Subgroups — #27294035

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	1	yes	maximal, characteristic
2	#20276514	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719908	—	2	1	no	—
6	C₄⋊#11719908	C₂²	1	2	yes	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	—	2	1	no	—
9	C₄×#11719908	—	2	1	no	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₅₃	—	2	1	no	—
13	C₈×Q₈×C₅₃	—	2	1	no	—
14	C₈×Q₈×C₅₃	C₄	1	2	yes	—
15	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	—	2	1	no	—
27	C₄×Q₈×C₅₃	—	2	1	no	—
28	C₄×Q₈×C₅₃	C₈	1	2	yes	—
29	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	—	2	1	no	—
37	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₅₃	—	2	1	no	—
45	C₂×Q₈×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	C₁₆	1	4	yes	complemented
47	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
70	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	#11719908	—	4	2	no	—
72	#11719908	—	4	2	no	—
73	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
74	Q₈×C₅₃	—	2	4	no	—
75	Q₈×C₅₃	—	2	2	no	—
76	Q₈×C₅₃	C₂×C₁₆	1	4	yes	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
88	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	—	4	2	no	—
96	C₈×C₅₃	#137	1	2	yes	—
97	#11637	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
101	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₄×C₅₃	—	2	4	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
108	C₄×C₅₃	#419	1	1	yes	characteristic
109	C₄×C₅₃	#421	1	1	yes	characteristic
110	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
111	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
112	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
113	#2162	—	53	1	no	—
114	#2162	—	53	1	no	—
115	Q₈⋊C₁₆	—	53	4	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	#1479	1	1	yes	characteristic
118	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₅₃	#2117	1	1	yes	characteristic
120	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
121	C₈×Q₈	—	2	1	no	—
122	C₈×Q₈	—	2	1	no	—
123	C₈×Q₈	Dic₅₃	1	2	yes	—
124	C₈×Q₈	C₂×D₅₃	1	2	yes	—
125	C₂×C₄⋊C₈	—	2	1	no	—
126	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
127	C₂×C₄×C₈	—	2	1	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₄⋊C₁₆	—	53	2	no	—
130	C₄⋊C₁₆	—	106	1	no	—
131	C₂²⋊C₁₆	—	106	1	no	—
132	C₂²⋊C₁₆	—	106	1	no	—
133	C₄×C₁₆	—	53	2	no	—
134	C₄×C₁₆	—	106	1	no	—
135	C₅₃	#11637	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	#88322	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	#88322	1	2	yes	—
143	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂×C₁₆	—	106	2	no	—
149	C₂×C₁₆	—	106	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	#11719908	1	4	yes	complemented
163	C₂×Q₈	C₂×#88322	1	2	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
167	C₂²×C₄	#11719917	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	C₂×#88322	1	2	yes	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂×#88322	1	1	yes	characteristic
184	C₄²	#11719917	1	1	yes	characteristic
185	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
187	C₁₆	—	212	2	no	—
188	C₁₆	—	212	2	no	—
189	C₂³	C₂²⋊#88322	1	1	yes	characteristic
190	Q₈	—	2	2	no	—
191	Q₈	—	2	4	no	—
192	Q₈	C₂×#11719908	1	4	yes	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
201	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
202	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
203	C₂×C₄	C₂×#11719908	1	2	yes	—
204	C₂×C₄	#13133157	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	#13133248	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂²⋊#11719908	1	2	yes	—
216	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂²⋊#11719908	1	2	yes	—
223	C₄	#16810632	1	1	yes	characteristic
224	C₄	#16810640	1	1	yes	characteristic
225	C₄	C₂²×#11719908	1	1	yes	characteristic
226	C₄	C₂×#13133157	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#20273110	1	1	yes	characteristic
229	C₂	C₂×C₂²⋊#11719908	1	1	yes	characteristic
230	C₂	#20276343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#11637
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#11719908	C₂	semidirect
46. C₂×Q₈×C₅₃	C₁₆	semidirect
135. C₅₃	#11637	semidirect
162. C₂×Q₈	#11719908	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
Q₈⋊#11719908	2
C₂×C₈×Q₈×C₅₃	2
#11637	53