Subgroups — #36712414

Order 22784.

552 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	#26033256	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033226	C₂	1	1	yes	maximal, characteristic
3	#26022509	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022506	C₂	1	2	yes	maximal, complemented
5	#26022296	C₂	1	2	yes	maximal, complemented
6	#26022293	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022290	C₂	1	2	yes	maximal, complemented
8	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15667956	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022101	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2208	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152323	—	2	1	no	—
13	#19152269	—	2	1	no	—
14	C₂×#15667957	C₂²	1	2	yes	—
15	C₂×#15667956	C₂²	1	1	yes	characteristic
16	C₂×#15667955	C₂²	1	2	yes	—
17	C₂×#15667955	C₂²	1	2	yes	—
18	#19151976	C₂²	1	2	yes	—
19	#19151975	C₂²	1	1	yes	characteristic
20	#19151973	C₂²	1	2	yes	—
21	#19151806	—	2	2	no	—
22	#19151805	—	2	1	no	—
23	#19151793	—	2	1	no	—
24	#19151792	—	2	2	no	—
25	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
27	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
28	#19151680	C₂²	1	2	yes	—
29	#19151425	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151401	C₂²	1	1	yes	characteristic
32	#19151400	C₂²	1	2	yes	—
33	#19151399	C₂²	1	1	yes	characteristic
34	C₄×#13083549	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	—
37	#494×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083574	—	4	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
43	C₂×#13083564	—	2	2	no	—
44	C₂×D₇₁₂	—	2	1	no	—
45	C₂×D₇₁₂	C₂³	1	2	yes	—
46	C₂×#13083559	—	2	2	no	—
47	#15668010	—	2	2	no	—
48	#15668009	—	2	1	no	—
49	#15668008	—	2	2	no	—
50	#15668003	—	4	1	no	—
51	#15668002	—	4	2	no	—
52	#15667986	—	4	1	no	—
53	#15667985	—	4	2	no	—
54	Q₈×Dic₈₉	—	2	2	no	—
55	C₄×Dic₁₇₈	—	2	1	no	—
56	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
57	#15667977	—	4	1	no	—
58	#15667974	—	2	2	no	—
59	#15667957	—	4	2	no	—
60	#15667956	—	2	2	no	—
61	#15667955	—	2	4	no	—
62	#15667955	—	4	2	no	—
63	C₂×#13083549	C₂³	1	1	yes	characteristic
64	C₂×#13083549	C₂³	1	1	yes	characteristic
65	#15667946	C₂³	1	1	yes	characteristic
66	C₄⋊#10623494	D₄	1	1	yes	characteristic
67	#15667941	D₄	1	2	yes	—
68	#15667937	—	2	1	no	—
69	#15667936	—	2	2	no	—
70	#15667932	—	2	2	no	—
71	C₈×Dic₈₉	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₈₉	—	2	1	no	—
73	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
74	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₈₉	—	2	1	no	—
76	C₄×D₄×C₈₉	—	2	1	no	—
77	SD₃₂×C₈₉	—	4	2	no	—
78	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#102×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	#13083574	—	8	4	no	—
83	C₂×Dic₁₇₈	—	2	1	no	—
84	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
85	C₂×D₃₅₆	—	2	2	no	—
86	C₂×D₃₅₆	—	4	1	no	—
87	C₂×D₃₅₆	—	4	1	no	—
88	C₂×C₂²⋊D₈₉	—	4	2	no	—
89	C₂×C₂²⋊D₈₉	—	4	2	no	—
90	C₂×C₄×D₈₉	—	4	2	no	—
91	Dic₃₅₆	—	2	1	no	—
92	Dic₃₅₆	—	2	1	no	—
93	#13083564	—	8	2	no	—
94	D₇₁₂	—	2	2	no	—
95	D₇₁₂	—	2	2	no	—
96	D₇₁₂	—	8	1	no	—
97	#13083559	—	8	2	no	—
98	C₂×#10623494	C₂×D₄	1	2	yes	—
99	C₄⋊Dic₈₉	—	2	1	no	—
100	C₄⋊Dic₈₉	—	4	2	no	—
101	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
102	#13083553	—	4	1	no	—
103	#13083553	—	4	2	no	—
104	#13083553	—	4	2	no	—
105	#13083553	D₈	1	2	yes	—
106	#13083551	—	4	2	no	—
107	C₄×Dic₈₉	—	4	1	no	—
108	C₄×Dic₈₉	—	4	2	no	—
109	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
110	#13083549	—	2	1	no	—
111	#13083549	#41	1	2	yes	complemented
