Subgroups — #36707881

Order 22784.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033386	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151719	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029053	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022153	C₂	1	4	yes	maximal, complemented
5	#26022146	C₂	1	4	yes	maximal, complemented
6	#26022026	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#19151375	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2148	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	C₂²	1	1	yes	characteristic
11	C₂×#15668020	—	2	1	no	—
12	#41×Dic₈₉	—	2	1	no	—
13	#19152039	C₂²	1	2	yes	—
14	#19152038	C₄	1	2	yes	complemented
15	#19152034	C₂²	1	2	yes	—
16	#19152033	C₄	1	2	yes	complemented
17	#19152028	C₄	1	4	yes	complemented
18	C₂×#15667952	C₂²	1	1	yes	characteristic
19	#19151719	—	2	2	no	—
20	#19151719	C₂²	1	4	yes	—
21	#19151710	C₂²	1	2	yes	complemented
22	#19151705	C₂²	1	2	yes	complemented
23	#19151699	—	2	2	no	—
24	#19151636	C₂²	1	1	yes	characteristic
25	C₂×#15667938	—	2	1	no	—
26	#19151594	—	2	1	no	—
27	C₂×#15667922	C₂²	1	1	yes	characteristic
28	#19151422	C₂²	1	4	yes	complemented
29	#19151409	C₂²	1	4	yes	—
30	#19151375	C₂²	1	4	yes	complemented
31	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
34	#362×C₈₉	C₂²	1	2	yes	complemented
35	#361×C₈₉	C₂²	1	2	yes	complemented
36	C₂×#13083574	—	2	1	no	—
37	#15668034	—	2	1	no	—
38	#15668034	—	2	1	no	—
39	#15668034	C₂×C₄	1	2	yes	complemented
40	#15668034	C₂³	1	2	yes	—
41	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic, complemented
42	C₂×#13083563	—	2	1	no	—
43	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic, complemented
44	#15668020	—	2	4	no	—
45	#15668020	—	4	2	no	—
46	C₂×#13083557	—	4	1	no	—
47	C₂²×#10623494	D₄	1	1	yes	characteristic
48	C₂×C₈×D₈₉	—	4	1	no	—
49	Q₈×Dic₈₉	—	2	2	no	—
50	D₄×Dic₈₉	—	2	2	no	—
51	D₄×Dic₈₉	—	4	2	no	—
52	#15667971	—	2	1	no	—
53	#15667971	—	4	1	no	—
54	C₂×C₄×Dic₈₉	—	4	1	no	—
55	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
56	#15667958	C₂×C₄	1	4	yes	complemented
57	#15667956	C₂×C₄	1	4	yes	complemented
58	#15667955	C₂×C₄	1	4	yes	complemented
59	#15667953	C₂×C₄	1	4	yes	complemented
60	#15667952	C₂×C₄	1	2	yes	complemented
61	#15667952	C₂³	1	2	yes	—
62	C₂×#13083549	C₂×C₄	1	2	yes	complemented
63	#15667946	—	2	1	no	—
64	#15667944	C₂³	1	2	yes	—
65	#15667938	—	2	4	no	—
66	#15667936	—	2	2	no	—
67	#15667933	—	2	2	no	—
68	#15667922	D₄	1	2	yes	—
69	#15667922	C₂³	1	2	yes	—
70	C₂×#41×C₈₉	—	2	1	no	—
71	#135×C₈₉	—	2	1	no	—
72	#135×C₈₉	—	2	1	no	—
73	#135×C₈₉	C₂×C₄	1	2	yes	—
74	#135×C₈₉	C₂×C₄	1	2	yes	—
75	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×SD₁₆×C₈₉	—	2	1	no	—
77	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₈₉	—	2	1	no	—
79	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	M₃₂×C₈₉	—	2	1	no	—
81	M₃₂×C₈₉	C₂³	1	2	yes	—
82	C₂×C₁₆×C₈₉	—	2	1	no	—
83	#108×C₈₉	—	2	1	no	—
84	#108×C₈₉	C₂³	1	2	yes	—
85	#13083574	—	2	2	no	—
86	#13083574	—	2	2	no	—
87	#13083574	—	4	1	no	—
88	#13083574	—	4	1	no	—
89	C₂×Dic₁₇₈	—	2	1	no	—
90	C₂×D₃₅₆	—	2	1	no	—
91	C₂×C₂²⋊D₈₉	—	4	1	no	—
92	C₂²×Dic₈₉	—	2	1	no	—
93	C₂²×Dic₈₉	—	4	2	no	—
94	C₂×C₄×D₈₉	—	4	1	no	—
