Subgroups — #36709776

Order 22784.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030746	C₂	1	1	yes	maximal, characteristic
2	#26022499	C₂	1	2	yes	maximal, complemented
3	#26022491	C₂	1	2	yes	maximal, complemented
4	#26022486	C₂	1	2	yes	maximal, complemented
5	#26022481	C₂	1	2	yes	maximal, complemented
6	#26021971	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021962	C₂	1	1	yes	maximal, characteristic
8	#26021957	C₂	1	1	yes	maximal, characteristic
9	C₂×#19151401	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021849	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668045	—	2	1	no	—
13	C₂×#15668034	C₂²	1	1	yes	characteristic
14	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
15	#19152174	—	2	1	no	—
16	C₂×#15667952	C₂²	1	1	yes	characteristic
17	C₂²×#13083549	C₂²	1	1	yes	characteristic
18	#19151992	—	2	2	no	—
19	#19151982	C₂²	1	2	yes	—
20	#19151978	C₂²	1	2	yes	—
21	#19151401	C₂²	1	2	yes	—
22	#19151401	C₂²	1	2	yes	—
23	#19151401	C₂²	1	1	yes	characteristic
24	#19151401	C₂²	1	1	yes	characteristic
25	#19151401	C₂²	1	2	yes	—
26	#19151401	C₂²	1	2	yes	—
27	#19151401	C₂²	1	2	yes	—
28	#19151398	C₂²	1	1	yes	characteristic
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	2	yes	—
33	#544×C₈₉	—	2	1	no	—
34	#501×C₈₉	—	2	2	no	—
35	#500×C₈₉	C₂²	1	2	yes	—
36	#499×C₈₉	C₂²	1	2	yes	—
37	#428×C₈₉	—	2	1	no	—
38	#427×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
40	#405×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	—	2	1	no	—
42	C₂²×Dic₁₇₈	—	2	1	no	—
43	#15668045	—	4	4	no	—
44	#15668034	—	2	2	no	—
45	#15668034	—	2	2	no	—
46	C₂×Dic₃₅₆	—	2	2	no	—
47	C₂×Dic₃₅₆	—	2	1	no	—
48	C₂×Dic₃₅₆	—	2	2	no	—
49	C₂×Dic₃₅₆	—	2	2	no	—
50	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
51	C₂×Dic₃₅₆	C₂³	1	2	yes	—
52	C₂×#13083563	—	2	2	no	—
53	C₂×#13083563	—	2	1	no	—
54	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
55	#15668012	—	2	2	no	—
56	#15668000	—	2	2	no	—
57	#15667996	—	4	2	no	—
58	#15667992	—	4	1	no	—
59	#15667992	—	4	1	no	—
60	#15667952	—	2	2	no	—
61	C₂×#13083549	—	2	2	no	—
62	C₂×#13083549	C₂³	1	1	yes	characteristic
63	C₂×#13083549	C₂³	1	2	yes	—
64	C₂×#13083549	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₈₉	—	2	2	no	—
68	C₄⋊Q₈×C₈₉	—	4	1	no	—
69	#125×C₈₉	—	2	2	no	—
70	#125×C₈₉	—	4	1	no	—
71	C₂²⋊Q₈×C₈₉	—	4	2	no	—
72	#117×C₈₉	—	2	1	no	—
73	#117×C₈₉	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
75	#107×C₈₉	—	2	1	no	—
76	#107×C₈₉	—	2	2	no	—
77	#107×C₈₉	C₂³	1	2	yes	—
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#106×C₈₉	—	2	1	no	—
81	#106×C₈₉	—	2	2	no	—
82	#97×C₈₉	D₄	1	2	yes	—
83	C₄×C₈×C₈₉	C₂³	1	2	yes	—
84	#13083574	—	4	2	no	—
