Subgroups — Q₈×#13133127 #27382760

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20274585	C₂	1	3	yes	maximal
4	#20274582	C₂	1	3	yes	maximal
5	#20274570	C₂	1	3	yes	maximal
6	C₄×#13133127	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₅₃	C₄	1	2	yes	—
9	Q₈×#88322	C₄	1	8	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	3	yes	—
11	C₂×#13133127	C₂²	1	3	yes	—
12	C₂×#13133127	C₂²	1	3	yes	—
13	C₂×#13133127	C₂²	1	3	yes	—
14	C₂×#13133127	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	3	yes	—
16	C₄⋊C₄×Dic₅₃	C₄	1	6	yes	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	3	yes	—
18	C₄²×Dic₅₃	C₂²	1	3	yes	—
19	#16810377	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₅₃	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂²×#88322	C₂³	1	3	yes	—
25	C₂²×#88322	C₂³	1	1	yes	characteristic
26	Q₈×Dic₅₃	—	2	8	no	—
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	6	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊#88322	C₂×C₄	1	12	yes	—
33	#13133127	—	2	6	no	—
34	#13133127	—	2	6	no	—
35	#13133127	—	2	6	no	—
36	#13133127	Q₈	1	4	yes	direct product factor
37	C₄×#88322	C₂×C₄	1	12	yes	—
38	#13133124	C₂×C₄	1	6	yes	—
39	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
46	C₄⋊C₈×C₅₃	C₂×C₄	1	12	yes	—
47	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	—
48	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
49	C₂×Q₈×C₅₃	M₁₆	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
54	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₅₃	—	2	3	no	—
56	C₂×C₈×C₅₃	—	2	3	no	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₄⋊C₄×C₅₃	C₄²	1	12	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
63	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
64	C₄⋊Dic₅₃	—	2	12	no	—
65	C₄×Dic₅₃	—	2	12	no	—
66	C₄×Dic₅₃	—	2	3	no	—
67	C₄×Dic₅₃	—	2	3	no	—
68	C₄×Dic₅₃	—	4	6	no	—
69	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
70	C₄×Dic₅₃	#41	1	2	yes	—
71	C₂×#88322	—	2	3	no	—
72	C₂×#88322	—	2	3	no	—
73	C₂×#88322	C₂²×C₄	1	6	yes	—
74	C₂×#88322	C₂²×C₄	1	2	yes	—
75	C₂×#88322	C₂×Q₈	1	2	yes	—
76	C₂×#88322	#41	1	2	yes	—
77	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	Q₈×C₅₃	#97	1	16	yes	complemented
79	C₂×C₄×C₅₃	—	2	6	no	—
80	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
82	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	—	2	2	no	—
91	C₂×Dic₅₃	—	2	6	no	—
92	C₂×Dic₅₃	—	2	6	no	—
93	C₂×Dic₅₃	—	2	6	no	—
94	C₂×Dic₅₃	C₄×Q₈	1	4	yes	—
95	C₈×C₅₃	—	2	6	no	—
96	C₈×C₅₃	C₄×Q₈	1	4	yes	complemented
97	#88322	—	2	6	no	—
98	#88322	C₄×Q₈	1	4	yes	complemented
99	Q₈×#97	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
102	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
103	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
104	C₂²×C₅₃	#568	1	2	yes	—
105	C₄×C₅₃	—	2	6	no	—
106	C₄×C₅₃	C₂×#97	1	12	yes	—
107	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
109	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
110	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
111	Dic₅₃	—	2	4	no	—
112	Dic₅₃	—	4	6	no	—
113	C₂×C₈×Q₈	—	53	1	no	—
114	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
115	C₄²×Q₈	—	53	1	no	—
116	#1770	—	53	3	no	—
117	#1770	—	53	3	no	—
118	#1767	—	53	3	no	—
119	C₄×#97	—	53	3	no	—
120	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
121	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
122	C₂×C₅₃	#2875	1	1	yes	characteristic
123	C₂×C₅₃	Q₈×M₁₆	1	2	yes	—
124	C₂×C₅₃	#2971	1	2	yes	—
125	C₂×C₄×Q₈	—	53	2	no	—
126	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
127	C₈×Q₈	—	53	8	no	—
128	C₈×Q₈	Dic₅₃	1	8	yes	complemented
129	C₂×C₄⋊C₈	—	53	3	no	—
130	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
131	C₂×#97	—	53	1	no	—
132	C₂×#97	—	53	3	no	—
133	C₂×#97	—	53	3	no	—
134	C₂×#97	—	53	3	no	—
