Subgroups — #36471295

Order 22784.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033354	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029153	C₂	1	1	yes	maximal, characteristic
3	#26021880	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151381	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018340	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018337	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2113	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152396	C₄	1	1	yes	characteristic
9	C₂×#15668018	C₂²	1	1	yes	characteristic
10	#19152016	—	2	2	no	—
11	#19152014	—	2	2	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	M₃₂×D₈₉	—	2	2	no	—
15	#19151718	—	2	2	no	—
16	#19151713	—	2	2	no	—
17	#19151646	C₄	1	1	yes	characteristic
18	#19151615	C₄	1	1	yes	characteristic
19	#19151614	C₄	1	1	yes	characteristic
20	#19151553	C₂²	1	1	yes	characteristic
21	#19151382	—	2	2	no	—
22	#19151381	—	2	2	no	—
23	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²⋊C₁₆×C₈₉	—	2	2	no	—
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	#15668043	C₈	1	2	yes	—
29	#15668042	—	2	1	no	—
30	#15668038	C₈	1	2	yes	—
31	#15668037	—	2	1	no	—
32	#15668032	—	2	2	no	—
33	#15668018	—	2	1	no	—
34	#15668018	—	2	1	no	—
35	#15668018	D₄	1	2	yes	—
36	#15668018	D₄	1	2	yes	—
37	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083557	—	2	1	no	—
40	C₂×C₈×D₈₉	—	2	1	no	—
41	#15667971	C₂×C₄	1	1	yes	characteristic
42	#15667952	—	2	1	no	—
43	#15667952	—	2	1	no	—
44	#15667952	—	2	1	no	—
45	#15667952	D₄	1	2	yes	—
46	#15667951	—	2	4	no	—
47	C₂×#13083549	—	2	1	no	—
48	C₂×#13083549	—	2	1	no	—
49	C₁₆×D₈₉	—	2	4	no	—
50	#15667947	—	2	2	no	—
51	#15667939	—	2	2	no	—
52	#15667938	—	2	2	no	—
53	#15667922	—	2	1	no	—
54	C₈×Dic₈₉	—	2	1	no	—
55	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
56	#131×C₈₉	—	2	1	no	—
57	#131×C₈₉	—	2	1	no	—
58	#131×C₈₉	—	2	1	no	—
59	#131×C₈₉	—	2	1	no	—
60	C₂×M₁₆×C₈₉	—	2	1	no	—
61	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	—	2	1	no	—
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	M₃₂×C₈₉	—	2	1	no	—
65	M₃₂×C₈₉	—	2	1	no	—
66	C₂×C₁₆×C₈₉	—	2	1	no	—
67	C₂×C₁₆×C₈₉	—	2	1	no	—
68	C₂×C₁₆×C₈₉	—	2	1	no	—
69	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
70	#13083574	—	2	1	no	—
71	#13083574	—	2	1	no	—
72	#13083574	C₂²⋊C₄	1	2	yes	—
73	#13083574	C₂×C₈	1	2	yes	—
74	C₂×Dic₁₇₈	M₁₆	1	1	yes	characteristic
75	C₂×D₃₅₆	M₁₆	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	#13083564	—	2	2	no	—
79	#13083562	—	2	2	no	—
80	#13083561	—	2	2	no	—
81	#13083559	—	2	2	no	—
82	#13083558	—	2	1	no	—
83	#13083558	—	2	1	no	—
84	#13083558	C₂²⋊C₄	1	2	yes	—
85	#13083558	C₂×C₈	1	2	yes	—
86	#13083557	—	2	2	no	—
87	#13083557	—	2	2	no	—
88	C₂×#10623494	—	2	1	no	—
89	C₂×#10623494	—	2	1	no	—
90	C₈×D₈₉	—	2	2	no	—
91	C₈×D₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₂²⋊Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	#13083549	—	2	2	no	—
96	#13083549	—	4	1	no	—
97	#41×C₈₉	—	2	1	no	—
98	#41×C₈₉	—	2	1	no	—
99	#41×C₈₉	—	2	1	no	—
100	#41×C₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
102	C₂×D₄×C₈₉	—	2	1	no	—
103	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	—	2	1	no	—
105	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	M₁₆×C₈₉	—	2	1	no	—
107	M₁₆×C₈₉	—	2	1	no	—
108	M₁₆×C₈₉	—	4	1	no	—
109	M₁₆×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	—	4	1	no	—
