Subgroups — #36711371

Order 22784.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032012	C₂	1	1	yes	maximal, characteristic
2	#26022477	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022443	C₂	1	2	yes	maximal, complemented
4	#26022391	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022350	C₂	1	2	yes	maximal, complemented
6	#26022292	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022190	C₂	1	2	yes	maximal, complemented
8	C₄×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021957	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₂²	1	2	yes	—
13	#19152269	—	2	1	no	—
14	C₂×#15667958	C₂²	1	1	yes	characteristic
15	C₂×#15667955	C₂²	1	2	yes	—
16	C₂²×#13083549	C₂²	1	2	yes	—
17	#19151975	C₂²	1	1	yes	characteristic
18	#19151834	C₂²	1	2	yes	—
19	#19151799	—	2	1	no	—
20	#19151745	—	2	2	no	—
21	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	#19151425	C₂²	1	2	yes	—
23	#19151423	C₂²	1	1	yes	characteristic
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151402	C₂²	1	1	yes	characteristic
26	#19151401	C₂²	1	2	yes	—
27	#19151401	C₂²	1	1	yes	characteristic
28	#19151398	C₂²	1	2	yes	—
29	#19151398	C₂²	1	2	yes	—
30	#19151398	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#556×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#485×C₈₉	—	2	2	no	—
36	#475×C₈₉	—	2	1	no	—
37	#467×C₈₉	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	#427×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083574	—	2	2	no	—
42	#15668034	—	2	4	no	—
43	#15668034	—	4	2	no	—
44	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
45	C₂×#13083563	—	2	2	no	—
46	C₂×D₇₁₂	—	2	1	no	—
47	C₂×D₇₁₂	C₂³	1	2	yes	—
48	#15668009	—	2	1	no	—
49	#15668008	—	4	1	no	—
50	#15667986	—	2	2	no	—
51	#15667985	—	4	2	no	—
52	C₄×Dic₁₇₈	—	2	1	no	—
53	#15667974	—	4	1	no	—
54	#15667958	—	2	2	no	—
55	#15667955	—	4	2	no	—
56	C₂×#13083549	—	4	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667937	—	2	1	no	—
62	#15667934	—	2	2	no	—
63	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	#126×C₈₉	—	4	2	no	—
67	#124×C₈₉	—	4	1	no	—
68	C₂²⋊Q₈×C₈₉	—	2	2	no	—
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	—	4	1	no	—
71	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
72	#117×C₈₉	—	2	2	no	—
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#106×C₈₉	—	2	2	no	—
77	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
78	#103×C₈₉	—	2	1	no	—
79	#103×C₈₉	—	2	2	no	—
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	#13083574	—	4	4	no	—
83	#13083574	—	8	2	no	—
84	C₂×Dic₁₇₈	—	2	1	no	—
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	#41	1	2	yes	—
93	#13083563	—	4	4	no	—
94	D₇₁₂	—	2	2	no	—
95	D₇₁₂	—	2	2	no	—
96	D₇₁₂	—	8	1	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	#13083553	—	4	1	no	—
99	#13083551	—	4	2	no	—
100	C₄×Dic₈₉	—	4	1	no	—
101	#13083549	—	2	1	no	—
102	#13083549	—	2	1	no	—
103	#13083549	—	2	2	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	C₂×Q₈×C₈₉	—	4	1	no	—
106	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
108	Q₁₆×C₈₉	—	2	1	no	—
109	Q₁₆×C₈₉	—	2	1	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	4	1	no	—
118	C₄⋊C₄×C₈₉	—	4	2	no	—
119	C₄⋊C₄×C₈₉	—	4	1	no	—