95	Dic₃₅₆	—	2	1	no	—
96	Dic₃₅₆	C₂²×C₄	1	2	yes	—
97	#13083563	—	2	2	no	—
98	#13083563	—	2	2	no	—
99	D₇₁₂	—	2	1	no	—
100	D₇₁₂	C₂²×C₄	1	2	yes	—
101	#13083557	—	4	4	no	—
102	C₂×#10623494	—	2	1	no	—
103	C₂×#10623494	—	2	1	no	—
104	C₂×#10623494	C₂×D₄	1	2	yes	—
105	C₈×D₈₉	—	4	4	no	—
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	—	4	2	no	—
108	C₂²⋊Dic₈₉	—	4	2	no	—
109	C₂²⋊Dic₈₉	—	4	1	no	—
110	C₄×Dic₈₉	—	2	1	no	—
111	C₄×Dic₈₉	—	4	2	no	—
112	C₄×Dic₈₉	—	4	1	no	—
113	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
114	#13083549	C₂²×C₄	1	4	yes	—
115	#41×C₈₉	—	2	2	no	—
116	#41×C₈₉	—	2	2	no	—
117	#41×C₈₉	—	4	1	no	—
118	#41×C₈₉	—	4	1	no	—
119	C₂×Q₈×C₈₉	—	2	1	no	—
120	C₂×D₄×C₈₉	—	2	1	no	—
121	C₂×D₄×C₈₉	—	4	1	no	—
122	C₂²×C₄×C₈₉	—	4	1	no	—
123	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	Q₁₆×C₈₉	—	2	1	no	—
125	Q₁₆×C₈₉	C₂²×C₄	1	2	yes	—
126	SD₁₆×C₈₉	—	2	2	no	—
127	SD₁₆×C₈₉	—	2	2	no	—
128	D₈×C₈₉	—	2	1	no	—
129	D₈×C₈₉	C₂²×C₄	1	2	yes	—
130	M₁₆×C₈₉	—	2	2	no	—
131	M₁₆×C₈₉	—	4	1	no	—
132	C₂×C₈×C₈₉	—	4	1	no	—
133	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
137	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
138	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
139	C₁₆×C₈₉	—	2	2	no	—
140	C₂³×C₈₉	—	4	1	no	—
141	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂²×D₈₉	—	4	1	no	—
143	Q₈×C₈₉	—	2	2	no	—
144	Q₈×C₈₉	—	4	1	no	—
145	D₄×C₈₉	—	2	2	no	—
146	D₄×C₈₉	—	4	2	no	—
147	D₄×C₈₉	—	4	2	no	—
148	D₄×C₈₉	—	4	1	no	—
149	C₂×C₄×C₈₉	—	4	1	no	—
150	C₂×C₄×C₈₉	—	4	2	no	—
151	C₂×C₄×C₈₉	—	4	2	no	—
152	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂²⋊D₈₉	—	4	2	no	—
159	C₂²⋊D₈₉	—	4	2	no	—
160	C₂×Dic₈₉	—	2	2	no	—
161	C₂×Dic₈₉	—	4	1	no	—
162	C₂×Dic₈₉	—	4	2	no	—
163	C₂×Dic₈₉	—	8	1	no	—
164	C₂×Dic₈₉	—	8	4	no	—
165	D₃₅₆	—	2	2	no	—
166	D₃₅₆	—	4	1	no	—
167	C₄×D₈₉	—	4	2	no	—
168	C₄×D₈₉	—	4	2	no	—
169	Dic₁₇₈	—	2	2	no	—
170	Dic₁₇₈	—	4	1	no	—
171	C₈×C₈₉	—	4	2	no	—
172	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
173	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
174	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
175	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
176	#10623494	—	2	2	no	—
177	C₂²×C₈₉	—	4	1	no	—
178	C₂²×C₈₉	—	4	2	no	—
179	C₂²×C₈₉	—	8	2	no	—
180	C₂²×C₈₉	#442	1	1	yes	characteristic
181	C₂²×C₈₉	#442	1	1	yes	characteristic
182	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂×D₈₉	—	4	2	no	—
184	C₂×D₈₉	—	8	2	no	—
185	C₄×C₈₉	—	4	2	no	—
186	C₄×C₈₉	#442	1	1	yes	characteristic
187	C₄×C₈₉	#442	1	1	yes	characteristic
188	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	Dic₈₉	—	4	2	no	—
191	Dic₈₉	—	4	2	no	—
192	Dic₈₉	—	8	2	no	—
193	#33468	—	89	1	no	maximal, Sylow
194	C₂×C₈₉	—	2	1	no	—
195	C₂×C₈₉	—	2	1	no	—
196	C₂×C₈₉	—	2	1	no	—
197	C₂×C₈₉	—	8	2	no	—
198	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
199	D₈₉	—	8	2	no	—
200	#3414	—	89	1	no	—
201	C₂×#444	—	89	1	no	—
202	#2943	—	89	1	no	—
203	AGL(1,ℤ/16)	—	89	4	no	—
204	#2180	—	89	4	no	—
205	#2155	—	89	1	no	—
206	#2148	D₈₉	1	1	yes	characteristic, complemented
207	#2145	—	89	1	no	—
208	C₂×#347	—	89	1	no	—
209	C₈₉	#33468	1	1	yes	characteristic, complemented, Sylow
