Subgroups — #36712430

Order 22784.

520 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	#26033241	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022511	C₂	1	2	yes	maximal, complemented
3	#26022507	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022311	C₂	1	2	yes	maximal, complemented
5	#26022307	C₂	1	2	yes	maximal, complemented
6	#26022302	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022284	C₂	1	1	yes	maximal, characteristic
8	Q₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#15667956	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3396	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152321	—	2	1	no	—
13	C₂×#15667958	C₂²	1	2	yes	—
14	C₂×#15667956	C₂²	1	2	yes	—
15	C₂×#15667956	C₂²	1	2	yes	—
16	C₂×#15667955	C₂²	1	1	yes	characteristic
17	C₂×#15667953	C₂²	1	2	yes	—
18	#19151982	C₂²	1	2	yes	—
19	#19151978	C₂²	1	1	yes	characteristic
20	#19151821	—	2	2	no	—
21	#19151804	—	2	1	no	—
22	C₄×Dic₃₅₆	C₂²	1	2	yes	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	#19151425	C₂²	1	1	yes	characteristic
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	2	yes	—
27	#19151402	C₂²	1	1	yes	characteristic
28	#19151401	C₂²	1	2	yes	—
29	#19151399	C₂²	1	2	yes	—
30	#19151398	C₂²	1	1	yes	characteristic
31	C₄×#13083549	C₂²	1	1	yes	characteristic
32	C₄×#13083549	C₂²	1	2	yes	—
33	#559×C₈₉	—	2	1	no	—
34	#495×C₈₉	C₂²	1	2	yes	—
35	#493×C₈₉	C₂²	1	1	yes	characteristic
36	#463×C₈₉	—	2	2	no	—
37	#460×C₈₉	—	2	1	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	—	2	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	2	yes	—
43	C₂×#13083563	—	2	2	no	—
44	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
45	#15668012	—	2	1	no	—
46	#15668007	—	4	2	no	—
47	#15668000	—	2	2	no	—
48	#15667988	—	4	1	no	—
49	C₄×Dic₁₇₈	—	2	2	no	—
50	C₄×D₃₅₆	—	2	1	no	—
51	#15667970	—	4	1	no	—
52	#15667958	—	4	2	no	—
53	#15667956	—	2	4	no	—
54	#15667956	—	4	2	no	—
55	#15667955	—	2	2	no	—
56	#15667953	—	4	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	2	yes	—
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667937	—	2	2	no	—
62	#15667934	—	2	1	no	—
63	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×SD₁₆×C₈₉	—	2	2	no	—
65	C₂×D₈×C₈₉	—	2	1	no	—
66	C₂×D₈×C₈₉	C₂³	1	2	yes	—
67	#127×C₈₉	—	2	1	no	—
68	#124×C₈₉	—	2	2	no	—
69	#124×C₈₉	—	4	1	no	—
70	#123×C₈₉	—	4	2	no	—
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
73	C₄×D₄×C₈₉	—	2	2	no	—
74	C₄×D₄×C₈₉	—	4	1	no	—
75	#107×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
78	#103×C₈₉	—	2	1	no	—
79	#102×C₈₉	—	2	2	no	—
80	C₄×C₈×C₈₉	C₂³	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	2	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	Dic₃₅₆	—	2	2	no	—
86	Dic₃₅₆	—	2	2	no	—
87	Dic₃₅₆	—	8	1	no	—
88	#13083563	—	8	2	no	—
89	D₇₁₂	—	2	1	no	—
90	D₇₁₂	—	2	1	no	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	4	1	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	#13083553	—	4	2	no	—
95	#13083551	—	4	2	no	—
96	#13083551	—	4	1	no	—
97	#13083551	—	4	2	no	—
98	C₄×Dic₈₉	—	4	2	no	—
99	#13083549	—	2	2	no	—
100	#13083549	—	2	1	no	—
101	#13083549	#41	1	2	yes	—
102	C₂×Q₈×C₈₉	—	2	1	no	—
103	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
104	C₂×D₄×C₈₉	—	2	2	no	—
105	C₂×D₄×C₈₉	—	4	1	no	—
106	C₂×D₄×C₈₉	—	4	1	no	—
107	C₂²×C₄×C₈₉	—	4	2	no	—
108	Q₁₆×C₈₉	—	2	1	no	—
109	Q₁₆×C₈₉	—	2	1	no	—
110	SD₁₆×C₈₉	—	8	2	no	—
111	D₈×C₈₉	—	2	2	no	—
112	D₈×C₈₉	—	2	2	no	—