85	#13083574	—	4	2	no	—
86	C₂×Dic₁₇₈	—	2	2	no	—
87	C₂×Dic₁₇₈	—	2	1	no	—
88	C₂×Dic₁₇₈	—	4	2	no	—
89	C₂×Dic₁₇₈	—	4	2	no	—
90	C₂×Dic₁₇₈	—	4	1	no	—
91	C₂×D₃₅₆	—	2	1	no	—
92	C₂×C₂²⋊D₈₉	—	4	1	no	—
93	C₂²×Dic₈₉	—	4	1	no	—
94	C₂×C₄×D₈₉	—	4	1	no	—
95	Dic₃₅₆	—	2	2	no	—
96	Dic₃₅₆	—	2	2	no	—
97	Dic₃₅₆	—	2	1	no	—
98	Dic₃₅₆	—	2	1	no	—
99	Dic₃₅₆	—	4	2	no	—
100	Dic₃₅₆	—	4	4	no	—
101	#13083563	—	4	2	no	—
102	#13083563	—	4	4	no	—
103	D₇₁₂	—	2	1	no	—
104	D₇₁₂	—	2	1	no	—
105	C₄⋊Dic₈₉	—	4	2	no	—
106	C₄⋊Dic₈₉	—	4	2	no	—
107	#13083551	—	4	2	no	—
108	#13083551	—	4	2	no	—
109	#13083549	—	2	1	no	—
110	#13083549	—	2	1	no	—
111	#13083549	—	2	2	no	—
112	C₂×Q₈×C₈₉	—	4	2	no	—
113	C₂²×C₄×C₈₉	—	4	1	no	—
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	M₁₆×C₈₉	SD₁₆	1	4	yes	—
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₈₉	#41	1	2	yes	—
119	C₂×C₈×C₈₉	#41	1	2	yes	—
120	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	—	2	1	no	—
124	C₄⋊C₄×C₈₉	—	4	1	no	—
125	C₄⋊C₄×C₈₉	—	4	2	no	—
126	C₄⋊C₄×C₈₉	—	4	2	no	—
127	C₄⋊C₄×C₈₉	—	4	2	no	—
128	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
129	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
130	C₂²⋊C₄×C₈₉	—	4	1	no	—
131	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
132	C₄²×C₈₉	—	4	1	no	—
133	C₄²×C₈₉	C₂×D₄	1	2	yes	—
134	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₈₉	—	4	1	no	—
136	Q₈×C₈₉	—	16	2	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	16	1	no	—
141	C₂×C₄×C₈₉	#124	1	2	yes	—
142	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₈₉	—	8	2	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	8	2	no	—
151	D₃₅₆	—	4	1	no	—
152	D₃₅₆	—	4	1	no	—
153	C₄×D₈₉	—	8	2	no	—
154	Dic₁₇₈	—	4	1	no	—
155	Dic₁₇₈	—	4	2	no	—
156	Dic₁₇₈	—	4	2	no	—
157	Dic₁₇₈	—	4	1	no	—
158	Dic₁₇₈	—	8	2	no	—
159	C₈×C₈₉	#124	1	2	yes	—
160	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
161	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
162	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	#486	1	2	yes	—
166	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₈₉	—	8	1	no	—
168	C₂×D₈₉	—	8	1	no	—
169	C₂×D₈₉	—	8	1	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	—	2	2	no	—
172	C₄×C₈₉	—	8	2	no	—
173	C₄×C₈₉	—	8	1	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	#486	1	2	yes	—
176	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
177	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
178	Dic₈₉	—	8	2	no	—
179	Dic₈₉	—	8	1	no	—
180	#33703	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	—	2	2	no	—
182	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
183	C₂×C₈₉	#3416	1	1	yes	characteristic
184	C₂×C₈₉	#3417	1	1	yes	characteristic
