Subgroups — #27282864

Order 13568.

606 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	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810355	C₂	1	4	yes	maximal
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	—	2	2	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	#16810529	C₄	1	2	yes	—
10	#16810521	—	2	2	no	—
11	#16810493	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₅₃	—	2	2	no	—
13	#16810355	—	2	8	no	—
14	C₂×C₄×C₈×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
18	C₂²×#88322	—	2	2	no	—
19	#13133217	—	2	2	no	—
20	#13133217	C₈	1	4	yes	complemented
21	C₄×Dic₁₀₆	—	2	4	no	—
22	C₄×Dic₁₀₆	—	2	4	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂×#11719919	—	2	2	no	—
27	C₂×C₄×Dic₅₃	—	2	2	no	—
28	C₄⋊#88322	—	2	4	no	—
29	C₄⋊#88322	—	4	4	no	—
30	#13133124	—	2	4	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₅₃	—	2	8	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	D₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	2	2	no	—
46	C₂²×Dic₅₃	—	2	2	no	—
47	C₂×Dic₁₀₆	—	2	4	no	—
48	C₂×Dic₁₀₆	—	2	4	no	—
49	C₂×Dic₁₀₆	—	2	2	no	—
50	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	C₂×C₈	1	2	yes	—
56	C₄²×C₅₃	M₁₆	1	2	yes	—
57	C₄²×C₅₃	SD₁₆	1	4	yes	—
58	C₄²×C₅₃	Q₁₆	1	4	yes	—
59	C₄²×C₅₃	C₂×D₄	1	2	yes	—
60	C₄²×C₅₃	#41	1	2	yes	—
61	C₄⋊Dic₅₃	—	2	8	no	—
62	C₄⋊Dic₅₃	—	2	4	no	—
63	C₄⋊Dic₅₃	—	4	4	no	—
64	#11719919	—	4	4	no	—
65	C₄×Dic₅₃	—	4	2	no	—
66	C₄×Dic₅₃	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	4	no	—
69	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₅₃	#103	1	4	yes	—
75	C₂×C₄×C₅₃	#103	1	4	yes	—
76	C₂×C₄×C₅₃	#104	1	4	yes	—
77	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₅₃	#121	1	2	yes	—
79	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#131	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
86	C₂×Dic₅₃	—	2	4	no	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	C₂×Dic₅₃	—	4	2	no	—
90	C₂×Dic₅₃	—	4	4	no	—
91	Dic₁₀₆	—	2	8	no	—
92	Dic₁₀₆	—	4	4	no	—
93	C₈×C₅₃	—	2	8	no	—
94	#88322	—	8	2	no	—
95	#9637	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#390	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₅₃	#408	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×#103	1	2	yes	—
100	C₂²×C₅₃	C₂×#104	1	2	yes	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	4	no	—
103	C₄×C₅₃	Q₈⋊C₈	1	8	yes	complemented
104	C₄×C₅₃	C₈×D₄	1	2	yes	—
105	C₄×C₅₃	#436	1	2	yes	—
106	C₄×C₅₃	#487	1	2	yes	—
107	C₄×C₅₃	#492	1	1	yes	characteristic
108	C₄×C₅₃	#494	1	1	yes	characteristic
109	Dic₅₃	—	4	4	no	—
110	Dic₅₃	—	8	2	no	—
111	C₄×C₄⋊Q₈	—	53	1	no	—
112	#1842	—	53	1	no	—
113	C₄²×C₈	D₅₃	1	1	yes	characteristic
114	C₂×Q₈⋊C₈	—	53	4	no	—
115	C₂×C₅₃	C₂×Q₈⋊C₈	1	2	yes	—
116	C₂×C₅₃	#1493	1	2	yes	—
117	C₂×C₅₃	#1954	1	1	yes	characteristic
118	C₂×C₅₃	#1956	1	1	yes	characteristic
119	C₂×C₅₃	#1962	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	—	53	1	no	—
121	C₂×C₄×Q₈	—	53	2	no	—
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₂×C₄⋊C₈	—	106	2	no	—
124	C₂×C₄×C₈	—	2	2	no	—
125	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
126	#391	—	53	2	no	—
127	#389	—	106	2	no	—
