Subgroups — #27293846

Order 13568.

386 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×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20272680	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₆	—	2	1	no	—
7	C₄×Dic₂₁₂	—	2	1	no	—
8	C₄×Dic₂₁₂	—	2	1	no	—
9	C₄×Dic₂₁₂	C₄	1	2	yes	—
10	C₄×Dic₂₁₂	C₂²	1	2	yes	—
11	#16810676	—	2	1	no	—
12	C₂×#13133151	C₄	1	1	yes	characteristic
13	C₂×#13133142	—	2	1	no	—
14	#16810376	C₂²	1	2	yes	—
15	#16810374	—	2	1	no	—
16	C₂×M₃₂×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#346	—	2	1	no	—
19	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₆	—	2	1	no	—
21	C₂×Dic₂₁₂	—	2	1	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	C₈	1	4	yes	complemented
24	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₆	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	4	1	no	—
28	C₄×Dic₁₀₆	—	4	1	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×#11719919	—	4	1	no	—
31	C₂×C₄×Dic₅₃	—	4	1	no	—
32	#13133151	—	2	1	no	—
33	#13133151	C₂×C₄	1	2	yes	—
34	C₄⋊#88322	—	2	2	no	—
35	#13133142	—	2	2	no	—
36	#13133142	—	2	2	no	—
37	C₂²⋊#88322	—	4	1	no	—
38	C₄×#88322	—	4	1	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	4	no	—
43	C₂×C₁₆×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	1	no	—
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂×Dic₁₀₆	—	2	4	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	4	1	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₃	D₈	1	1	yes	characteristic
66	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄⋊Dic₅₃	—	2	2	no	—
69	C₄⋊Dic₅₃	—	4	1	no	—
70	#11719919	—	4	2	no	—
71	#11719919	—	4	2	no	—
72	C₄×Dic₅₃	—	4	2	no	—
73	C₄×Dic₅₃	—	4	2	no	—
74	C₂×#88322	—	4	2	no	—
75	Dic₂₁₂	—	2	4	no	—
76	Dic₂₁₂	—	2	2	no	—
77	Dic₂₁₂	C₂×C₈	1	4	yes	—
78	C₁₆×C₅₃	—	2	4	no	—
79	C₂³×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	4	1	no	—
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#104	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×Dic₅₃	—	4	4	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	8	2	no	—
94	Dic₁₀₆	—	2	4	no	—
95	Dic₁₀₆	—	4	4	no	—
96	Dic₁₀₆	—	4	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
100	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	#88322	—	8	2	no	—
103	#11594	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
107	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
112	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
114	C₄×C₅₃	#510	1	2	yes	—
115	Dic₅₃	—	4	4	no	—
116	Dic₅₃	—	8	2	no	—
117	C₂×C₄×Q₁₆	—	53	1	no	—
118	C₄×M₃₂	D₅₃	1	1	yes	characteristic
119	#1564	—	53	1	no	—
120	#1332	—	53	4	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
123	C₂×C₅₃	#2141	1	1	yes	characteristic
124	C₂×C₅₃	#2143	1	1	yes	characteristic
125	C₂²×Q₁₆	—	53	1	no	—
126	C₂×C₄×Q₈	—	106	1	no	—
127	C₂×M₃₂	—	2	1	no	—
128	C₄×Q₁₆	—	53	2	no	—
129	C₄×Q₁₆	—	53	2	no	—
130	C₄×Q₁₆	—	106	1	no	—
131	C₄×Q₁₆	—	106	1	no	—
132	#431	—	106	1	no	—
133	C₂×#107	—	53	1	no	—
134	C₂×#103	—	106	1	no	—
135	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
136	#346	—	2	1	no	—
137	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
138	#335	—	53	2	no	—
