Subgroups — Q₈×#13133140 #27400695

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20275675	C₂	1	3	yes	maximal
4	#20275663	C₂	1	3	yes	maximal
5	#20274717	C₂	1	3	yes	maximal
6	C₄×#13133140	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₀₆	—	2	4	no	—
9	Q₈×Dic₁₀₆	C₄	1	8	yes	complemented
10	C₂×#13133217	C₂²	1	3	yes	—
11	C₂×#13133216	—	2	3	no	—
12	C₂×Q₈×Dic₅₃	—	2	1	no	—
13	C₂×C₄×Dic₁₀₆	C₂²	1	3	yes	—
14	Q₈×#88322	—	2	4	no	—
15	C₂×C₄⋊#88322	C₂²	1	3	yes	—
16	C₂×#13133140	C₂²	1	3	yes	—
17	C₂×#13133140	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	3	yes	—
19	#16810385	C₂²	1	6	yes	—
20	#16810385	C₂²	1	3	yes	—
21	#16810385	C₂²	1	3	yes	—
22	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#389	—	2	3	no	—
25	C₅₃×#384	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
27	C₂²×Dic₁₀₆	—	2	3	no	—
28	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	3	yes	—
30	C₂²×#88322	C₂³	1	1	yes	characteristic
31	#13133217	—	2	6	no	—
32	#13133217	C₂×C₄	1	12	yes	—
33	#13133216	—	2	24	no	—
34	Q₈×Dic₅₃	—	2	8	no	—
35	C₄×Dic₁₀₆	—	2	6	no	—
36	C₄×Dic₁₀₆	C₂×C₄	1	12	yes	—
37	C₂×C₄⋊Dic₅₃	—	2	3	no	—
38	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
39	C₂×#11719919	—	2	3	no	—
40	C₂×C₄×Dic₅₃	—	2	3	no	—
41	C₄⋊#88322	—	2	6	no	—
42	#13133140	—	2	6	no	—
43	#13133140	Q₈	1	4	yes	direct product factor
44	C₄×#88322	—	2	6	no	—
45	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₅₃	—	4	4	no	—
47	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₅₃	D₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
51	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	—	2	3	no	—
55	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
56	#95×C₅₃	—	2	3	no	—
57	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
58	C₂×Q₈×C₅₃	SD₁₆	1	4	yes	—
59	C₂×Q₈×C₅₃	Q₁₆	1	4	yes	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	3	no	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	2	1	no	—
66	C₂²×Dic₅₃	—	2	3	no	—
67	C₂×Dic₁₀₆	—	2	1	no	—
68	C₂×Dic₁₀₆	—	2	1	no	—
69	C₂×Dic₁₀₆	—	2	1	no	—
70	C₂×Dic₁₀₆	—	2	12	no	—
71	C₂×Dic₁₀₆	—	8	3	no	—
72	C₂×Dic₁₀₆	—	8	3	no	—
73	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₀₆	C₂×Q₈	1	2	yes	—
75	C₂×Dic₁₀₆	#41	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₄⋊C₄×C₅₃	—	2	3	no	—
78	C₄⋊C₄×C₅₃	—	2	3	no	—
79	C₄⋊C₄×C₅₃	—	4	6	no	—
80	C₄⋊C₄×C₅₃	—	8	3	no	—
81	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
82	C₄⋊C₄×C₅₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₅₃	—	4	6	no	—
84	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
85	C₄²×C₅₃	C₂×D₄	1	6	yes	—
86	C₄⋊Dic₅₃	—	2	12	no	—
87	C₄⋊Dic₅₃	—	2	3	no	—
88	C₄⋊Dic₅₃	C₂²×C₄	1	6	yes	—
89	#11719919	—	2	12	no	—
90	C₄×Dic₅₃	—	2	12	no	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	—	2	3	no	—
94	C₂×#88322	—	2	3	no	—
95	C₂×#88322	—	2	3	no	—
96	C₂×#88322	C₂×Q₈	1	2	yes	—
97	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₅₃	#103	1	16	yes	complemented
99	C₂×C₄×C₅₃	—	2	3	no	—
100	C₂×C₄×C₅₃	—	2	3	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	3	no	—
105	C₂×C₄×C₅₃	—	4	3	no	—
106	C₂×C₄×C₅₃	—	4	3	no	—
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
110	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₅₃	—	2	2	no	—