112	C₂×C₈×C₈₉	—	4	1	no	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
118	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₁₆×C₈₉	—	4	1	no	—
121	C₂³×C₈₉	—	2	1	no	—
122	C₂³×C₈₉	#131	1	1	yes	characteristic
123	C₂²×D₈₉	—	2	1	no	—
124	Q₈×C₈₉	—	2	1	no	—
125	Q₈×C₈₉	—	2	1	no	—
126	D₄×C₈₉	—	2	1	no	—
127	D₄×C₈₉	—	2	1	no	—
128	D₄×C₈₉	—	4	1	no	—
129	D₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
139	C₂²⋊D₈₉	—	2	2	no	—
140	C₂²⋊D₈₉	—	2	2	no	—
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	—	4	1	no	—
143	D₃₅₆	—	2	1	no	—
144	D₃₅₆	C₂²⋊C₈	1	2	yes	—
145	C₄×D₈₉	—	2	2	no	—
146	C₄×D₈₉	—	2	2	no	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	C₂²⋊C₈	1	2	yes	—
149	C₈×C₈₉	—	4	1	no	—
150	C₈×C₈₉	#120	1	1	yes	characteristic
151	C₈×C₈₉	#121	1	1	yes	characteristic
152	C₈×C₈₉	#121	1	1	yes	characteristic
153	C₈×C₈₉	#123	1	1	yes	characteristic
154	#10623494	—	2	2	no	—
155	#10623494	—	2	2	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	—	4	2	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	#408	1	1	yes	characteristic
160	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
161	C₂²×C₈₉	#435	1	1	yes	characteristic
162	C₂×D₈₉	—	2	2	no	—
163	C₂×D₈₉	—	8	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	#386	1	1	yes	characteristic
166	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
168	C₄×C₈₉	#435	1	1	yes	characteristic
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	8	1	no	—
171	#12979	—	89	1	no	maximal, Sylow
172	C₂×C₈₉	—	2	1	no	—
173	C₂×C₈₉	—	2	1	no	—
174	C₂×C₈₉	—	2	1	no	—
175	C₂×C₈₉	—	8	1	no	—
176	C₂×C₈₉	#1869	1	1	yes	characteristic
177	D₈₉	—	4	2	no	—
178	#3407	—	89	1	no	—
179	#2955	—	89	1	no	—
180	#2117	—	89	1	no	—
181	C₂×#350	—	89	1	no	—
182	#2113	D₈₉	1	1	yes	characteristic, complemented
183	#1379	—	89	1	no	—
184	#1378	—	89	1	no	—
185	C₈₉	#12979	1	1	yes	characteristic, complemented, Sylow
186	#575	—	89	1	no	—
187	C₂×#131	—	89	1	no	—
188	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
189	#504	—	178	2	no	—
190	#504	—	178	2	no	—
191	C₂×M₃₂	—	89	1	no	—
192	C₂×M₃₂	—	89	1	no	—
193	C₂×M₃₂	—	178	2	no	—
194	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
195	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
196	#444	—	178	2	no	—
197	#443	—	178	2	no	—
198	#430	—	89	1	no	—
199	#421	—	89	1	no	—
200	#421	—	89	1	no	—
201	#405	—	89	1	no	—
202	#350	—	178	2	no	—
203	#350	—	178	2	no	—
204	C₂²⋊C₁₆	—	2	2	no	—
205	C₂×#41	—	89	1	no	—
206	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
207	#137	—	89	2	no	—
208	#137	—	178	1	no	—
209	AGL(1,ℤ/8)	—	89	2	no	—
210	AGL(1,ℤ/8)	—	178	1	no	—
211	#135	—	178	2	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	2	1	no	—
216	#131	—	89	2	no	—
217	#131	—	89	2	no	—
218	#131	—	178	1	no	—
219	#131	—	178	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	89	1	no	—
222	C₂×M₁₆	—	89	1	no	—
223	C₂×M₁₆	—	178	1	no	—
224	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	178	1	no	—
227	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
228	#117	—	89	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	—	89	2	no	—
232	M₃₂	—	178	4	no	—
233	M₃₂	—	178	1	no	—
234	M₃₂	—	178	1	no	—
235	M₃₂	—	178	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	178	4	no	—
240	C₂×C₁₆	—	178	1	no	—
241	C₂×C₁₆	—	178	1	no	—
242	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
