Subgroups — #36585321

Order 22784.

608 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029190	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029156	C₂	1	1	yes	maximal, characteristic
3	#26021833	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021828	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021704	C₂	1	2	yes	maximal, complemented
6	#26021559	C₂	1	1	yes	maximal, characteristic
7	#26021540	C₂	1	1	yes	maximal, characteristic
8	#26021423	C₂	1	2	yes	maximal, complemented
9	#26020098	C₂	1	2	yes	maximal, complemented
10	#26020053	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152398	C₂²	1	1	yes	characteristic
13	C₂×#15668018	C₂²	1	2	yes	—
14	#19152151	C₂²	1	1	yes	characteristic
15	#19151728	—	2	2	no	—
16	#19151726	—	2	2	no	—
17	#19151721	—	2	2	no	—
18	#19151716	—	2	2	no	—
19	#19151647	C₂²	1	1	yes	characteristic
20	#19151626	C₂²	1	1	yes	characteristic
21	#19151625	C₂²	1	1	yes	characteristic
22	#19151621	C₂²	1	1	yes	characteristic
23	#19151616	C₂²	1	2	yes	—
24	C₂×#15667938	C₂²	1	2	yes	—
25	#19151592	C₂²	1	2	yes	—
26	#19151580	C₂²	1	2	yes	—
27	#19151571	C₂²	1	2	yes	—
28	#19151553	C₂²	1	1	yes	characteristic
29	C₂×#15667922	C₂²	1	1	yes	characteristic
30	#19151370	C₂²	1	2	yes	—
31	#19151368	C₂²	1	2	yes	—
32	#19151362	C₂²	1	2	yes	—
33	#19151357	C₂²	1	2	yes	—
34	#405×C₈₉	C₂²	1	2	yes	—
35	#405×C₈₉	C₂²	1	1	yes	characteristic
36	#338×C₈₉	C₂²	1	1	yes	characteristic
37	H(3,ℤ/4)×C₈₉	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#13083574	C₂³	1	2	yes	—
40	#15668045	—	2	2	no	—
41	#15668040	—	2	2	no	—
42	#15668034	—	2	2	no	—
43	#15668018	—	2	2	no	—
44	#15668018	—	2	2	no	—
45	#15668018	—	2	2	no	—
46	#15668018	—	2	2	no	—
47	C₂×#13083558	C₂³	1	2	yes	—
48	C₂×#13083558	C₂³	1	1	yes	characteristic
49	C₂×#13083557	—	2	2	no	—
50	C₂²×#10623494	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₈₉	—	2	2	no	—
52	#15668011	—	2	1	no	—
53	#15668003	—	2	1	no	—
54	#15667989	—	2	2	no	—
55	C₄×Dic₁₇₈	—	2	2	no	—
56	#15667971	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
58	#15667948	—	2	2	no	—
59	C₄⋊#10623494	—	2	2	no	—
60	#15667941	—	2	2	no	—
61	#15667941	—	2	2	no	—
62	#15667940	—	2	4	no	—
63	#15667938	—	2	2	no	—
64	#15667938	—	2	2	no	—
65	#15667935	—	2	2	no	—
66	#15667932	—	2	2	no	—
67	#15667930	—	2	2	no	—
68	#15667928	—	2	2	no	—
69	#15667926	—	2	4	no	—
70	#15667922	—	2	1	no	—
71	#15667922	—	2	1	no	—
72	#15667922	—	2	1	no	—
73	C₈×Dic₈₉	—	2	1	no	—
74	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	#117×C₈₉	C₂³	1	2	yes	—
78	#117×C₈₉	C₂³	1	1	yes	characteristic
79	#97×C₈₉	—	2	2	no	—
80	#97×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	—	2	2	no	—
82	C₄×C₈×C₈₉	D₄	1	2	yes	—
83	#13083574	—	2	2	no	—
84	#13083574	—	2	2	no	—
85	#13083574	—	2	2	no	—
86	#13083574	—	2	2	no	—
87	C₂×Dic₁₇₈	—	2	2	no	—
88	C₂×Dic₁₇₈	#41	1	2	yes	—
89	C₂×D₃₅₆	#41	1	2	yes	—
90	C₂×C₂²⋊D₈₉	—	2	2	no	—
91	C₂²×Dic₈₉	—	2	1	no	—
92	C₂×C₄×D₈₉	—	2	2	no	—
93	Dic₃₅₆	—	2	2	no	—
94	#13083563	—	2	4	no	—
95	D₇₁₂	—	2	2	no	—
96	#13083558	—	2	2	no	—
97	#13083558	—	2	2	no	—
98	#13083558	—	2	2	no	—
99	#13083557	—	4	2	no	—
100	#13083557	—	4	2	no	—
101	C₂×#10623494	—	2	1	no	—
102	C₂×#10623494	—	2	1	no	—
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₈×D₈₉	—	4	2	no	—
106	C₈×D₈₉	—	4	2	no	—
107	C₄⋊Dic₈₉	—	2	2	no	—
108	C₄⋊Dic₈₉	—	2	1	no	—
109	#13083553	—	2	2	no	—
110	#13083553	—	2	2	no	—
111	C₂²⋊Dic₈₉	#41	1	2	yes	—
112	C₄×Dic₈₉	—	2	1	no	—
113	C₄×Dic₈₉	—	2	1	no	—
114	C₄×Dic₈₉	#41	1	1	yes	characteristic
