2	1	no	—
113	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
114	SD₁₆×C₈₉	—	2	4	no	—
115	D₈×C₈₉	—	2	1	no	—
116	D₈×C₈₉	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
122	C₂×C₈×C₈₉	#41	1	2	yes	—
123	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₄⋊C₄×C₈₉	—	4	1	no	—
127	C₄⋊C₄×C₈₉	—	4	1	no	—
128	C₂²⋊C₄×C₈₉	—	4	1	no	—
129	C₂²⋊C₄×C₈₉	—	4	1	no	—
130	C₂²⋊C₄×C₈₉	—	4	1	no	—
131	C₄²×C₈₉	—	4	1	no	—
132	C₄²×C₈₉	—	4	1	no	—
133	C₄²×C₈₉	—	4	1	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	—	4	1	no	—
139	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₈₉	—	2	2	no	—
141	Q₈×C₈₉	—	4	1	no	—
142	D₄×C₈₉	—	2	2	no	—
143	D₄×C₈₉	—	4	4	no	—
144	D₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	4	4	no	—
149	C₂×C₄×C₈₉	—	4	1	no	—
150	C₂×C₄×C₈₉	—	4	1	no	—
151	C₂×C₄×C₈₉	—	8	2	no	—
152	C₂×C₄×C₈₉	—	8	2	no	—
153	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
154	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₈₉	#137	1	2	yes	—
160	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×Dic₈₉	—	4	2	no	—
162	C₂×Dic₈₉	—	4	2	no	—
163	Dic₁₇₈	—	8	2	no	—
164	Dic₁₇₈	—	8	2	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	—	4	2	no	—
173	C₂²×C₈₉	—	8	1	no	—
174	C₂²×C₈₉	—	8	1	no	—
175	C₂²×C₈₉	#474	1	2	yes	—
176	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	4	2	no	—
181	C₄×C₈₉	—	8	1	no	—
182	C₄×C₈₉	—	8	1	no	—
183	C₄×C₈₉	#474	1	2	yes	—
184	C₄×C₈₉	#508	1	2	yes	—
185	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
187	Dic₈₉	—	8	2	no	—
188	Dic₈₉	—	8	2	no	—
189	#33626	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	—	2	2	no	—
191	C₂×C₈₉	—	8	2	no	—
192	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
193	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
194	C₂×C₈₉	#3414	1	1	yes	characteristic
195	#3160	—	89	1	no	—
196	C₄×#135	D₈₉	1	1	yes	characteristic
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	C₂×D₈₉	1	1	yes	characteristic
208	#533	—	178	1	no	—
209	C₄×#41	—	2	1	no	—
210	C₂²×C₁₆	—	89	2	no	—
211	#500	—	89	1	no	—
212	#500	—	89	1	no	—
213	#499	—	89	2	no	—
214	#499	—	178	1	no	—
215	#498	—	178	1	no	—
216	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
217	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
218	C₄×SD₁₆	—	2	1	no	—
219	C₄×SD₁₆	—	2	1	no	—
220	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
221	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
222	#427	—	89	2	no	—
223	#427	—	178	1	no	—
224	#427	C₂×D₈₉	1	1	yes	characteristic
225	#416	—	2	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	—	2	1	no	—
239	#135	—	2	2	no	—
240	#135	Dic₁₇₈	1	4	yes	complemented
241	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₄⋊Q₈	—	178	1	no	—
247	C₄⋊Q₈	—	178	1	no	—
248	#125	—	178	2	no	—
249	C₂²⋊Q₈	—	356	1	no	—
250	C₂²⋊Q₈	—	356	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	—	356	1	no	—
254	C₄×Q₈	—	356	1	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	4	1	no	—
258	C₄×D₄	—	4	1	no	—
259	#117	—	2	1	no	—
260	#117	—	4	1	no	—
261	#117	—	178	2	no	—
262	C₂×C₄²	—	4	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	—	89	2	no	—
268	#107	—	89	2	no	—
269	#107	—	178	1	no	—
270	#107	—	178	1	no	—
271	#107	C₂²×D₈₉	1	1	yes	characteristic
272	#107	C₂²×D₈₉	1	1	yes	characteristic
273	#106	—	2	1	no	—
274	#106	—	178	2	no	—
275	#106	—	178	2	no	—
276	#103	—	2	1	no	—
277	#103	—	2	1	no	—
278	#102	—	2	1	no	—
279	#102	—	2	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	—	2	4	no	—
285	#41	—	4	2	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	356	1	no	—
288	C₂×Q₈	—	356	1	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	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	—	2	1	no	—
298	Q₁₆	C₂×Dic₁₇₈	1	2	yes	—
299	SD₁₆	—	2	4	no	—
300	D₈	—	2	1	no	—
301	D₈	C₂×Dic₁₇₈	1	2	yes	—
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	C₂×Dic₁₇₈	1	2	yes	—
307	C₂×C₈	#13083574	1	2	yes	—
308	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	356	2	no	—
314	C₄⋊C₄	—	356	2	no	—
315	C₄⋊C₄	—	356	1	no	—
316	C₄⋊C₄	—	356	1	no	—
317	C₄⋊C₄	—	178	2	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	4	1	no	—
321	C₂²⋊C₄	—	4	1	no	—
322	C₂²⋊C₄	—	356	2	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	4	1	no	—
325	C₄²	—	4	1	no	—
326	C₄²	—	356	2	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	—	4	1	no	—
334	C₂³	C₂×#13083559	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	4	1	no	—
337	Q₈	—	712	2	no	—
338	Q₈	—	712	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	4	4	no	—
341	D₄	—	4	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	356	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	#15667991	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	#15668042	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	#15667991	1	2	yes	—
363	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
364	C₈	C₂×#13083574	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	8	1	no	—
370	C₂²	#19151878	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	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	—	712	2	no	—
379	C₄	—	712	2	no	—
380	C₄	#19151878	1	2	yes	—
381	C₄	#19152028	1	2	yes	—
382	C₄	C₂×#15667991	1	1	yes	characteristic
383	C₄	C₂×#15668042	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	8	2	no	—
386	C₂	C₂×#19151878	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	C₄×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#135×C₈₉	C₂²
hypercenter	C₄×#135	D₈₉
hypocenter	C₈₉	#33626
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #26030766	C₂	semidirect
2. #26022416	C₂	semidirect
3. #26022409	C₂	semidirect
4. #26022404	C₂	semidirect
5. #26022399	C₂	semidirect
6. #26022394	C₂	semidirect
7. #26022389	C₂	semidirect
8. #26022034	C₂	semidirect
9. C₂×C₄⋊#13083549	C₂	semidirect
62. #135×C₈₉	Q₈	semidirect
206. C₈₉	#33626	semidirect
240. #135	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26022416	2
#26022409	2
#26022404	2
#26022399	2
#26022394	2
#26022389	2
#26022034	2
C₂×C₄⋊#13083549	2
#26021959	2
C₄×#135×C₈₉	2
#33626	89