112	C₂×Q₈×C₈₉	—	2	2	no	—
113	C₂×D₄×C₈₉	—	2	1	no	—
114	C₂²×C₄×C₈₉	—	4	1	no	—
115	Q₁₆×C₈₉	—	2	2	no	—
116	Q₁₆×C₈₉	—	2	2	no	—
117	Q₁₆×C₈₉	—	8	1	no	—
118	D₈×C₈₉	—	2	1	no	—
119	D₈×C₈₉	—	2	1	no	—
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₄⋊C₄×C₈₉	—	4	1	no	—
124	C₄⋊C₄×C₈₉	—	4	1	no	—
125	C₂²⋊C₄×C₈₉	—	4	1	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₈₉	—	2	2	no	—
128	C₂³×C₈₉	—	4	1	no	—
129	C₂²×D₈₉	—	4	2	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	Q₈×C₈₉	—	4	2	no	—
132	D₄×C₈₉	—	4	1	no	—
133	D₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	—	8	2	no	—
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₈₉	—	8	4	no	—
141	C₂²⋊D₈₉	—	16	2	no	—
142	C₂×Dic₈₉	—	4	1	no	—
143	C₂×Dic₈₉	—	4	1	no	—
144	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
145	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
146	D₃₅₆	—	4	2	no	—
147	D₃₅₆	—	4	2	no	—
148	D₃₅₆	—	8	2	no	—
149	D₃₅₆	—	16	1	no	—
150	C₄×D₈₉	—	8	4	no	—
151	Dic₁₇₈	—	4	1	no	—
152	Dic₁₇₈	—	4	1	no	—
153	Dic₁₇₈	SD₃₂	1	4	yes	complemented
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
157	#10623494	—	2	2	no	—
158	C₂²×C₈₉	—	8	1	no	—
159	C₂²×C₈₉	—	8	1	no	—
160	C₂²×C₈₉	—	8	1	no	—
161	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₈₉	—	8	2	no	—
163	C₂×D₈₉	—	8	2	no	—
164	C₂×D₈₉	—	8	2	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	8	2	no	—
167	C₄×C₈₉	—	8	1	no	—
168	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
169	C₄×C₈₉	#550	1	1	yes	characteristic
170	C₄×C₈₉	#576	1	1	yes	characteristic
171	Dic₈₉	—	2	2	no	—
172	Dic₈₉	—	8	1	no	—
173	Dic₈₉	—	8	1	no	—
174	Dic₈₉	C₂×SD₃₂	1	4	yes	—
175	#33959	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	16	1	no	—
177	C₂×C₈₉	#3391	1	1	yes	characteristic
178	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
179	C₂×C₈₉	#3416	1	1	yes	characteristic
180	D₈₉	—	16	2	no	—
181	#3395	—	89	1	no	—
182	#3391	—	89	1	no	—
183	#2248	—	89	2	no	—
184	#2248	—	89	1	no	—
185	#2208	—	89	2	no	—
186	#2208	D₈₉	1	1	yes	characteristic, complemented
187	#2207	—	89	2	no	—
188	#2207	—	89	1	no	—
189	Q₈×C₁₆	—	89	1	no	—
190	C₄×SD₃₂	—	89	1	no	—
191	C₄×SD₃₂	—	89	2	no	—
192	C₈₉	#33959	1	1	yes	characteristic, complemented, Sylow
193	#557	—	178	1	no	—
194	#550	—	178	1	no	—
195	C₂×SD₃₂	—	89	2	no	—
196	C₂×SD₃₂	—	89	2	no	—
197	C₂×SD₃₂	—	89	1	no	—
198	C₂×SD₃₂	—	89	2	no	—
199	C₂×SD₃₂	C₂×D₈₉	1	2	yes	—
200	#494	—	89	2	no	—
201	#494	C₂×D₈₉	1	1	yes	characteristic
202	#493	—	89	2	no	—
203	#493	—	89	1	no	—
204	#462	—	178	1	no	—
205	#462	—	178	2	no	—
206	#459	—	178	2	no	—
207	#459	—	178	1	no	—
208	C₈×Q₈	—	89	1	no	—
209	C₄×Q₁₆	—	89	2	no	—
210	C₄×Q₁₆	—	89	1	no	—
211	C₄×D₈	—	89	2	no	—
212	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
213	#367	—	89	1	no	—
214	C₄⋊C₁₆	—	89	2	no	—
215	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
216	#358	—	89	1	no	—
217	#358	C₂×D₈₉	1	2	yes	—
218	#357	—	89	1	no	—
219	#357	—	89	2	no	—
220	C₄×C₁₆	—	89	2	no	—
221	C₄×C₁₆	—	89	1	no	—
222	C₂×#41	—	356	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	89	1	no	—
225	C₂×Q₁₆	—	178	2	no	—
226	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
227	C₂×D₈	—	89	2	no	—
228	C₂×D₈	—	178	2	no	—
229	C₂×D₈	—	178	1	no	—
230	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	178	2	no	—
233	#127	—	178	2	no	—
234	#127	—	178	1	no	—
235	#125	—	356	2	no	—
236	#125	—	356	1	no	—
237	#121	—	356	2	no	—
238	#121	—	356	1	no	—
239	C₄×Q₈	—	89	1	no	—
240	C₄×Q₈	—	356	1	no	—
241	C₄×Q₈	—	178	2	no	—
242	C₄×Q₈	—	178	1	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	—	178	2	no	—
245	SD₃₂	—	4	2	no	—
