Subgroups — #27537288

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	2	yes	maximal, complemented
2	#20283428	C₂	1	1	yes	maximal, characteristic
3	#20276542	C₂	1	8	yes	maximal, complemented
4	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276328	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133223	C₂²	1	2	yes	—
8	#16811173	C₄	1	1	yes	characteristic
9	#16811039	—	2	8	no	—
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂×#13133156	C₂²	1	4	yes	—
12	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
13	#16810702	C₄	1	4	yes	—
14	C₈×D₂₁₂	C₂²	1	4	yes	—
15	#16810647	C₄	1	1	yes	characteristic
16	#16810589	C₄	1	2	yes	—
17	C₄⋊#11719908	C₂²	1	4	yes	—
18	#16810402	C₂²	1	8	yes	—
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#346	C₂²	1	4	yes	—
23	C₂×#11719945	C₂×C₄	1	2	yes	—
24	#13133223	—	2	4	no	—
25	#13133223	—	2	4	no	—
26	C₂×#11719917	C₂×C₄	1	2	yes	—
27	C₂×#11719912	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₅₃	C₂³	1	4	yes	—
29	#13133217	C₈	1	1	yes	characteristic
30	#13133214	C₈	1	1	yes	characteristic
31	#13133205	C₈	1	2	yes	—
32	#13133191	C₈	1	4	yes	—
33	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
34	#13133157	—	2	4	no	—
35	#13133156	—	2	8	no	—
36	C₂×#11719908	C₂³	1	4	yes	—
37	C₁₆×D₅₃	—	2	8	no	—
38	C₄⋊#88322	C₂×C₄	1	4	yes	—
39	#13133129	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	D₄	1	4	yes	complemented
45	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
46	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₅₃	C₂×C₈	1	4	yes	—
52	#11719945	—	2	4	no	—
53	#11719945	—	2	4	no	—
54	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
55	C₂×D₂₁₂	C₂×C₈	1	4	yes	—
56	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
57	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₅₃	#41	1	2	yes	—
63	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
67	#11719921	C₂×C₈	1	8	yes	—
68	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
69	#11719917	—	2	4	no	—
70	C₂×#88322	C₂²×C₄	1	4	yes	—
71	#11719912	—	2	8	no	—
72	C₈×D₅₃	—	2	4	no	—
73	C₈×D₅₃	—	2	4	no	—
74	C₁₆×C₅₃	—	2	2	no	—
75	C₁₆×C₅₃	C₂×D₄	1	4	yes	—
76	#11719908	—	2	4	no	—
77	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#131	1	2	yes	—
85	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
86	C₂²⋊D₅₃	—	2	8	no	—
87	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
88	D₂₁₂	—	2	4	no	—
89	D₂₁₂	—	4	4	no	—
90	C₄×D₅₃	—	2	4	no	—
91	C₄×D₅₃	—	2	4	no	—
92	Dic₁₀₆	—	2	4	no	—
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	C₄×D₄	1	2	yes	—
95	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
97	#88322	—	2	4	no	—
98	#33435	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	C₈×D₄	1	2	yes	—
101	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
102	C₂×D₅₃	—	2	8	no	—
103	C₂×D₅₃	—	2	4	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	C₈×D₄	1	2	yes	—
107	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
109	Dic₅₃	—	2	4	no	—
110	C₂×#504	—	53	2	no	—
111	#2931	—	53	1	no	—
112	#2168	—	53	8	no	—
113	#2150	—	53	1	no	—
114	#2113	—	53	2	no	—
115	C₂×#346	D₅₃	1	1	yes	characteristic, complemented
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
118	C₂×C₅₃	#3407	1	2	yes	—
119	D₅₃	—	4	4	no	—
120	C₂×#131	—	53	2	no	—
121	#532	—	53	1	no	—
122	#504	—	106	8	no	—
123	C₂×M₃₂	—	53	2	no	—
124	C₂×M₃₂	—	53	4	no	—
125	C₂²×C₁₆	—	53	4	no	—
126	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
127	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