118	C₂×Dic₅₃	—	2	2	no	—
119	C₂×Dic₅₃	—	2	2	no	—
120	C₂×Dic₅₃	—	2	6	no	—
121	C₂×Dic₅₃	—	4	3	no	—
122	C₂×Dic₅₃	—	4	3	no	—
123	Dic₁₀₆	—	2	2	no	—
124	Dic₁₀₆	—	2	2	no	—
125	Dic₁₀₆	—	2	2	no	—
126	Dic₁₀₆	—	8	12	no	—
127	Dic₁₀₆	C₄×Q₈	1	4	yes	complemented
128	#88322	—	2	2	no	—
129	#88322	—	4	3	no	—
130	Q₈×#103	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₅₃	#548	1	1	yes	characteristic
133	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
134	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
135	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
136	C₂²×C₅₃	#576	1	1	yes	characteristic
137	C₂²×C₅₃	#577	1	1	yes	characteristic
138	C₄×C₅₃	—	2	6	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	—	8	3	no	—
141	C₄×C₅₃	C₂×#103	1	12	yes	—
142	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₄×C₅₃	#519	1	1	yes	characteristic
144	C₄×C₅₃	#548	1	1	yes	characteristic
145	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
146	Dic₅₃	—	2	4	no	—
147	Dic₅₃	—	4	6	no	—
148	C₂×Q₈²	—	53	1	no	—
149	C₂×C₈×Q₈	—	53	1	no	—
150	Q₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
151	#1990	—	53	3	no	—
152	#1987	—	53	3	no	—
153	#1803	—	53	3	no	—
154	C₄×#103	—	53	3	no	—
155	C₂×C₅₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₅₃	C₂²×#103	1	1	yes	characteristic
157	C₂×C₅₃	#2897	1	1	yes	characteristic
158	C₂×C₅₃	#3362	1	1	yes	characteristic
159	C₂×C₅₃	#3363	1	1	yes	characteristic
160	C₂×C₅₃	Q₈×SD₁₆	1	1	yes	characteristic
161	C₂×C₅₃	Q₈×Q₁₆	1	1	yes	characteristic
162	Q₈²	—	53	8	no	—
163	Q₈²	—	106	4	no	—
164	C₂×C₄⋊Q₈	—	53	3	no	—
165	C₂×C₄⋊Q₈	—	106	3	no	—
166	C₂×C₄×Q₈	—	2	1	no	—
167	C₂×C₄×Q₈	—	53	3	no	—
168	C₂×C₄×Q₈	—	106	1	no	—
169	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
170	C₈×Q₈	—	106	4	no	—
171	C₂×C₄⋊C₈	—	53	3	no	—
172	C₂×#103	—	53	1	no	—
173	C₂×#103	—	53	3	no	—
174	C₂×C₄×C₈	—	53	3	no	—
175	#389	—	2	3	no	—
176	#384	C₂×D₅₃	1	3	yes	—
177	C₄×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
178	#340	—	53	6	no	—
179	#340	—	53	3	no	—
180	#340	—	53	3	no	—
181	C₅₃	Q₈×#103	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₈	—	53	1	no	—
183	C₂²×Q₈	—	106	3	no	—
184	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂²×C₈	—	53	1	no	—
186	C₂²×C₈	—	53	3	no	—
187	C₄⋊Q₈	—	53	12	no	—
188	C₄⋊Q₈	—	106	24	no	—
189	C₄⋊Q₈	—	106	6	no	—
190	C₄×Q₈	—	4	4	no	—
191	C₄×Q₈	—	53	12	no	—
192	C₄×Q₈	—	106	8	no	—
193	C₄×Q₈	—	106	6	no	—
194	C₄×Q₈	C₄×D₅₃	1	4	yes	—
195	C₄×Q₈	D₂₁₂	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	—	2	3	no	—
198	C₂×C₄⋊C₄	—	53	3	no	—
199	C₂×C₄⋊C₄	—	106	3	no	—
200	C₂×C₄⋊C₄	—	106	3	no	—
201	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
202	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
203	C₂×C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
204	C₂×C₄²	—	2	3	no	—
205	C₂×C₄²	—	106	3	no	—
206	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
207	C₄⋊C₈	—	106	6	no	—
208	#103	—	53	4	no	—
209	#103	—	106	6	no	—
210	C₄×C₈	—	106	6	no	—
211	#95	—	2	3	no	—
212	C₂×Q₈	—	53	2	no	—
213	C₂×Q₈	—	53	2	no	—
214	C₂×Q₈	—	53	2	no	—
215	C₂×Q₈	—	106	12	no	—
216	C₂×Q₈	—	106	1	no	—
217	C₂×Q₈	—	106	1	no	—
218	C₂×Q₈	—	106	1	no	—