185	D₈₉	—	16	1	no	—
186	#3164	D₈₉	1	1	yes	characteristic, complemented
187	#3156	—	89	1	no	—
188	#2245	—	89	2	no	—
189	#2243	—	89	2	no	—
190	#2242	—	89	2	no	—
191	#2241	—	89	2	no	—
192	#2141	—	89	1	no	—
193	#2139	—	89	1	no	—
194	#2138	—	89	1	no	—
195	C₂×#358	—	89	1	no	—
196	#2108	—	89	1	no	—
197	C₈₉	#33703	1	1	yes	characteristic, complemented, Sylow
198	C₂×#137	—	178	1	no	—
199	C₂×#135	—	89	1	no	—
200	C₂²×Q₁₆	—	89	1	no	—
201	#544	—	2	1	no	—
202	#536	—	178	1	no	—
203	C₂×M₃₂	—	89	1	no	—
204	C₂²×C₁₆	—	89	1	no	—
205	#501	—	2	2	no	—
206	#500	C₂×D₈₉	1	2	yes	—
207	#499	C₂×D₈₉	1	2	yes	—
208	#497	—	178	2	no	—
209	#495	—	89	2	no	—
210	#494	—	89	2	no	—
211	#428	—	2	1	no	—
212	#427	C₂×D₈₉	1	1	yes	characteristic
213	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
214	#405	C₂×D₈₉	1	1	yes	characteristic
215	#358	—	89	2	no	—
216	#358	—	89	2	no	—
217	#358	—	89	1	no	—
218	#358	—	89	1	no	—
219	#358	—	89	2	no	—
220	#358	—	89	2	no	—
221	#358	—	89	2	no	—
222	#357	—	89	1	no	—
223	#357	—	89	1	no	—
224	#357	—	89	2	no	—
225	#346	—	89	2	no	—
226	C₄×C₁₆	—	89	2	no	—
227	C₂×#41	—	178	1	no	—
228	C₂²×Q₈	—	178	1	no	—
229	#137	—	356	4	no	—
230	#135	—	178	2	no	—
231	#135	—	178	2	no	—
232	C₂×Q₁₆	—	89	2	no	—
233	C₂×Q₁₆	—	89	1	no	—
234	C₂×Q₁₆	—	178	1	no	—
235	C₂×Q₁₆	—	178	2	no	—
236	C₂×Q₁₆	—	178	2	no	—
237	C₂×Q₁₆	—	178	2	no	—
238	C₂×SD₁₆	—	178	1	no	—
239	C₂×SD₁₆	—	178	2	no	—
240	C₂×D₈	—	89	1	no	—
241	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₄⋊Q₈	—	2	2	no	—
244	C₄⋊Q₈	—	4	1	no	—
245	C₄⋊Q₈	—	178	2	no	—
246	#125	—	2	2	no	—
247	#125	—	4	1	no	—
248	#124	—	178	2	no	—
249	#123	—	356	2	no	—
250	C₂²⋊Q₈	—	4	2	no	—
251	C₂²⋊Q₈	—	356	1	no	—
252	C₂²⋊Q₈	—	356	1	no	—
253	#117	—	2	1	no	—
254	#117	C₂²⋊D₈₉	1	1	yes	characteristic
255	C₂×C₄⋊C₄	—	2	1	no	—
256	M₃₂	—	178	2	no	—
257	C₂×C₁₆	—	89	1	no	—
258	C₂×C₁₆	—	89	2	no	—
259	C₂×C₁₆	—	89	1	no	—
260	C₂×C₁₆	—	178	2	no	—
261	#107	—	2	1	no	—
262	#107	—	2	2	no	—
263	#107	C₂²×D₈₉	1	2	yes	—
264	#107	C₂²×D₈₉	1	1	yes	characteristic
265	#107	C₂²×D₈₉	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	2	2	no	—
268	#97	C₂²⋊D₈₉	1	2	yes	—
269	C₄×C₈	C₂²×D₈₉	1	2	yes	—
270	#41	—	356	2	no	—
271	#41	—	356	2	no	—
272	C₂×Q₈	—	4	2	no	—
273	C₂×Q₈	—	356	1	no	—
274	C₂×Q₈	—	356	2	no	—
275	C₂×Q₈	—	356	2	no	—
276	C₂×Q₈	—	178	2	no	—
277	C₂×Q₈	—	178	1	no	—
278	C₂×D₄	—	356	1	no	—
279	C₂×D₄	—	178	1	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	356	1	no	—
282	C₂²×C₄	—	356	1	no	—
283	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
284	Q₁₆	—	356	2	no	—
285	Q₁₆	—	356	4	no	—
286	Q₁₆	—	178	2	no	—
287	Q₁₆	—	178	2	no	—
288	Q₁₆	—	178	1	no	—
289	Q₁₆	—	178	1	no	—
290	SD₁₆	—	356	2	no	—