128	#436	—	53	4	no	—
129	C₈×D₄	—	53	4	no	—
130	#423	—	53	1	no	—
131	#408	—	53	2	no	—
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	C₄⋊C₁₆	—	53	4	no	—
134	C₂²⋊C₁₆	—	53	8	no	—
135	#346	C₂×D₅₃	1	4	yes	—
136	C₅₃	#33435	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	—	53	2	no	—
138	#131	—	106	4	no	—
139	#131	—	106	4	no	—
140	C₂×M₁₆	—	53	2	no	—
141	C₂×M₁₆	—	53	4	no	—
142	C₂²×C₈	—	53	4	no	—
143	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
145	C₄⋊Q₈	—	53	1	no	—
146	#127	—	53	1	no	—
147	#124	—	53	2	no	—
148	#121	—	53	4	no	—
149	C₄×D₄	—	53	4	no	—
150	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
151	M₃₂	—	106	4	no	—
152	M₃₂	—	106	8	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	53	4	no	—
155	C₂×C₁₆	—	106	8	no	—
156	C₂×C₁₆	D₂₁₂	1	4	yes	complemented
157	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
158	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
159	C₄⋊C₈	—	53	4	no	—
160	C₂²⋊C₈	—	53	8	no	—
161	C₄×C₈	C₄×D₅₃	1	2	yes	—
162	C₄×C₈	C₂²×D₅₃	1	2	yes	—
163	#41	—	106	4	no	—
164	#41	—	106	4	no	—
165	C₂×Q₈	—	53	2	no	—
166	C₂×D₄	—	53	4	no	—
167	C₂×D₄	—	53	2	no	—
168	C₂×D₄	—	53	4	no	—
169	C₂²×C₄	—	53	4	no	—
170	C₂²×C₄	C₈×D₅₃	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
172	M₁₆	—	106	4	no	—
173	M₁₆	—	106	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	106	4	no	—
177	C₂×C₈	—	106	4	no	—
178	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
179	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
181	C₂×C₈	#11719945	1	2	yes	—
182	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
183	C₄⋊C₄	—	53	4	no	—
184	C₂²⋊C₄	—	53	8	no	—
185	C₄²	C₈×D₅₃	1	1	yes	characteristic
186	C₄²	C₈×D₅₃	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₅₃	1	2	yes	—
188	C₁₆	—	2	2	no	—
189	C₁₆	—	106	4	no	—
190	C₁₆	C₂×D₂₁₂	1	4	yes	—
191	C₂³	—	53	4	no	—
192	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
193	Q₈	—	106	4	no	—
194	D₄	—	106	8	no	—
195	D₄	—	106	4	no	—
196	D₄	—	212	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	53	4	no	—
199	C₂×C₄	—	106	4	no	—
200	C₂×C₄	—	106	4	no	—
201	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
203	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
204	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
205	C₂×C₄	#13133223	1	2	yes	—
206	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	106	4	no	—
209	C₈	C₄×D₂₁₂	1	2	yes	—
210	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
211	C₈	C₂×#11719945	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	106	8	no	—
214	C₂²	—	106	4	no	—
215	C₂²	C₈×D₂₁₂	1	2	yes	—
216	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	106	4	no	—
220	C₄	C₈×D₂₁₂	1	2	yes	—
221	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
222	C₄	C₂×#13133223	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	212	4	no	—
225	C₂	C₂×C₈×D₂₁₂	1	1	yes	characteristic
226	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×#346	C₂
prime radical	#16811173	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#33435
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#16811039	C₂	semidirect
3. #20276542	C₂	semidirect
4. #20276471	C₂	semidirect
5. #20276328	C₂	semidirect
6. C₂×C₅₃×#346	C₂	semidirect
44. C₂×C₁₆×C₅₃	D₄	semidirect
115. C₂×#346	D₅₃	semidirect
136. C₅₃	#33435	semidirect
156. C₂×C₁₆	D₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811039	2
#20283428	2
#20276542	2
#20276471	2
#20276328	2
C₂×C₅₃×#346	2
#33435	53