210	C₂×#135	—	89	1	no	—
211	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
212	C₂×#131	—	178	1	no	—
213	C₄×#41	—	178	1	no	—
214	#510	—	89	2	no	—
215	#510	—	89	2	no	—
216	#509	—	89	2	no	—
217	#509	—	89	2	no	—
218	#508	—	89	4	no	—
219	C₂×M₃₂	—	89	1	no	—
220	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
221	#444	—	89	4	no	—
222	#444	—	178	2	no	—
223	#442	—	89	2	no	—
224	#441	—	89	2	no	—
225	#440	—	178	2	no	—
226	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
227	#427	—	89	1	no	—
228	C₂×#104	—	178	1	no	—
229	#416	—	178	1	no	—
230	C₂×#97	—	89	1	no	—
231	#365	—	89	4	no	—
232	#362	C₂×D₈₉	1	2	yes	complemented
233	#361	C₂×D₈₉	1	2	yes	complemented
234	#360	—	89	4	no	—
235	#347	—	89	4	no	—
236	C₂×#41	—	2	1	no	—
237	C₂×#41	—	178	1	no	—
238	#135	—	2	1	no	—
239	#135	—	2	1	no	—
240	#135	—	89	2	no	—
241	#135	—	89	2	no	—
242	#135	—	178	1	no	—
243	#135	—	178	1	no	—
244	#135	C₄×D₈₉	1	2	yes	—
245	#135	C₄×D₈₉	1	2	yes	—
246	C₂×Q₁₆	—	89	1	no	—
247	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
248	C₂×SD₁₆	—	2	1	no	—
249	C₂×SD₁₆	—	178	1	no	—
250	C₂×D₈	—	89	1	no	—
251	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
252	#131	—	356	2	no	—
253	#131	—	178	4	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	356	1	no	—
256	C₂²×C₈	—	89	1	no	—
257	C₂²×C₈	—	356	1	no	—
258	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₄×Q₈	—	178	2	no	—
260	C₄×D₄	—	356	2	no	—
261	C₄×D₄	—	178	2	no	—
262	#117	—	356	1	no	—
263	#117	—	178	1	no	—
264	C₂×C₄²	—	89	1	no	—
265	C₂×C₄²	—	356	1	no	—
266	Q₃₂	—	89	4	no	—
267	SD₃₂	—	89	4	no	—
268	SD₃₂	—	89	4	no	—
269	D₁₆	—	89	4	no	—
270	M₃₂	—	2	1	no	—
271	M₃₂	—	89	2	no	—
272	M₃₂	—	89	2	no	—
273	M₃₂	C₂²×D₈₉	1	2	yes	—
274	C₂×C₁₆	—	2	1	no	—
275	C₂×C₁₆	—	89	2	no	—
276	#108	—	2	1	no	—
277	#108	C₂²×D₈₉	1	2	yes	—
278	#107	—	178	1	no	—
279	#106	—	89	2	no	—
280	#104	—	178	4	no	—
281	#103	—	178	2	no	—
282	#102	—	178	2	no	—
283	#97	—	89	2	no	—
284	#97	—	89	2	no	—
285	#41	—	2	2	no	—
286	#41	—	2	2	no	—
287	#41	—	4	1	no	—
288	#41	—	4	1	no	—
289	#41	—	356	1	no	—
290	#41	—	356	1	no	—
291	#41	—	178	2	no	—
292	#41	—	178	2	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	178	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	356	1	no	—
298	C₂×D₄	—	178	1	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	—	356	1	no	—
301	C₂²×C₄	—	356	2	no	—
302	C₂²×C₄	—	178	1	no	—
303	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
304	Q₁₆	—	2	1	no	—
305	Q₁₆	—	89	2	no	—
306	Q₁₆	—	178	1	no	—
307	Q₁₆	C₂×C₄×D₈₉	1	2	yes	—
308	SD₁₆	—	2	2	no	—
309	SD₁₆	—	2	2	no	—
310	SD₁₆	—	178	2	no	—
311	SD₁₆	—	178	2	no	—
312	D₈	—	2	1	no	—
313	D₈	—	89	2	no	—
314	D₈	—	178	1	no	—
315	D₈	C₂×C₄×D₈₉	1	2	yes	—
316	M₁₆	—	2	2	no	—
317	M₁₆	—	4	1	no	—
318	M₁₆	—	356	4	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	89	2	no	—
321	C₂×C₈	—	356	4	no	—
322	C₂×C₈	—	178	1	no	—
323	C₂×C₈	—	178	1	no	—
324	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
326	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
327	C₂×C₈	#13083573	1	1	yes	characteristic
328	C₂×C₈	#13083573	1	1	yes	characteristic
329	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
330	C₄⋊C₄	—	356	2	no	—
331	C₄⋊C₄	—	178	2	no	—