113	D₈×C₈₉	—	8	1	no	—
114	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	4	1	no	—
119	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
121	C₂²⋊C₄×C₈₉	—	4	2	no	—
122	C₂²⋊C₄×C₈₉	—	4	2	no	—
123	C₂²⋊C₄×C₈₉	—	4	2	no	—
124	C₄²×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	2	yes	—
127	C₂³×C₈₉	—	4	2	no	—
128	C₂²×D₈₉	—	4	1	no	—
129	Q₈×C₈₉	—	2	2	no	—
130	Q₈×C₈₉	—	4	1	no	—
131	Q₈×C₈₉	—	4	1	no	—
132	D₄×C₈₉	—	4	2	no	—
133	D₄×C₈₉	—	4	2	no	—
134	D₄×C₈₉	—	8	2	no	—
135	D₄×C₈₉	—	16	1	no	—
136	C₂×C₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	16	2	no	—
139	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	D₃₅₆	—	4	1	no	—
147	D₃₅₆	—	4	1	no	—
148	C₄×D₈₉	—	8	2	no	—
149	Dic₁₇₈	—	4	2	no	—
150	Dic₁₇₈	—	4	2	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	2	2	no	—
153	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₈₉	—	8	2	no	—
156	C₂²×C₈₉	—	8	2	no	—
157	C₂²×C₈₉	—	8	2	no	—
158	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₈₉	—	8	1	no	—
160	C₂×D₈₉	—	8	1	no	—
161	C₂×D₈₉	—	8	1	no	—
162	C₄×C₈₉	—	2	2	no	—
163	C₄×C₈₉	—	2	2	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	8	1	no	—
166	C₄×C₈₉	—	8	1	no	—
167	C₄×C₈₉	#508	1	2	yes	—
168	C₄×C₈₉	#550	1	1	yes	characteristic
169	C₄×C₈₉	#576	1	1	yes	characteristic
170	Dic₈₉	—	8	2	no	—
171	Dic₈₉	—	8	1	no	—
172	#33961	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	—	16	2	no	—
174	C₂×C₈₉	#3391	1	1	yes	characteristic
175	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
176	C₂×C₈₉	#3416	1	1	yes	characteristic
177	D₈₉	—	16	1	no	—
178	#3396	D₈₉	1	1	yes	characteristic, complemented
179	#3393	—	89	1	no	—
180	#2249	—	89	2	no	—
181	#2248	—	89	1	no	—
182	#2211	—	89	2	no	—
183	#2210	—	89	2	no	—
184	#2209	—	89	1	no	—
185	#2206	—	89	1	no	—
186	Q₈×C₁₆	—	89	1	no	—
187	C₄×SD₃₂	—	89	1	no	—
188	C₄×SD₃₂	—	89	2	no	—
189	C₈₉	#33961	1	1	yes	characteristic, complemented, Sylow
190	#559	—	2	1	no	—
191	#556	—	178	1	no	—
192	C₂×Q₃₂	—	89	2	no	—
193	C₂×SD₃₂	—	89	2	no	—
194	C₂×SD₃₂	—	89	1	no	—
195	C₂×SD₃₂	—	89	2	no	—
196	C₂×D₁₆	—	89	2	no	—
197	#495	—	89	2	no	—
198	#495	C₂×D₈₉	1	2	yes	—
199	#494	—	89	1	no	—
200	#493	C₂×D₈₉	1	1	yes	characteristic
201	#464	—	178	2	no	—
202	#463	—	2	2	no	—
203	#461	—	178	1	no	—
204	#460	—	2	1	no	—
205	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
206	C₄×Q₁₆	—	89	2	no	—
207	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₄×D₈	—	89	1	no	—
209	C₄×D₈	C₂×D₈₉	1	2	yes	—
210	#367	—	89	1	no	—
211	C₄⋊C₁₆	—	89	1	no	—
212	C₄⋊C₁₆	—	89	2	no	—
213	#358	—	89	1	no	—
214	#358	—	89	2	no	—
215	#357	—	89	2	no	—
216	#357	—	89	1	no	—
217	C₄×C₁₆	—	89	1	no	—
218	C₄×C₁₆	—	89	2	no	—
219	C₂×Q₁₆	—	89	2	no	—
220	C₂×Q₁₆	—	178	1	no	—
221	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	178	2	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	89	1	no	—
226	C₂×D₈	C₂²×D₈₉	1	2	yes	—
227	C₄⋊Q₈	—	178	1	no	—
228	#127	—	2	1	no	—
229	#126	—	356	2	no	—
230	#124	—	2	2	no	—
231	#124	—	4	1	no	—
232	#124	—	178	2	no	—
233	#123	—	4	2	no	—
234	C₂²⋊Q₈	—	356	1	no	—
235	C₄×Q₈	—	2	1	no	—
236	C₄×Q₈	—	178	2	no	—
237	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	4	1	no	—
240	C₄×D₄	—	178	1	no	—
241	#117	—	356	1	no	—
242	Q₃₂	—	356	2	no	—
243	SD₃₂	—	356	2	no	—
244	SD₃₂	—	178	4	no	—
245	SD₃₂	—	178	2	no	—
246	D₁₆	—	356	2	no	—
247	C₂×C₁₆	—	89	1	no	—
248	C₂×C₁₆	—	89	2	no	—
249	C₂×C₁₆	—	89	1	no	—
250	#107	—	89	1	no	—
251	#107	C₂²×D₈₉	1	1	yes	characteristic