246	SD₃₂	—	356	2	no	—
247	SD₃₂	—	356	2	no	—
248	SD₃₂	—	178	2	no	—
249	SD₃₂	—	178	4	no	—
250	C₂×C₁₆	—	89	1	no	—
251	C₂×C₁₆	—	89	1	no	—
252	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
253	#107	—	89	1	no	—
254	#107	C₂²×D₈₉	1	1	yes	characteristic
255	C₄⋊C₈	—	89	2	no	—
256	C₄⋊C₈	—	89	1	no	—
257	#103	—	178	2	no	—
258	#103	—	178	1	no	—
259	#102	—	2	1	no	—
260	#102	—	178	2	no	—
261	C₄×C₈	—	89	2	no	—
262	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	#41	—	712	4	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	89	1	no	—
266	C₂×Q₈	—	178	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	356	2	no	—
269	C₂×D₄	—	356	2	no	—
270	C₂×D₄	—	356	1	no	—
271	C₂×D₄	—	356	1	no	—
272	C₂×D₄	—	178	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	356	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	8	1	no	—
278	Q₁₆	—	178	1	no	—
279	Q₁₆	—	178	1	no	—
280	Q₁₆	—	712	2	no	—
281	D₈	—	2	1	no	—
282	D₈	—	2	1	no	—
283	D₈	—	178	2	no	—
284	D₈	—	178	2	no	—
285	D₈	—	712	1	no	—
286	D₈	—	712	2	no	—
287	C₂×C₈	—	89	2	no	—
288	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	89	2	no	—
294	C₄⋊C₄	—	89	1	no	—
295	C₄⋊C₄	—	356	2	no	—
296	C₄⋊C₄	—	356	2	no	—
297	C₄⋊C₄	—	356	2	no	—
298	C₄⋊C₄	—	356	1	no	—
299	C₄⋊C₄	—	178	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	356	2	no	—
302	C₄²	—	89	2	no	—
303	C₄²	—	356	2	no	—
304	C₄²	—	356	1	no	—
305	C₄²	D₄×D₈₉	1	1	yes	characteristic
306	C₁₆	—	2	2	no	—
307	C₁₆	—	89	2	no	—
308	C₁₆	—	178	1	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	356	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	89	4	no	—
314	Q₈	—	356	1	no	—
315	Q₈	—	356	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	356	2	no	—
319	D₄	—	356	2	no	—
320	D₄	—	1424	1	no	—
321	D₄	—	1424	2	no	—
322	D₄	—	712	4	no	—
323	D₄	—	712	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	89	2	no	—
328	C₂×C₄	—	89	2	no	—
329	C₂×C₄	—	356	1	no	—
330	C₂×C₄	—	356	1	no	—
331	C₂×C₄	—	712	4	no	—
332	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
333	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	178	2	no	—
337	C₈	#41×D₈₉	1	1	yes	characteristic
338	C₈	#15668057	1	1	yes	characteristic
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	712	2	no	—
343	C₂²	—	712	2	no	—
344	C₂²	—	712	2	no	—
345	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	89	4	no	—
350	C₄	—	178	2	no	—
351	C₄	—	712	1	no	—
352	C₄	—	712	1	no	—
353	C₄	SD₃₂×D₈₉	1	2	yes	—
354	C₄	#19152266	1	1	yes	characteristic
355	C₄	#19152402	1	1	yes	characteristic
356	C₂	—	16	1	no	—
357	C₂	—	1424	2	no	—
358	C₂	#26033223	1	1	yes	characteristic
359	C₂	C₂×SD₃₂×D₈₉	1	1	yes	characteristic
360	C₂	#26033397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2208	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26033226	C₂
hypercenter	#2208	D₈₉
hypocenter	C₈₉	#33959
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2208

Complements

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

N	H ≅ G/N	type
1. #26033256	C₂	semidirect
3. #26022509	C₂	semidirect
4. #26022506	C₂	semidirect
5. #26022296	C₂	semidirect
6. #26022293	C₂	semidirect
7. #26022290	C₂	semidirect
8. #26022154	C₂	semidirect
9. C₄×#15667956	C₂	semidirect
10. #26022101	C₂	semidirect
11. C₈₉×#2208	C₂	semidirect
111. #13083549	#41	semidirect
153. Dic₁₇₈	SD₃₂	semidirect
186. #2208	D₈₉	semidirect
192. C₈₉	#33959	semidirect

Maximal subgroups

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

subgroup	index
#26033256	2
#26033226	2
#26022509	2
#26022506	2
#26022296	2
#26022293	2
#26022290	2
#26022154	2
C₄×#15667956	2
#26022101	2
C₈₉×#2208	2
#33959	89