115	C₄×Dic₈₉	#41	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₈₉	—	2	2	no	—
118	M₁₆×C₈₉	—	2	2	no	—
119	M₁₆×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₈₉	#41	1	2	yes	—
125	C₂×C₈×C₈₉	#41	1	2	yes	—
126	C₄⋊C₄×C₈₉	—	2	1	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₂²⋊C₄×C₈₉	—	2	1	no	—
129	C₂²⋊C₄×C₈₉	—	2	2	no	—
130	C₄²×C₈₉	—	2	2	no	—
131	C₄²×C₈₉	C₂×D₄	1	2	yes	—
132	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
133	C₂²×D₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
138	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
140	C₂²⋊D₈₉	—	4	2	no	—
141	C₂²⋊D₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	2	1	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	8	1	no	—
147	D₃₅₆	—	2	2	no	—
148	D₃₅₆	—	2	2	no	—
149	C₄×D₈₉	—	4	2	no	—
150	C₄×D₈₉	—	4	2	no	—
151	Dic₁₇₈	—	2	2	no	—
152	Dic₁₇₈	—	2	2	no	—
153	Dic₁₇₈	—	8	2	no	—
154	C₈×C₈₉	—	2	2	no	—
155	C₈×C₈₉	—	4	2	no	—
156	C₈×C₈₉	#124	1	1	yes	characteristic
157	C₈×C₈₉	#124	1	2	yes	—
158	C₈×C₈₉	#124	1	1	yes	characteristic
159	#10623494	—	4	1	no	—
160	#10623494	—	4	1	no	—
161	#10623494	—	4	2	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	#525	1	1	yes	characteristic
164	C₂²×C₈₉	#529	1	2	yes	—
165	C₂×D₈₉	—	4	2	no	—
166	C₂×D₈₉	—	8	2	no	—
167	C₄×C₈₉	—	4	2	no	—
168	C₄×C₈₉	—	8	2	no	—
169	C₄×C₈₉	#525	1	1	yes	characteristic
170	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
171	C₄×C₈₉	#529	1	2	yes	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	—	4	2	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	4	2	no	—
176	#23834	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	—	2	2	no	—
179	C₂×C₈₉	#2398	1	1	yes	characteristic
180	D₈₉	—	8	2	no	—
181	#2961	—	89	1	no	—
182	#2955	—	89	1	no	—
183	#2103	—	89	1	no	—
184	#2102	—	89	1	no	—
185	#2080	—	89	2	no	—
186	#2053	—	89	1	no	—
187	#2050	—	89	1	no	—
188	#2031	—	89	2	no	—
189	#1765	—	89	2	no	—
190	#1757	—	89	2	no	—
191	#1747	D₈₉	1	1	yes	characteristic, complemented
192	C₈₉	#23834	1	1	yes	characteristic, complemented, Sylow
193	#575	—	89	1	no	—
194	C₂×#131	—	89	2	no	—
195	#533	—	89	1	no	—
196	#446	—	178	2	no	—
197	#446	—	178	2	no	—
198	#444	—	178	2	no	—
199	#443	—	178	2	no	—
200	#430	—	89	1	no	—
201	#424	—	89	1	no	—
202	#424	—	89	1	no	—
203	#423	—	89	1	no	—
204	#421	—	89	2	no	—
205	C₂×#104	—	89	2	no	—
206	#416	—	89	2	no	—
207	#413	—	89	2	no	—
208	#410	—	89	2	no	—
209	#405	—	89	1	no	—
210	#405	C₂×D₈₉	1	2	yes	—
211	#405	C₂×D₈₉	1	1	yes	characteristic
212	C₂×#97	—	89	1	no	—
213	#344	—	89	2	no	—
214	#344	—	89	2	no	—
215	#341	—	89	2	no	—
216	#339	—	89	2	no	—
217	#338	C₂×D₈₉	1	1	yes	characteristic
218	H(3,ℤ/4)	C₂×D₈₉	1	1	yes	characteristic
219	H(3,ℤ/4)	C₂×D₈₉	1	2	yes	—
220	C₂×#41	—	89	2	no	—
221	#137	—	178	2	no	—
222	AGL(1,ℤ/8)	—	178	2	no	—
223	#135	—	178	2	no	—
224	#131	—	178	2	no	—
225	#131	—	178	2	no	—
226	#131	—	178	2	no	—
227	#131	—	178	2	no	—
228	C₂×M₁₆	—	89	2	no	—
229	C₂×M₁₆	—	89	1	no	—
230	C₂×M₁₆	—	178	2	no	—
231	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
232	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
233	C₂²×C₈	—	89	1	no	—
234	C₂²×C₈	—	178	2	no	—
235	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₄⋊Q₈	—	178	1	no	—
237	#125	—	178	1	no	—
238	C₂²⋊Q₈	—	178	2	no	—
239	C₄×Q₈	—	178	2	no	—
240	#117	—	89	1	no	—
241	#117	C₂²×D₈₉	1	2	yes	—
242	#117	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₄²	—	89	1	no	—
244	#108	—	178	2	no	—
245	C₄⋊C₈	—	178	2	no	—
246	C₄⋊C₈	—	178	2	no	—