291	SD₁₆	—	356	4	no	—
292	D₈	—	178	1	no	—
293	D₈	—	178	1	no	—
294	M₁₆	#13083561	1	4	yes	—
295	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
297	C₂×C₈	#13083575	1	2	yes	—
298	C₂×C₈	#13083575	1	2	yes	—
299	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	356	2	no	—
308	C₄⋊C₄	—	356	2	no	—
309	C₄⋊C₄	#13083559	1	1	yes	characteristic
310	C₄⋊C₄	#13083559	1	1	yes	characteristic
311	C₂²⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	356	2	no	—
313	C₂²⋊C₄	—	356	2	no	—
314	C₂²⋊C₄	#13083559	1	2	yes	—
315	C₄²	—	4	1	no	—
316	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
317	C₁₆	—	178	1	no	—
318	C₁₆	—	178	1	no	—
319	C₁₆	—	178	2	no	—
320	C₂³	—	356	1	no	—
321	C₂³	C₂×#13083559	1	1	yes	characteristic
322	Q₈	—	16	2	no	—
323	Q₈	—	356	2	no	—
324	Q₈	—	356	1	no	—
325	Q₈	—	356	1	no	—
326	Q₈	—	356	2	no	—
327	Q₈	—	712	2	no	—
328	D₄	—	356	1	no	—
329	D₄	—	356	1	no	—
330	D₄	—	712	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	16	1	no	—
335	C₂×C₄	—	356	2	no	—
336	C₂×C₄	—	356	1	no	—
337	C₂×C₄	—	712	2	no	—
338	C₂×C₄	—	712	2	no	—
339	C₂×C₄	#15668001	1	2	yes	—
340	C₂×C₄	C₂×#13083559	1	2	yes	—
341	C₂×C₄	C₂×#13083559	1	2	yes	—
342	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
343	C₂×C₄	C₂×#13083561	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
345	C₈	#15668001	1	2	yes	—
346	C₈	C₂×#13083561	1	4	yes	—
347	C₈	C₂×#13083575	1	1	yes	characteristic
348	C₈	C₂×#13083575	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	712	1	no	—
352	C₂²	—	712	1	no	—
353	C₂²	—	712	1	no	—
354	C₂²	#19151946	1	2	yes	—
355	C₂²	C₂²×#13083559	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	712	1	no	—
362	C₄	—	712	2	no	—
363	C₄	#19151946	1	2	yes	—
364	C₄	C₂×#15668001	1	1	yes	characteristic
365	C₄	C₂²×#13083561	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	1424	1	no	—
368	C₂	C₂×#19151946	1	1	yes	characteristic
369	C₂	#26033399	1	1	yes	characteristic
370	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3164	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15668034	C₂²
hypercenter	#3164	D₈₉
hypocenter	C₈₉	#33703
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #26022499	C₂	semidirect
3. #26022491	C₂	semidirect
4. #26022486	C₂	semidirect
5. #26022481	C₂	semidirect
6. #26021971	C₂	semidirect
9. C₂×#19151401	C₂	semidirect
10. #26021849	C₂	semidirect
11. C₈₉×#3164	C₂	semidirect
186. #3164	D₈₉	semidirect
197. C₈₉	#33703	semidirect

Maximal subgroups

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

subgroup	index
#26030746	2
#26022499	2
#26022491	2
#26022486	2
#26022481	2
#26021971	2
#26021962	2
#26021957	2
C₂×#19151401	2
#26021849	2
C₈₉×#3164	2
#33703	89