Subgroups — #36712152

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022488	C₂	1	2	yes	maximal, complemented
3	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022404	C₂	1	2	yes	maximal, complemented
5	#26022389	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022304	C₂	1	2	yes	maximal
7	#26022294	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022114	C₂	1	2	yes	maximal, complemented
10	C₄×#15667953	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#3393	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₃₅₆	—	2	1	no	—
13	C₂×#15667955	C₂²	1	2	yes	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151879	—	2	1	no	—
18	#19151879	—	2	2	no	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₄×D₇₁₂	C₂²	1	2	yes	—
21	#19151423	C₂²	1	1	yes	characteristic
22	#19151423	C₂²	1	2	yes	—
23	C₄⋊#13083549	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151402	C₂²	1	2	yes	—
26	#19151402	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151399	C₂²	1	1	yes	characteristic
29	#19151399	C₂²	1	2	yes	—
30	#19151398	C₂²	1	1	yes	characteristic
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#556×C₈₉	—	2	1	no	—
34	#495×C₈₉	C₂²	1	2	yes	—
35	#494×C₈₉	C₂²	1	1	yes	characteristic
36	#464×C₈₉	—	2	2	no	—
37	#461×C₈₉	—	2	1	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₈₉	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	1	no	—
44	#15668012	—	2	2	no	—
45	#15667992	—	4	1	no	—
46	#15667992	—	4	2	no	—
47	C₄×D₃₅₆	—	2	1	no	—
48	C₄×D₃₅₆	—	2	2	no	—
49	#15667955	—	2	4	no	—
50	#15667953	—	2	2	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₂×#13083549	C₂³	1	2	yes	—
54	#15667946	—	2	1	no	—
55	#15667946	—	2	2	no	—
56	#15667946	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	2	yes	—
58	#15667934	—	2	1	no	—
59	#15667934	—	2	2	no	—
60	C₂×Q₁₆×C₈₉	—	2	1	no	—
61	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₈₉	—	2	2	no	—
63	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	—	2	1	no	—
65	#126×C₈₉	—	4	2	no	—
66	#124×C₈₉	—	2	2	no	—
67	C₂²⋊Q₈×C₈₉	—	4	1	no	—
68	C₄×Q₈×C₈₉	—	2	2	no	—
69	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄×D₄×C₈₉	—	2	1	no	—
71	#117×C₈₉	—	4	1	no	—
72	#107×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
75	#103×C₈₉	—	2	2	no	—
76	#102×C₈₉	—	2	1	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	Q₈×D₈₉	—	8	4	no	—
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	D₇₁₂	—	2	1	no	—
86	D₇₁₂	#41	1	2	yes	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	#13083551	—	4	2	no	—
92	#13083551	—	4	1	no	—
93	#13083549	—	2	1	no	—
94	#13083549	—	2	2	no	—
95	#13083549	—	2	1	no	—
96	C₂×Q₈×C₈₉	—	2	2	no	—
97	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
98	C₂×D₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	—	4	1	no	—
100	Q₁₆×C₈₉	—	2	2	no	—
101	Q₁₆×C₈₉	—	2	2	no	—
102	Q₁₆×C₈₉	—	8	1	no	—
103	SD₁₆×C₈₉	—	8	2	no	—
104	D₈×C₈₉	—	2	1	no	—
105	D₈×C₈₉	—	2	1	no	—
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₄⋊C₄×C₈₉	—	4	1	no	—
111	C₄⋊C₄×C₈₉	—	4	2	no	—
112	C₄⋊C₄×C₈₉	—	4	1	no	—
113	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₂²⋊C₄×C₈₉	—	4	2	no	—
116	C₂²⋊C₄×C₈₉	—	4	1	no	—
117	C₂²⋊C₄×C₈₉	—	4	2	no	—
118	C₄²×C₈₉	—	4	2	no	—
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₉	C₂×D₄	1	2	yes	—
121	C₂³×C₈₉	—	4	1	no	—
122	C₂²×D₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	2	2	no	—
124	Q₈×C₈₉	—	4	2	no	—
125	Q₈×C₈₉	—	4	2	no	—
126	D₄×C₈₉	—	4	1	no	—
127	D₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	—	4	1	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	D₃₅₆	—	2	2	no	—
139	D₃₅₆	—	4	1	no	—
140	C₄×D₈₉	—	8	4	no	—
141	C₄×D₈₉	—	8	2	no	—
142	Dic₁₇₈	—	8	4	no	—
143	Dic₁₇₈	—	8	2	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₈₉	—	8	1	no	—
149	C₂²×C₈₉	—	8	1	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₈₉	—	4	2	no	—
153	C₂×D₈₉	—	8	1	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	—	8	1	no	—
160	C₄×C₈₉	#509	1	2	yes	—
161	C₄×C₈₉	#548	1	1	yes	characteristic
162	C₄×C₈₉	#577	1	1	yes	characteristic
163	Dic₈₉	—	8	1	no	—
164	Dic₈₉	—	8	2	no	—
165	#33934	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	16	1	no	—
167	C₂×C₈₉	#3358	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
169	C₂×C₈₉	#3417	1	1	yes	characteristic
170	D₈₉	—	8	2	no	—
171	#3393	D₈₉	1	1	yes	characteristic, complemented
172	Q₈×D₈	—	89	1	no	—
173	#2243	—	89	2	no	—
174	#2241	—	89	1	no	—
175	#2227	—	89	2	no	—
176	#2224	—	89	1	no	—
177	#2210	—	89	2	no	—