139	#334	—	106	1	no	—
140	C₅₃	#11594	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	106	1	no	—
142	C₂×Q₁₆	—	53	2	no	—
143	C₂×Q₁₆	—	53	4	no	—
144	C₂×Q₁₆	—	106	2	no	—
145	C₂×Q₁₆	—	106	1	no	—
146	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₄×Q₈	—	106	2	no	—
149	C₄×Q₈	—	106	2	no	—
150	C₄×Q₈	—	212	1	no	—
151	C₄×Q₈	—	212	1	no	—
152	C₂×C₄⋊C₄	—	106	1	no	—
153	C₂×C₄⋊C₄	—	212	1	no	—
154	C₂×C₄²	—	212	1	no	—
155	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
156	M₃₂	—	2	4	no	—
157	C₂×C₁₆	—	2	2	no	—
158	#107	—	53	2	no	—
159	#107	—	106	1	no	—
160	C₄⋊C₈	—	106	2	no	—
161	#103	—	106	2	no	—
162	#103	—	106	2	no	—
163	C₂²⋊C₈	—	212	1	no	—
164	C₄×C₈	—	212	1	no	—
165	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
166	C₄×C₈	D₂₁₂	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	C₂×Q₈	—	106	4	no	—
170	C₂×Q₈	—	106	2	no	—
171	C₂×Q₈	—	212	1	no	—
172	C₂×Q₈	—	212	2	no	—
173	C₂²×C₄	—	212	1	no	—
174	C₂²×C₄	—	212	1	no	—
175	C₂²×C₄	#11719913	1	1	yes	characteristic
176	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
177	C₂²×C₄	#11719921	1	1	yes	characteristic
178	Q₁₆	—	53	4	no	—
179	Q₁₆	—	106	4	no	—
180	Q₁₆	—	106	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	C₈×D₅₃	1	2	yes	—
184	C₂×C₈	#11719912	1	2	yes	—
185	C₂×C₈	#11719921	1	1	yes	characteristic
186	C₂×C₈	#11719921	1	1	yes	characteristic
187	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₄⋊C₄	—	106	2	no	—
190	C₄⋊C₄	—	212	1	no	—
191	C₄⋊C₄	—	212	2	no	—
192	C₄⋊C₄	—	212	2	no	—
193	C₄²	—	212	2	no	—
194	C₄²	—	212	2	no	—
195	C₄²	#11719913	1	1	yes	characteristic
196	C₄²	D₄₂₄	1	1	yes	characteristic
197	C₄²	#11719921	1	1	yes	characteristic
198	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
199	C₁₆	—	2	4	no	—
200	C₂³	#13133139	1	1	yes	characteristic
201	Q₈	—	106	4	no	—
202	Q₈	—	212	4	no	—
203	Q₈	—	212	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	424	2	no	—
207	C₂×C₄	—	212	2	no	—
208	C₂×C₄	—	212	4	no	—
209	C₂×C₄	—	212	2	no	—
210	C₂×C₄	#13133129	1	2	yes	—
211	C₂×C₄	#13133139	1	1	yes	characteristic
212	C₂×C₄	#13133139	1	1	yes	characteristic
213	C₂×C₄	#13133139	1	1	yes	characteristic
214	C₂×C₄	#13133145	1	2	yes	—
215	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
217	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	424	2	no	—
221	C₈	#13133129	1	2	yes	—
222	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
223	C₈	C₂×#11719912	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#16810352	1	2	yes	—
227	C₂²	C₂×#13133139	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	424	2	no	—
232	C₄	—	212	4	no	—
233	C₄	#16810352	1	2	yes	—
234	C₄	C₂×#13133129	1	1	yes	characteristic
235	C₄	C₂×#13133145	1	1	yes	characteristic
236	C₄	#16811066	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	C₂×#16810352	1	1	yes	characteristic
239	C₂	#20276436	1	1	yes	characteristic
240	C₂	#20276444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#11594
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272680	C₂	semidirect
23. C₂×Dic₂₁₂	C₈	semidirect
140. C₅₃	#11594	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
#20273463	2
#20272680	2
C₄×M₃₂×C₅₃	2
#11594	53