120	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
121	C₂²⋊C₄×C₈₉	—	4	2	no	—
122	C₂²⋊C₄×C₈₉	—	4	2	no	—
123	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
124	C₄²×C₈₉	—	4	2	no	—
125	C₄²×C₈₉	—	4	1	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₈₉	C₂×D₈	1	2	yes	—
128	C₂²×D₈₉	—	4	2	no	—
129	Q₈×C₈₉	—	4	1	no	—
130	Q₈×C₈₉	—	4	1	no	—
131	Q₈×C₈₉	—	8	2	no	—
132	D₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₈₉	—	8	4	no	—
143	C₂²⋊D₈₉	—	16	2	no	—
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	—	4	1	no	—
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	—	2	2	no	—
152	Dic₁₇₈	—	4	1	no	—
153	C₈×C₈₉	—	2	2	no	—
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	C₂²×C₈₉	—	2	2	no	—
158	C₂²×C₈₉	—	2	2	no	—
159	C₂²×C₈₉	—	2	2	no	—
160	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
161	C₂×D₈₉	—	8	2	no	—
162	C₂×D₈₉	—	8	2	no	—
163	C₂×D₈₉	—	8	2	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	2	no	—
166	C₄×C₈₉	—	8	2	no	—
167	C₄×C₈₉	—	8	1	no	—
168	C₄×C₈₉	—	8	1	no	—
169	C₄×C₈₉	#508	1	2	yes	—
170	C₄×C₈₉	#546	1	1	yes	characteristic
171	C₄×C₈₉	#577	1	1	yes	characteristic
172	Dic₈₉	—	4	2	no	—
173	Dic₈₉	—	8	1	no	—
174	#33871	—	89	1	no	maximal, Sylow
175	C₂×C₈₉	—	4	2	no	—
176	C₂×C₈₉	#3294	1	1	yes	characteristic
177	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
178	C₂×C₈₉	#3416	1	1	yes	characteristic
179	D₈₉	—	16	2	no	—
180	#3342	D₈₉	1	1	yes	characteristic, complemented
181	#3287	—	89	1	no	—
182	#2240	—	89	1	no	—
183	#2234	—	89	2	no	—
184	#2225	—	89	1	no	—
185	#2217	—	89	2	no	—
186	#2207	—	89	1	no	—
187	#2189	—	89	2	no	—
188	C₄×Q₃₂	—	89	1	no	—
189	D₄×C₁₆	—	89	1	no	—
190	#2138	—	89	2	no	—
191	C₈₉	#33871	1	1	yes	characteristic, complemented, Sylow
192	C₂×#135	—	89	2	no	—
193	#556	—	2	1	no	—
194	#550	—	178	1	no	—
195	C₂×Q₃₂	—	89	1	no	—
196	C₂×SD₃₂	—	89	2	no	—
197	C₂²×C₁₆	—	89	2	no	—
198	#500	C₂×D₈₉	1	1	yes	characteristic
199	#493	—	89	1	no	—
200	#485	—	2	2	no	—
201	#475	—	2	1	no	—
202	#467	C₂×D₈₉	1	2	yes	—
203	#466	—	89	2	no	—
204	#460	—	178	1	no	—
205	#449	—	178	2	no	—
206	C₄×Q₁₆	—	89	1	no	—
207	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
209	#427	C₂×D₈₉	1	2	yes	—
210	#367	—	89	2	no	—
211	#366	—	89	1	no	—
212	C₄⋊C₁₆	—	89	1	no	—
213	#358	—	89	1	no	—
214	#358	—	89	1	no	—
215	#358	—	89	2	no	—
216	#357	—	89	2	no	—
217	#357	—	89	2	no	—
218	#357	—	89	2	no	—
219	C₂²⋊C₁₆	—	89	2	no	—
220	C₄×C₁₆	—	89	1	no	—
221	C₂×#41	—	178	2	no	—
222	#135	—	356	2	no	—
223	#135	—	178	4	no	—
224	C₂×Q₁₆	—	89	1	no	—
225	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
226	C₂×SD₁₆	—	178	2	no	—
227	C₂×D₈	—	89	2	no	—
228	C₂×D₈	—	178	1	no	—
229	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
230	C₄⋊Q₈	—	2	1	no	—
231	#127	—	356	1	no	—
232	#127	—	178	1	no	—
233	#126	—	4	2	no	—
234	#124	—	4	1	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	#121	—	356	2	no	—
237	#121	—	178	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	4	1	no	—
240	C₄×Q₈	—	178	1	no	—
241	C₄×D₄	—	356	1	no	—
242	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
243	#117	—	2	2	no	—
244	Q₃₂	—	178	2	no	—
245	SD₃₂	—	356	2	no	—
246	C₂×C₁₆	—	89	1	no	—
247	C₂×C₁₆	—	89	1	no	—
248	C₂×C₁₆	—	89	2	no	—
249	C₂×C₁₆	—	356	2	no	—
