Subgroups — #27293902

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283649	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272693	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
5	#16811180	C₄	1	1	yes	characteristic
6	C₂²×#11719908	—	2	1	no	—
7	#16810753	—	2	1	no	—
8	#16810753	C₄	1	2	yes	—
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	C₂²	1	2	yes	—
11	#16810649	—	2	1	no	—
12	#16810647	C₄	1	1	yes	characteristic
13	#16810578	—	2	1	no	—
14	C₄×#11719908	—	2	1	no	—
15	C₄×#11719908	C₂²	1	2	yes	—
16	C₂×M₃₂×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×C₄⋊C₁₆	—	2	1	no	—
19	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
20	C₂×#11719917	—	2	1	no	—
21	#13133217	C₈	1	1	yes	characteristic
22	#13133217	C₈	1	1	yes	characteristic
23	#13133209	—	2	1	no	—
24	#13133196	—	2	1	no	—
25	#13133196	C₈	1	2	yes	—
26	C₄×Dic₁₀₆	—	2	1	no	—
27	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
28	#13133176	—	2	1	no	—
29	C₂×#11719908	—	2	4	no	—
30	C₂×#11719908	—	2	2	no	—
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	C₂×C₄	1	2	yes	—
33	#13133146	—	2	2	no	—
34	#13133146	—	2	2	no	—
35	#13133127	—	2	2	no	—
36	C₈×Dic₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₅₃	—	4	2	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂×Dic₁₀₆	—	2	1	no	—
50	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	—	2	1	no	—
64	C₄⋊Dic₅₃	—	2	1	no	—
65	C₄⋊Dic₅₃	—	2	1	no	—
66	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
67	#11719919	—	2	2	no	—
68	#11719919	—	2	2	no	—
69	C₄×Dic₅₃	—	2	2	no	—
70	#11719917	—	4	2	no	—
71	C₂×#88322	—	2	2	no	—
72	C₁₆×C₅₃	—	4	2	no	—
73	#11719908	—	2	4	no	—
74	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#103	1	2	yes	—
81	C₂×C₄×C₅₃	#104	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₅₃	—	2	2	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	Dic₁₀₆	—	2	4	no	—
88	Dic₁₀₆	—	4	2	no	—
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	#103	1	2	yes	—
92	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	#88322	—	4	2	no	—
95	#11608	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
99	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
103	C₄×C₅₃	#350	1	2	yes	—
104	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
105	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
106	C₄×C₅₃	#504	1	2	yes	—
107	Dic₅₃	—	4	2	no	—
108	Dic₅₃	—	4	2	no	—
109	#2960	—	53	1	no	—
110	#2150	D₅₃	1	1	yes	characteristic
111	C₂×C₄×C₁₆	—	53	1	no	—
112	Q₈⋊C₁₆	—	53	4	no	—
113	C₂×C₅₃	—	2	2	no	—
114	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
115	C₂×C₅₃	#2113	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
117	#533	—	53	1	no	—
118	C₂×M₃₂	—	2	1	no	—
119	C₂²×C₁₆	—	106	1	no	—
120	#446	—	53	2	no	—
121	#446	—	106	1	no	—
122	C₈×Q₈	—	53	2	no	—
123	C₈×Q₈	—	106	1	no	—
124	#424	—	106	1	no	—
125	#423	—	53	1	no	—
126	#405	—	106	1	no	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	1	no	—
129	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
130	C₄×C₁₆	—	53	2	no	—
131	C₄×C₁₆	—	106	1	no	—
132	C₅₃	#11608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	106	1	no	—
134	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	53	1	no	—
137	C₄⋊Q₈	—	53	1	no	—
138	#125	—	106	1	no	—
139	C₂²⋊Q₈	—	53	2	no	—
140	C₂²⋊Q₈	—	106	1	no	—
141	C₄×Q₈	—	53	2	no	—
142	C₄×Q₈	—	106	1	no	—
143	#117	—	106	1	no	—
144	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
145	M₃₂	—	4	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	106	4	no	—
148	C₂×C₁₆	—	106	2	no	—
149	C₄⋊C₈	—	53	2	no	—
150	C₄⋊C₈	—	106	2	no	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	106	1	no	—
153	#97	—	106	2	no	—
154	C₄×C₈	—	106	2	no	—
155	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
156	C₄×C₈	D₂₁₂	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×Q₈	—	53	2	no	—
160	C₂×Q₈	—	106	1	no	—
161	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
162	C₂²×C₄	#11719912	1	1	yes	characteristic
163	C₂²×C₄	#11719921	1	1	yes	characteristic
164	M₁₆	—	212	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	106	2	no	—
167	C₂×C₈	#11719921	1	1	yes	characteristic
168	C₂×C₈	#11719921	1	1	yes	characteristic
169	C₂×C₈	#11719923	1	2	yes	—
170	C₂×C₈	#11719925	1	2	yes	—
171	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₄⋊C₄	—	53	2	no	—
174	C₄⋊C₄	—	106	1	no	—
175	C₄⋊C₄	—	106	1	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	—	106	1	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₂²⋊C₄	—	106	2	no	—
180	C₄²	—	106	2	no	—
181	C₄²	C₈×D₅₃	1	1	yes	characteristic
182	C₄²	#11719912	1	1	yes	characteristic
183	C₄²	#11719921	1	1	yes	characteristic
184	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₁₆	—	4	2	no	—
186	C₁₆	—	106	4	no	—
187	C₂³	#13133129	1	1	yes	characteristic
188	Q₈	—	106	4	no	—
189	Q₈	—	212	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	106	2	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	#13133129	1	1	yes	characteristic
195	C₂×C₄	#13133129	1	1	yes	characteristic
196	C₂×C₄	#13133129	1	1	yes	characteristic
197	C₂×C₄	#13133140	1	2	yes	—
198	C₂×C₄	#13133143	1	2	yes	—
199	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	212	2	no	—
205	C₈	#13133140	1	2	yes	—
206	C₈	C₂×#11719923	1	1	yes	characteristic
207	C₈	C₂×#11719925	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#16810353	1	2	yes	—
211	C₂²	C₂×#13133129	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	212	2	no	—
215	C₄	—	212	2	no	—
216	C₄	#16810353	1	2	yes	—
217	C₄	#16810408	1	2	yes	—
218	C₄	C₂×#13133140	1	1	yes	characteristic
219	C₄	C₂×#13133143	1	1	yes	characteristic
220	C₄	#16811041	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₂×#16810353	1	1	yes	characteristic
223	C₂	#20276330	1	1	yes	characteristic
224	C₂	C₂×#16810408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#11608
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #20272693	C₂	semidirect
132. C₅₃	#11608	semidirect

Maximal subgroups

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

subgroup	index
#20283649	2
C₂×C₄×#11719908	2
#20272693	2
C₅₃×#2150	2
#11608	53