252	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
253	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
254	#103	—	2	1	no	—
255	#103	—	178	2	no	—
256	#102	—	2	2	no	—
257	#102	—	178	1	no	—
258	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₈₉	1	2	yes	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	178	2	no	—
262	C₂×Q₈	#13083559	1	1	yes	characteristic
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	178	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	356	1	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	178	2	no	—
272	Q₁₆	—	178	2	no	—
273	Q₁₆	—	712	1	no	—
274	SD₁₆	—	8	2	no	—
275	SD₁₆	—	712	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	8	1	no	—
279	D₈	—	178	1	no	—
280	D₈	—	178	1	no	—
281	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
282	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
283	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	356	1	no	—
287	C₄⋊C₄	—	356	2	no	—
288	C₄⋊C₄	—	356	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₄⋊C₄	#13083559	1	1	yes	characteristic
291	C₄⋊C₄	#13083559	1	2	yes	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	356	2	no	—
296	C₂²⋊C₄	—	356	1	no	—
297	C₂²⋊C₄	—	356	2	no	—
298	C₄²	—	4	1	no	—
299	C₄²	—	356	2	no	—
300	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
301	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
302	C₁₆	—	89	2	no	—
303	C₁₆	—	178	2	no	—
304	C₁₆	—	178	1	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	356	1	no	—
307	Q₈	—	2	2	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	356	2	no	—
311	Q₈	—	356	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	16	1	no	—
316	D₄	—	356	1	no	—
317	D₄	—	356	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	—	356	1	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	712	2	no	—
324	C₂×C₄	C₂×#13083559	1	2	yes	—
325	C₂×C₄	C₂×#13083559	1	2	yes	—
326	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
327	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	2	no	—
331	C₈	#41×D₈₉	1	1	yes	characteristic
332	C₈	#15668058	1	1	yes	characteristic
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	712	1	no	—
337	C₂²	—	712	1	no	—
338	C₂²	—	712	1	no	—
339	C₂²	C₂²×#13083559	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	712	2	no	—
346	C₄	—	712	1	no	—
347	C₄	#19152028	1	2	yes	—
348	C₄	#19152267	1	1	yes	characteristic
349	C₄	#19152401	1	1	yes	characteristic
350	C₂	—	16	2	no	—
351	C₂	—	1424	1	no	—
352	C₂	#26033224	1	1	yes	characteristic
353	C₂	C₂×#19152028	1	1	yes	characteristic
354	C₂	#26033399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3396	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26022284	C₂
hypercenter	#3396	D₈₉
hypocenter	C₈₉	#33961
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3396

Complements

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

N	H ≅ G/N	type
1. #26033241	C₂	semidirect
2. #26022511	C₂	semidirect
3. #26022507	C₂	semidirect
4. #26022311	C₂	semidirect
5. #26022307	C₂	semidirect
6. #26022302	C₂	semidirect
8. Q₈×#13083549	C₂	semidirect
9. C₄×#15667955	C₂	semidirect
10. C₄×#15667956	C₂	semidirect
11. C₈₉×#3396	C₂	semidirect
178. #3396	D₈₉	semidirect
189. C₈₉	#33961	semidirect

Maximal subgroups

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

subgroup	index
#26033241	2
#26022511	2
#26022507	2
#26022311	2
#26022307	2
#26022302	2
#26022284	2
Q₈×#13083549	2
C₄×#15667955	2
C₄×#15667956	2
C₈₉×#3396	2
#33961	89