135	C₂×C₄×C₈	—	53	3	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
137	C₄×C₄⋊C₄	—	53	6	no	—
138	C₄×C₄⋊C₄	—	53	3	no	—
139	C₄³	—	53	3	no	—
140	#336	—	53	6	no	—
141	C₅₃	Q₈×#97	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
143	C₂²×C₈	—	53	1	no	—
144	C₂²×C₈	—	53	3	no	—
145	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
146	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
147	C₄×Q₈	—	106	8	no	—
148	C₄×Q₈	C₄×D₅₃	1	4	yes	—
149	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
150	C₂×C₄⋊C₄	—	53	6	no	—
151	C₂×C₄⋊C₄	C₄×D₅₃	1	3	yes	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	—	53	6	no	—
154	C₂×C₄²	—	53	3	no	—
155	C₂×C₄²	—	53	6	no	—
156	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
157	C₄⋊C₈	—	53	12	no	—
158	C₄⋊C₈	C₂×Dic₅₃	1	12	yes	—
159	#97	—	53	4	no	—
160	#97	—	106	6	no	—
161	#97	—	106	6	no	—
162	#97	—	106	6	no	—
163	C₄×C₈	—	53	12	no	—
164	C₄×C₈	C₂×Dic₅₃	1	12	yes	—
165	#95	—	53	6	no	—
166	C₂×Q₈	#11719912	1	8	yes	—
167	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
168	C₂²×C₄	—	53	2	no	—
169	C₂²×C₄	—	53	6	no	—
170	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
171	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
172	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	53	6	no	—
179	C₂×C₈	—	106	3	no	—
180	C₂×C₈	—	106	3	no	—
181	C₂×C₈	Q₈×D₅₃	1	2	yes	—
182	C₂×C₈	#11719949	1	2	yes	—
183	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
184	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
185	C₄⋊C₄	—	106	12	no	—
186	C₄⋊C₄	C₄×Dic₅₃	1	12	yes	—
187	C₄²	—	53	2	no	—
188	C₄²	—	53	2	no	—
189	C₄²	—	106	12	no	—
190	C₄²	—	106	3	no	—
191	C₄²	—	106	3	no	—
192	C₄²	—	212	6	no	—
193	C₄²	C₂×C₄×D₅₃	1	6	yes	—
194	C₄²	C₂²×Dic₅₃	1	6	yes	—
195	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
196	Q₈	#13133127	1	16	yes	direct product factor
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	53	4	no	—
199	C₂×C₄	—	106	2	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	—	106	6	no	—
202	C₂×C₄	—	106	6	no	—
203	C₂×C₄	—	106	6	no	—
204	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
205	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
206	C₂×C₄	C₂×#11719912	1	12	yes	—
207	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
210	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
211	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
213	C₈	—	2	6	no	—
214	C₈	—	53	4	no	—
215	C₈	—	106	6	no	—
216	C₈	Q₈×Dic₅₃	1	4	yes	complemented
217	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
218	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
219	C₂²	C₄×#11719949	1	1	yes	characteristic
220	C₂²	C₂²×#11719912	1	2	yes	—
221	C₂²	#16811388	1	2	yes	—
222	C₄	—	2	6	no	—
223	C₄	—	106	4	no	—
224	C₄	—	212	6	no	—
225	C₄	C₂×#13133127	1	12	yes	—
226	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
227	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
228	C₄	C₄×#11719949	1	1	yes	characteristic
229	C₄	#41×Dic₅₃	1	1	yes	characteristic
230	C₂	C₄×Q₈×Dic₅₃	1	1	yes	characteristic
231	C₂	C₂²×#13133127	1	1	yes	characteristic
232	C₂	#20283141	1	1	yes	characteristic
233	C₂	Q₈×#11719912	1	2	yes	—
234	C₂	#20283734	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	Q₈×#97
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×#88322	C₄	semidirect
21. C₈×Q₈×C₅₃	C₄	semidirect
36. #13133127	Q₈	direct (N × H)
78. Q₈×C₅₃	#97	semidirect
96. C₈×C₅₃	C₄×Q₈	semidirect
98. #88322	C₄×Q₈	semidirect
128. C₈×Q₈	Dic₅₃	semidirect
141. C₅₃	Q₈×#97	semidirect
196. Q₈	#13133127	direct (N × H)
216. C₈	Q₈×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#88322	2
C₄×Q₈×Dic₅₃	2
#20274585	2
#20274582	2
#20274570	2
C₄×#13133127	2
C₂×C₈×Q₈×C₅₃	2
Q₈×#97	53

Subgroups — Q8×#13133127 #27382760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13133127 #27382760