243	#108	—	178	2	no	—
244	#104	—	178	2	no	—
245	#104	—	178	2	no	—
246	#97	—	178	1	no	—
247	C₄×C₈	—	178	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	—	2	1	no	—
252	#41	—	89	2	no	—
253	#41	—	89	2	no	—
254	#41	—	178	1	no	—
255	#41	—	178	1	no	—
256	C₂×Q₈	—	89	1	no	—
257	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	89	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
265	Q₁₆	—	178	2	no	—
266	SD₁₆	—	178	2	no	—
267	SD₁₆	—	178	2	no	—
268	D₈	—	178	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	89	2	no	—
274	M₁₆	—	89	2	no	—
275	M₁₆	—	178	1	no	—
276	M₁₆	—	178	1	no	—
277	M₁₆	—	178	2	no	—
278	M₁₆	—	178	2	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	178	1	no	—
283	C₂×C₈	—	178	1	no	—
284	C₂×C₈	—	178	2	no	—
285	C₂×C₈	—	178	2	no	—
286	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
287	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₂×C₈	#13083573	1	1	yes	characteristic
291	C₂×C₈	#13083574	1	1	yes	characteristic
292	C₄⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₄²	—	178	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	4	1	no	—
297	C₁₆	—	356	1	no	—
298	C₁₆	—	178	2	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	178	1	no	—
301	C₂³	#15668018	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	89	2	no	—
305	Q₈	—	178	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	1	no	—
310	D₄	—	89	2	no	—
311	D₄	—	178	2	no	—
312	D₄	—	178	2	no	—
313	D₄	—	178	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	356	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	—	178	2	no	—
320	C₂×C₄	—	178	2	no	—
321	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
322	C₂×C₄	#15667972	1	1	yes	characteristic
323	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
325	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
326	C₂×C₄	#15668020	1	1	yes	characteristic
327	C₈	—	4	1	no	—
328	C₈	—	178	2	no	—
329	C₈	—	178	2	no	—
330	C₈	#15667980	1	1	yes	characteristic
331	C₈	#15667983	1	1	yes	characteristic
332	C₈	#15667985	1	1	yes	characteristic
333	C₈	#15667994	1	1	yes	characteristic
334	C₂²	—	4	1	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	178	2	no	—
338	C₂²	—	712	1	no	—
339	C₂²	#19151563	1	1	yes	characteristic
340	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
341	C₂²	#19151672	1	1	yes	characteristic
342	C₄	—	4	1	no	—
343	C₄	—	178	2	no	—
344	C₄	—	712	1	no	—
345	C₄	#19151474	1	1	yes	characteristic
346	C₄	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
347	C₄	#19151661	1	1	yes	characteristic
348	C₄	#19151667	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	8	1	no	—
353	C₂	—	356	2	no	—
354	C₂	#26020541	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667980
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2113	C₂
socle	C₂×C₈₉	#1869
prime radical	#19152396	C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2113	D₈₉
hypocenter	C₈₉	#12979
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
1. #26033354	C₂	semidirect
3. #26021880	C₂	semidirect
4. C₂×#19151381	C₂	semidirect
5. #26018340	C₂	semidirect
6. #26018337	C₂	semidirect
7. C₈₉×#2113	C₂	semidirect
182. #2113	D₈₉	semidirect
185. C₈₉	#12979	semidirect

Maximal subgroups

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

subgroup	index
#26033354	2
#26029153	2
#26021880	2
C₂×#19151381	2
#26018340	2
#26018337	2
C₈₉×#2113	2
#12979	89