332	C₂²⋊C₄	—	356	1	no	—
333	C₂²⋊C₄	—	356	2	no	—
334	C₄²	—	89	2	no	—
335	C₄²	—	356	2	no	—
336	C₄²	—	356	1	no	—
337	C₄²	—	178	1	no	—
338	C₁₆	—	2	2	no	—
339	C₁₆	—	89	4	no	—
340	C₂³	—	4	1	no	—
341	C₂³	—	356	1	no	—
342	C₂³	#15668039	1	1	yes	characteristic
343	Q₈	—	2	2	no	—
344	Q₈	—	4	1	no	—
345	Q₈	—	356	1	no	—
346	Q₈	—	178	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	1	no	—
351	D₄	—	356	2	no	—
352	D₄	—	356	2	no	—
353	D₄	—	356	1	no	—
354	D₄	—	178	2	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	356	1	no	—
359	C₂×C₄	—	356	2	no	—
360	C₂×C₄	—	356	2	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	178	2	no	—
363	C₂×C₄	—	712	1	no	—
364	C₂×C₄	—	712	4	no	—
365	C₂×C₄	#15667972	1	1	yes	characteristic
366	C₂×C₄	#15667972	1	1	yes	characteristic
367	C₂×C₄	#15668036	1	1	yes	characteristic
368	C₂×C₄	#15668036	1	1	yes	characteristic
369	C₂×C₄	#15668039	1	1	yes	characteristic
370	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
371	C₈	—	4	2	no	—
372	C₈	—	178	2	no	—
373	C₈	#15667972	1	1	yes	characteristic
374	C₈	#15667972	1	1	yes	characteristic
375	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
376	C₈	C₂×#13083573	1	1	yes	characteristic
377	C₂²	—	4	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	356	2	no	—
381	C₂²	—	712	2	no	—
382	C₂²	#19151707	1	1	yes	characteristic
383	C₂²	#19151707	1	1	yes	characteristic
384	C₂²	C₂×#15668039	1	1	yes	characteristic
385	C₄	—	4	2	no	—
386	C₄	—	356	2	no	—
387	C₄	—	356	2	no	—
388	C₄	—	712	2	no	—
389	C₄	#19151707	1	1	yes	characteristic
390	C₄	#19151707	1	1	yes	characteristic
391	C₄	C₂×#15667972	1	1	yes	characteristic
392	C₄	C₂×#15668036	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	8	2	no	—
397	C₂	—	712	2	no	—
398	C₂	C₂×#19151707	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151707
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2148	C₂
socle	C₂×C₈₉	C₂×#442
prime radical	#26033386	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2148	D₈₉
hypocenter	C₈₉	#33468
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2148

Complements

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

N	H ≅ G/N	type
2. C₂×#19151719	C₂	semidirect
3. #26029053	C₂	semidirect
4. #26022153	C₂	semidirect
5. #26022146	C₂	semidirect
6. #26022026	C₂	semidirect
7. #26021989	C₂	semidirect
8. C₂×#19151375	C₂	semidirect
9. C₈₉×#2148	C₂	semidirect
14. #19152038	C₄	semidirect
16. #19152033	C₄	semidirect
17. #19152028	C₄	semidirect
21. #19151710	C₂²	semidirect
22. #19151705	C₂²	semidirect
28. #19151422	C₂²	semidirect
30. #19151375	C₂²	semidirect
34. #362×C₈₉	C₂²	semidirect
35. #361×C₈₉	C₂²	semidirect
39. #15668034	C₂×C₄	semidirect
41. C₂×Dic₃₅₆	C₂×C₄	semidirect
43. C₂×D₇₁₂	C₂×C₄	semidirect
56. #15667958	C₂×C₄	semidirect
57. #15667956	C₂×C₄	semidirect
58. #15667955	C₂×C₄	semidirect
59. #15667953	C₂×C₄	semidirect
60. #15667952	C₂×C₄	semidirect
62. C₂×#13083549	C₂×C₄	semidirect
206. #2148	D₈₉	semidirect
209. C₈₉	#33468	semidirect
232. #362	C₂×D₈₉	semidirect
233. #361	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033386	2
C₂×#19151719	2
#26029053	2
#26022153	2
#26022146	2
#26022026	2
#26021989	2
C₂×#19151375	2
C₈₉×#2148	2
#33468	89