250	#107	—	2	1	no	—
251	#107	—	89	1	no	—
252	#107	C₂²×D₈₉	1	1	yes	characteristic
253	#107	C₂²×D₈₉	1	2	yes	—
254	#106	—	2	2	no	—
255	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
256	#103	—	2	1	no	—
257	#103	—	2	2	no	—
258	#103	—	178	1	no	—
259	#102	—	178	2	no	—
260	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
261	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	#41	—	356	4	no	—
263	#41	—	712	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	178	1	no	—
267	C₂×D₄	—	356	2	no	—
268	C₂×D₄	—	356	2	no	—
269	C₂×D₄	—	356	1	no	—
270	C₂×D₄	—	356	1	no	—
271	C₂×D₄	—	178	2	no	—
272	C₂×D₄	#13083559	1	1	yes	characteristic
273	C₂²×C₄	—	356	2	no	—
274	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	89	2	no	—
278	Q₁₆	—	178	1	no	—
279	SD₁₆	—	356	4	no	—
280	D₈	—	178	2	no	—
281	D₈	—	178	2	no	—
282	D₈	—	712	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
287	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	356	1	no	—
294	C₄⋊C₄	—	178	1	no	—
295	C₄⋊C₄	#13083559	1	1	yes	characteristic
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	356	2	no	—
299	C₂²⋊C₄	#13083559	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	356	1	no	—
303	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
304	C₁₆	—	178	1	no	—
305	C₁₆	—	178	1	no	—
306	C₁₆	—	178	2	no	—
307	C₂³	—	356	2	no	—
308	C₂³	C₂×#13083559	1	2	yes	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	Q₈	—	356	1	no	—
313	Q₈	—	178	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	356	2	no	—
316	D₄	—	356	2	no	—
317	D₄	—	1424	1	no	—
318	D₄	—	1424	2	no	—
319	D₄	—	712	4	no	—
320	D₄	—	712	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	356	1	no	—
327	C₂×C₄	—	356	1	no	—
328	C₂×C₄	—	712	4	no	—
329	C₂×C₄	C₂×#13083559	1	2	yes	—
330	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
331	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	2	no	—
335	C₈	#41×D₈₉	1	1	yes	characteristic
336	C₈	#15668058	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	712	2	no	—
341	C₂²	—	712	2	no	—
342	C₂²	—	712	2	no	—
343	C₂²	C₂²×#13083559	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	356	2	no	—
350	C₄	—	712	1	no	—
351	C₄	#19152028	1	2	yes	—
352	C₄	#19152236	1	1	yes	characteristic
353	C₄	#19152406	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	1424	2	no	—
356	C₂	#26032059	1	1	yes	characteristic
357	C₂	C₂×#19152028	1	1	yes	characteristic
358	C₂	#26033396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3342	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26032012	C₂
hypercenter	#3342	D₈₉
hypocenter	C₈₉	#33871
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #26022477	C₂	semidirect
3. #26022443	C₂	semidirect
4. #26022391	C₂	semidirect
5. #26022350	C₂	semidirect
6. #26022292	C₂	semidirect
7. #26022190	C₂	semidirect
8. C₄×#15667958	C₂	semidirect
9. D₄×#13083549	C₂	semidirect
10. #26021957	C₂	semidirect
11. C₈₉×#3342	C₂	semidirect
180. #3342	D₈₉	semidirect
191. C₈₉	#33871	semidirect

Maximal subgroups

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

subgroup	index
#26032012	2
#26022477	2
#26022443	2
#26022391	2
#26022350	2
#26022292	2
#26022190	2
C₄×#15667958	2
D₄×#13083549	2
#26021957	2
C₈₉×#3342	2
#33871	89