178	#2208	—	89	1	no	—
179	#2183	—	89	1	no	—
180	#2175	—	89	2	no	—
181	C₄×D₁₆	—	89	1	no	—
182	C₈₉	#33934	1	1	yes	characteristic, complemented, Sylow
183	#556	—	2	1	no	—
184	D₄×Q₈	—	178	1	no	—
185	C₂×SD₃₂	—	89	2	no	—
186	C₂×D₁₆	—	89	1	no	—
187	#500	—	89	1	no	—
188	#500	—	89	2	no	—
189	#495	C₂×D₈₉	1	2	yes	—
190	#494	C₂×D₈₉	1	1	yes	characteristic
191	#474	—	178	1	no	—
192	#474	—	178	2	no	—
193	#464	—	2	2	no	—
194	#461	—	2	1	no	—
195	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
196	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
197	C₄×D₈	—	89	1	no	—
198	C₄×D₈	—	89	2	no	—
199	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
200	#366	—	89	1	no	—
201	#366	—	89	2	no	—
202	C₄⋊C₁₆	—	89	1	no	—
203	C₄⋊C₁₆	—	89	2	no	—
204	#358	—	89	2	no	—
205	#358	—	89	2	no	—
206	#358	—	89	2	no	—
207	#357	—	89	1	no	—
208	#357	—	89	2	no	—
209	#357	—	89	1	no	—
210	#346	—	89	2	no	—
211	C₄×C₁₆	—	89	1	no	—
212	C₂²×Q₈	—	356	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×D₈	—	89	1	no	—
217	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	178	1	no	—
220	C₄⋊Q₈	—	178	2	no	—
221	#126	—	4	2	no	—
222	#124	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	356	1	no	—
225	C₂²⋊Q₈	—	356	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	178	1	no	—
230	C₄×D₄	—	178	2	no	—
231	#117	—	4	1	no	—
232	SD₃₂	—	178	4	no	—
233	D₁₆	—	178	2	no	—
234	C₂×C₁₆	—	89	1	no	—
235	C₂×C₁₆	—	89	1	no	—
236	C₂×C₁₆	—	89	2	no	—
237	#107	—	89	1	no	—
238	#107	—	89	2	no	—
239	#107	—	178	1	no	—
240	#107	—	178	2	no	—
241	#107	C₂²×D₈₉	1	1	yes	characteristic
242	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
244	#103	—	2	2	no	—
245	#102	—	2	1	no	—
246	#102	—	178	1	no	—
247	#102	—	178	2	no	—
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₉	1	2	yes	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	356	2	no	—
252	C₂×Q₈	—	356	1	no	—
253	C₂×Q₈	—	712	4	no	—
254	C₂×Q₈	#13083559	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	356	2	no	—
259	C₂²×C₄	—	356	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	8	1	no	—
263	SD₁₆	—	8	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	89	2	no	—
267	D₈	—	178	1	no	—
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
270	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	356	2	no	—
276	C₄⋊C₄	—	356	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	2	no	—
279	C₄⋊C₄	#13083559	1	1	yes	characteristic
280	C₄⋊C₄	#13083559	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	356	2	no	—
285	C₂²⋊C₄	—	356	1	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
289	C₁₆	—	178	1	no	—
290	C₁₆	—	178	2	no	—
291	C₁₆	—	178	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	356	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	712	4	no	—
298	Q₈	—	712	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	356	1	no	—
302	D₄	—	178	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	356	1	no	—
308	C₂×C₄	—	712	4	no	—
309	C₂×C₄	—	712	2	no	—
310	C₂×C₄	C₂×#13083559	1	2	yes	—
311	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
312	C₂×C₄	C₂×#13083559	1	2	yes	—
313	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	#41×D₈₉	1	1	yes	characteristic
318	C₈	#15668059	1	1	yes	characteristic
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	356	2	no	—
323	C₂²	—	712	1	no	—
324	C₂²	—	712	1	no	—
325	C₂²	C₂²×#13083559	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	712	2	no	—
332	C₄	—	712	1	no	—
333	C₄	#19152033	1	2	yes	—
334	C₄	#19152255	1	1	yes	characteristic
335	C₄	#19152409	1	1	yes	characteristic
336	C₂	—	16	1	no	—
337	C₂	—	712	2	no	—
338	C₂	#26032883	1	1	yes	characteristic
339	C₂	C₂×#19152033	1	1	yes	characteristic
340	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3393	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15667953	C₂²
hypercenter	#3393	D₈₉
hypocenter	C₈₉	#33934
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3393

Complements

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

N	H ≅ G/N	type
1. Q₈×D₇₁₂	C₂	semidirect
2. #26022488	C₂	semidirect
3. #26022479	C₂	semidirect
4. #26022404	C₂	semidirect
5. #26022389	C₂	semidirect
7. #26022294	C₂	semidirect
8. #26022161	C₂	semidirect
9. #26022114	C₂	semidirect
11. C₈₉×#3393	C₂	semidirect
171. #3393	D₈₉	semidirect
182. C₈₉	#33934	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₇₁₂	2
#26022488	2
#26022479	2
#26022404	2
#26022389	2
#26022304	2
#26022294	2
#26022161	2
#26022114	2
C₄×#15667953	2
C₈₉×#3393	2
#33934	89