Subgroups — #28311376

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811544	C₂	1	2	yes	maximal
2	#20289754	C₂	1	2	yes	maximal
3	#20289753	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	#16811544	—	2	4	no	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#11719910	—	2	4	no	—
15	C₂×#11719910	—	2	4	no	—
16	C₂×#11719910	—	2	4	no	—
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂²×#88322	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×#88322	—	2	2	no	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	D₄	1	2	yes	—
30	C₈×Dic₅₃	Q₈	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
48	C₄²×C₅₃	—	2	1	no	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄×Dic₅₃	—	2	2	no	—
51	C₄×Dic₅₃	—	2	1	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	2	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	C₁₆	1	4	yes	complemented
56	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₅₃	M₁₆	1	2	yes	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	M₁₆	1	2	yes	—
61	C₂×#88322	C₂×D₄	1	1	yes	characteristic
62	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
63	C₂×#88322	#41	1	2	yes	—
64	C₂×#88322	#41	1	2	yes	—
65	#11719910	—	4	4	no	—
66	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
72	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	—	2	1	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
86	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
87	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
88	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
89	C₂×Dic₅₃	M₃₂	1	2	yes	—
90	C₂×Dic₅₃	M₃₂	1	2	yes	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	—	2	2	no	—
93	#88322	—	2	2	no	—
94	#88322	#124	1	2	yes	—
95	#88322	#125	1	2	yes	—
96	#12879	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂²×C₅₃	#431	1	1	yes	characteristic
99	C₂²×C₅₃	#431	1	1	yes	characteristic
100	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	#382	1	1	yes	characteristic
107	C₄×C₅₃	#423	1	1	yes	characteristic
108	C₄×C₅₃	#432	1	1	yes	characteristic
109	C₄×C₅₃	#432	1	1	yes	characteristic
110	Dic₅₃	—	2	2	no	—
111	Dic₅₃	—	2	2	no	—
112	Dic₅₃	—	2	2	no	—
113	Dic₅₃	C₄⋊C₁₆	1	4	yes	—
114	C₂×C₄⋊C₁₆	—	53	2	no	—
115	C₄²×C₈	—	53	1	no	—
116	#1374	—	53	2	no	—
117	#1374	—	53	2	no	—
118	C₂×C₅₃	#1831	1	1	yes	characteristic
119	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
120	C₂×C₅₃	#2150	1	1	yes	characteristic
121	C₂×C₅₃	#2162	1	1	yes	characteristic
122	C₂×C₅₃	#2162	1	1	yes	characteristic
123	C₂×C₅₃	#2163	1	1	yes	characteristic
124	C₂×C₅₃	#2163	1	1	yes	characteristic
125	C₂²×C₁₆	—	53	4	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	C₂×C₄×C₈	#6157	1	1	yes	characteristic
132	C₄³	—	53	1	no	—
133	C₄⋊C₁₆	—	106	4	no	—
134	C₅₃	#12879	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
138	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
139	C₂×C₄²	—	53	1	no	—
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₄²	—	53	1	no	—
142	C₂×C₄²	—	53	1	no	—
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
146	C₂×C₁₆	—	106	4	no	—
147	C₂×C₁₆	—	106	4	no	—
148	C₂×C₁₆	—	106	4	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	106	2	no	—
153	C₄×C₈	—	106	2	no	—
154	C₄×C₈	—	106	2	no	—
155	C₄×C₈	—	106	2	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
162	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	1	no	—
174	C₂×C₈	C₄⋊#6157	1	2	yes	—
175	C₂×C₈	#11719940	1	2	yes	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	53	4	no	—
179	C₄²	—	53	2	no	—
180	C₄²	—	53	2	no	—
181	C₄²	—	106	2	no	—
182	C₄²	—	106	2	no	—
183	C₄²	—	106	2	no	—
184	C₄²	—	106	1	no	—
185	C₄²	—	106	1	no	—
186	C₁₆	—	212	4	no	—
187	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	106	1	no	—
199	C₂×C₄	—	106	1	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	1	no	—
203	C₂×C₄	—	106	1	no	—
204	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
205	C₂×C₄	#13133292	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
207	C₂×C₄	#13133296	1	1	yes	characteristic
208	C₂×C₄	#13133296	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
210	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
211	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	—	53	2	no	—
215	C₈	—	53	2	no	—
216	C₈	—	106	2	no	—
217	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
218	C₂²	#16811640	1	1	yes	characteristic
219	C₂²	#16811641	1	1	yes	characteristic
220	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
221	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
222	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
223	C₂²	C₂×#13133292	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	53	4	no	—
227	C₄	—	106	2	no	—
228	C₄	—	106	2	no	—
229	C₄	—	106	2	no	—
230	C₄	#16811567	1	1	yes	characteristic
231	C₄	#16811626	1	1	yes	characteristic
232	C₄	#16811643	1	1	yes	characteristic
233	C₄	#16811646	1	1	yes	characteristic
234	C₂	#20290631	1	1	yes	characteristic
235	C₂	D₅₃⋊(C₄⋊C₁₆)	1	1	yes	characteristic
236	C₂	#20291048	1	1	yes	characteristic
237	C₂	#20291073	1	1	yes	characteristic
238	C₂	#20291074	1	1	yes	characteristic
239	C₂	#20291077	1	1	yes	characteristic
240	C₂	#20291078	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12879
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
55. C₄×Dic₅₃	C₁₆	semidirect
134. C₅₃	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811544	2
#20289754	2
#20289753	2
C₄×C₈×Dic₅₃	2
#12879	53