128	#384	—	53	1	no	—
129	C₄×C₄⋊C₄	—	106	2	no	—
130	C₄³	C₂×D₅₃	1	1	yes	characteristic
131	Q₈⋊C₈	—	106	8	no	—
132	C₅₃	#9637	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	53	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	106	2	no	—
136	C₄⋊Q₈	—	53	4	no	—
137	C₄⋊Q₈	—	106	2	no	—
138	C₄×Q₈	—	106	4	no	—
139	C₄×Q₈	—	106	4	no	—
140	C₂×C₄⋊C₄	—	53	2	no	—
141	C₂×C₄⋊C₄	—	106	2	no	—
142	C₂×C₄⋊C₄	—	106	2	no	—
143	C₂×C₄⋊C₄	—	106	2	no	—
144	C₂×C₄²	—	106	2	no	—
145	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₄²	D₂₁₂	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
149	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
150	C₄⋊C₈	—	106	4	no	—
151	C₄⋊C₈	—	212	4	no	—
152	C₄×C₈	—	2	8	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	D₂₁₂	1	4	yes	—
155	#95	—	106	4	no	—
156	C₂×Q₈	—	53	4	no	—
157	C₂×Q₈	—	106	2	no	—
158	C₂×Q₈	—	106	4	no	—
159	C₂×Q₈	—	106	4	no	—
160	C₂²×C₄	—	106	2	no	—
161	C₂²×C₄	—	106	2	no	—
162	C₂²×C₄	#11719921	1	2	yes	—
163	C₂²×C₄	#11719921	1	2	yes	—
164	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	—	212	4	no	—
171	C₄⋊C₄	—	106	8	no	—
172	C₄⋊C₄	—	106	4	no	—
173	C₄⋊C₄	—	212	4	no	—
174	C₄⋊C₄	—	212	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	212	2	no	—
178	C₄²	—	212	2	no	—
179	C₄²	C₈×D₅₃	1	2	yes	—
180	C₄²	#11719912	1	2	yes	—
181	C₄²	#11719913	1	4	yes	—
182	C₄²	Dic₂₁₂	1	4	yes	—
183	C₄²	C₂×D₂₁₂	1	2	yes	—
184	C₄²	#11719945	1	2	yes	—
185	C₂³	C₂×#11719921	1	1	yes	characteristic
186	Q₈	—	106	8	no	—
187	Q₈	—	212	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	106	4	no	—
192	C₂×C₄	—	212	2	no	—
193	C₂×C₄	—	212	4	no	—
194	C₂×C₄	—	212	2	no	—
195	C₂×C₄	—	212	2	no	—
196	C₂×C₄	#13133129	1	4	yes	—
197	C₂×C₄	#13133142	1	4	yes	—
198	C₂×C₄	#13133142	1	4	yes	—
199	C₂×C₄	#13133145	1	4	yes	—
200	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
201	C₂×C₄	#13133191	1	2	yes	—
202	C₂×C₄	#13133205	1	1	yes	characteristic
203	C₂×C₄	#13133214	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
206	C₂×C₄	#13133223	1	2	yes	—
207	C₂×C₄	C₂×#11719913	1	2	yes	—
208	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
209	C₈	—	2	8	no	—
210	C₈	—	424	2	no	—
211	C₂²	#16810525	1	1	yes	characteristic
212	C₂²	C₂×#13133129	1	1	yes	characteristic
213	C₂²	#16810589	1	1	yes	characteristic
214	C₂²	C₂×#13133142	1	2	yes	—
215	C₂²	C₂×#13133145	1	2	yes	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	424	2	no	—
219	C₄	—	212	4	no	—
220	C₄	#16810355	1	8	yes	complemented
221	C₄	C₈×D₂₁₂	1	2	yes	—
222	C₄	#16810702	1	2	yes	—
223	C₄	#16810974	1	2	yes	—
224	C₄	#16810996	1	1	yes	characteristic
225	C₄	#16811003	1	1	yes	characteristic
226	C₂	C₂×#16810355	1	2	yes	—
227	C₂	#20273171	1	2	yes	—
228	C₂	#20275509	1	1	yes	characteristic
229	C₂	#20275517	1	1	yes	characteristic
230	C₂	#20275543	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#9637
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
20. #13133217	C₈	semidirect
103. C₄×C₅₃	Q₈⋊C₈	semidirect
132. C₅₃	#9637	semidirect
220. C₄	#16810355	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
#20274867	2
C₂×#16810355	2
C₄²×C₈×C₅₃	2
#9637	53