219	C₂×Q₈	—	424	3	no	—
220	C₂×Q₈	—	424	3	no	—
221	C₂×Q₈	#11719921	1	4	yes	—
222	C₂×Q₈	#11719923	1	4	yes	—
223	C₂×Q₈	#11719925	1	4	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	3	no	—
226	C₂²×C₄	—	106	1	no	—
227	C₂²×C₄	—	106	3	no	—
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
230	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	106	1	no	—
233	C₂×C₈	—	106	1	no	—
234	C₂×C₈	—	106	3	no	—
235	C₂×C₈	—	106	3	no	—
236	C₂×C₈	—	106	3	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	3	no	—
239	C₄⋊C₄	—	2	3	no	—
240	C₄⋊C₄	—	4	6	no	—
241	C₄⋊C₄	—	8	3	no	—
242	C₄⋊C₄	—	53	6	no	—
243	C₄⋊C₄	—	106	12	no	—
244	C₄⋊C₄	—	106	12	no	—
245	C₄⋊C₄	—	106	3	no	—
246	C₄⋊C₄	#11719921	1	12	yes	—
247	C₄⋊C₄	Q₈×D₅₃	1	2	yes	—
248	C₄²	—	4	6	no	—
249	C₄²	—	106	12	no	—
250	C₄²	C₂×C₄×D₅₃	1	6	yes	—
251	C₄²	C₂×D₂₁₂	1	6	yes	—
252	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
253	Q₈	—	53	4	no	—
254	Q₈	—	106	2	no	—
255	Q₈	—	106	2	no	—
256	Q₈	—	106	2	no	—
257	Q₈	—	424	12	no	—
258	Q₈	#13133140	1	16	yes	direct product factor
259	C₂×C₄	—	2	3	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	3	no	—
265	C₂×C₄	—	4	3	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	106	2	no	—
270	C₂×C₄	—	106	6	no	—
271	C₂×C₄	—	212	3	no	—
272	C₂×C₄	—	212	3	no	—
273	C₂×C₄	C₂×#11719921	1	6	yes	—
274	C₂×C₄	C₂×#11719921	1	6	yes	—
275	C₂×C₄	C₂×#11719923	1	6	yes	—
276	C₂×C₄	C₂×#11719925	1	6	yes	—
277	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
280	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
281	C₂×C₄	#13133264	1	1	yes	characteristic
282	C₂×C₄	#13133265	1	1	yes	characteristic
283	C₈	—	106	2	no	—
284	C₈	—	212	3	no	—
285	C₂²	C₂²×#11719921	1	1	yes	characteristic
286	C₂²	#16811280	1	1	yes	characteristic
287	C₂²	Q₈×C₂²⋊D₅₃	1	1	yes	characteristic
288	C₂²	C₂²×#11719923	1	1	yes	characteristic
289	C₂²	C₂²×#11719925	1	1	yes	characteristic
290	C₂²	#16811426	1	1	yes	characteristic
291	C₂²	#16811431	1	1	yes	characteristic
292	C₄	—	2	6	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	3	no	—
295	C₄	—	106	4	no	—
296	C₄	—	212	6	no	—
297	C₄	C₂×#13133140	1	12	yes	—
298	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
299	C₄	#16811099	1	1	yes	characteristic
300	C₄	#16811285	1	1	yes	characteristic
301	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
302	C₂	Q₈×#11719921	1	1	yes	characteristic
303	C₂	C₂²×#13133140	1	1	yes	characteristic
304	C₂	#20283238	1	1	yes	characteristic
305	C₂	#20287382	1	1	yes	characteristic
306	C₂	#20287389	1	1	yes	characteristic
307	C₂	Q₈×#11719923	1	1	yes	characteristic
308	C₂	Q₈×#11719925	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	Q₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	Q₈×#103
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Q₈×Dic₁₀₆	C₄	semidirect
43. #13133140	Q₈	direct (N × H)
98. Q₈×C₅₃	#103	semidirect
127. Dic₁₀₆	C₄×Q₈	semidirect
181. C₅₃	Q₈×#103	semidirect
258. Q₈	#13133140	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₁₀₆	2
C₂×Q₈×#88322	2
#20275675	2
#20275663	2
#20274717	2
C₄×#13133140	2
Q₈×C₄⋊C₄×C₅₃	2
Q₈×#103	53

Subgroups — Q8×#13133140 #27400695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13133140 #27400695