247	C₄⋊C₈	—	178	2	no	—
248	#104	—	178	2	no	—
249	#104	—	178	2	no	—
250	#104	—	178	4	no	—
251	#103	—	178	2	no	—
252	#102	—	178	2	no	—
253	#101	—	178	2	no	—
254	#100	—	178	2	no	—
255	#99	—	178	4	no	—
256	#97	—	2	2	no	—
257	#97	—	178	1	no	—
258	#97	—	178	1	no	—
259	#97	—	178	1	no	—
260	#97	D₃₅₆	1	2	yes	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	178	1	no	—
263	C₄×C₈	D₃₅₆	1	2	yes	—
264	#41	—	178	2	no	—
265	#41	—	178	2	no	—
266	#41	—	178	2	no	—
267	#41	—	178	2	no	—
268	C₂×Q₈	—	89	2	no	—
269	C₂×Q₈	—	178	2	no	—
270	C₂×D₄	—	89	2	no	—
271	C₂×D₄	—	178	2	no	—
272	C₂²×C₄	—	178	2	no	—
273	C₂²×C₄	—	178	1	no	—
274	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
275	Q₁₆	—	178	2	no	—
276	SD₁₆	—	178	4	no	—
277	D₈	—	178	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	356	2	no	—
282	M₁₆	—	356	2	no	—
283	M₁₆	—	178	2	no	—
284	M₁₆	—	178	2	no	—
285	M₁₆	—	178	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	356	2	no	—
288	C₂×C₈	—	356	2	no	—
289	C₂×C₈	—	178	1	no	—
290	C₂×C₈	—	178	1	no	—
291	C₂×C₈	—	178	2	no	—
292	C₂×C₈	—	178	2	no	—
293	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
294	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
295	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
296	C₂×C₈	#13083574	1	2	yes	—
297	C₂×C₈	#13083574	1	2	yes	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄⋊C₄	—	178	2	no	—
302	C₄⋊C₄	—	178	2	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	89	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	89	1	no	—
309	C₄²	—	89	1	no	—
310	C₄²	—	178	1	no	—
311	C₄²	—	178	1	no	—
312	C₄²	C₂×D₃₅₆	1	2	yes	—
313	C₂³	—	178	2	no	—
314	C₂³	#41×D₈₉	1	1	yes	characteristic
315	Q₈	—	178	2	no	—
316	Q₈	—	178	2	no	—
317	Q₈	—	712	2	no	—
318	D₄	—	356	2	no	—
319	D₄	—	356	2	no	—
320	D₄	—	178	2	no	—
321	D₄	—	178	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	712	1	no	—
331	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
332	C₂×C₄	#41×D₈₉	1	2	yes	—
333	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂×#13083574	1	2	yes	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	356	1	no	—
338	C₈	—	356	1	no	—
339	C₈	—	356	2	no	—
340	C₈	#15668000	1	1	yes	characteristic
341	C₈	#15668000	1	2	yes	—
342	C₈	#15668000	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	356	2	no	—
345	C₂²	—	712	2	no	—
346	C₂²	#19152110	1	1	yes	characteristic
347	C₂²	#19152126	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	356	1	no	—
351	C₄	—	356	1	no	—
352	C₄	—	356	2	no	—
353	C₄	—	356	2	no	—
354	C₄	#19152110	1	1	yes	characteristic
355	C₄	C₂×#15668000	1	1	yes	characteristic
356	C₄	#19152126	1	2	yes	—
357	C₂	—	2	1	no	—
358	C₂	—	2	2	no	—
359	C₂	—	712	2	no	—
360	C₂	#26023538	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#15668000
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1747	C₂
socle	C₂×C₈₉	#2398
prime radical	#19152398	C₂²
hypercenter	#1747	D₈₉
hypocenter	C₈₉	#23834
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
1. #26029190	C₂	semidirect
3. #26021833	C₂	semidirect
4. #26021828	C₂	semidirect
5. #26021704	C₂	semidirect
8. #26021423	C₂	semidirect
9. #26020098	C₂	semidirect
10. #26020053	C₂	semidirect
11. C₈₉×#1747	C₂	semidirect
191. #1747	D₈₉	semidirect
192. C₈₉	#23834	semidirect

Maximal subgroups

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

subgroup	index
#26029190	2
#26029156	2
#26021833	2
#26021828	2
#26021704	2
#26021559	2
#26021540	2
#26021423	2
#26020098	2
#26020053	2
C₈₉×#1747	2
#23834	89