Subgroups — #27539451

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
3	#20276961	C₂	1	4	yes	maximal, complemented
4	#20276943	C₂	1	4	yes	maximal, complemented
5	#20276436	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	—	2	1	no	—
8	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
9	#16811180	—	2	1	no	—
10	C₂×#13133217	—	2	1	no	—
11	#16811028	C₂²	1	2	yes	—
12	#16811025	—	2	1	no	—
13	#16811025	C₂²	1	2	yes	—
14	#16811021	—	2	1	no	—
15	#16811003	—	2	1	no	—
16	#16811003	—	2	1	no	—
17	#16811003	C₂²	1	2	yes	—
18	#16811003	C₂²	1	2	yes	—
19	#16810661	—	2	1	no	—
20	#16810661	C₂²	1	2	yes	—
21	#16810428	C₂²	1	16	yes	—
22	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#346	C₂²	1	2	yes	—
25	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
26	C₂²×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₂₁₂	—	2	2	no	—
28	C₂×Dic₂₁₂	—	2	2	no	—
29	C₂×Dic₂₁₂	—	2	2	no	—
30	C₂×Dic₂₁₂	—	2	8	no	—
31	C₂×Dic₂₁₂	—	4	4	no	—
32	C₂×Dic₂₁₂	C₂³	1	4	yes	—
33	#13133217	—	2	2	no	—
34	#13133217	—	2	2	no	—
35	#13133217	—	2	2	no	—
36	#13133217	—	4	2	no	—
37	#13133209	—	2	2	no	—
38	#13133196	—	4	2	no	—
39	C₄×Dic₁₀₆	—	4	2	no	—
40	#13133176	—	2	2	no	—
41	C₂×C₄⋊Dic₅₃	—	4	2	no	—
42	#13133151	—	2	2	no	—
43	#13133151	C₂³	1	4	yes	—
44	#13133149	—	2	2	no	—
45	#13133149	—	2	2	no	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	M₃₂×C₅₃	—	2	4	no	—
50	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	D₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	4	2	no	—
58	C₂×Dic₁₀₆	—	2	4	no	—
59	C₂×Dic₁₀₆	—	4	4	no	—
60	C₂×Dic₁₀₆	—	4	4	no	—
61	C₂×Dic₁₀₆	—	4	2	no	—
62	C₂×Dic₁₀₆	—	4	2	no	—
63	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	4	yes	—
68	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₄²×C₅₃	D₈	1	2	yes	—
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	2	4	no	—
74	C₄⋊Dic₅₃	—	4	4	no	—
75	C₄⋊Dic₅₃	—	4	4	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	#11719919	—	4	4	no	—
78	C₄×Dic₅₃	—	4	2	no	—
79	Dic₂₁₂	—	2	8	no	—
80	Dic₂₁₂	—	4	4	no	—
81	Dic₂₁₂	—	4	4	no	—
82	Dic₂₁₂	—	4	4	no	—
83	C₁₆×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	#124	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₅₃	—	4	4	no	—
93	C₂×Dic₅₃	—	4	4	no	—
94	C₂×Dic₅₃	—	8	4	no	—
95	Dic₁₀₆	—	4	4	no	—
96	Dic₁₀₆	—	4	4	no	—
97	Dic₁₀₆	—	8	4	no	—
98	Dic₁₀₆	—	8	4	no	—
99	C₈×C₅₃	#124	1	2	yes	—
100	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
101	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
102	C₈×C₅₃	C₂×#41	1	2	yes	—
103	#33705	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	#486	1	2	yes	—
106	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	#486	1	2	yes	—
109	C₄×C₅₃	#510	1	4	yes	—
110	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
111	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
112	Dic₅₃	—	8	4	no	—
113	Dic₅₃	—	8	4	no	—
114	#3160	—	53	1	no	—
115	C₂×#494	—	53	1	no	—
116	#2245	—	53	4	no	—
117	#2241	—	53	4	no	—
118	#2141	—	53	4	no	—
119	C₄×M₃₂	D₅₃	1	1	yes	characteristic
120	C₂×C₅₃	—	2	2	no	—
121	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#510	1	2	yes	—
123	C₂²×Q₁₆	—	53	2	no	—
124	C₂²×Q₁₆	—	106	1	no	—
125	#533	—	106	1	no	—
126	C₂×C₄⋊Q₈	—	106	1	no	—
127	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
128	#500	—	53	2	no	—
129	#499	—	53	2	no	—
130	#499	—	106	1	no	—
131	#498	—	106	1	no	—
132	#494	—	53	2	no	—
133	#494	—	53	2	no	—
134	#494	—	106	1	no	—
135	#494	—	106	1	no	—
136	#427	—	53	2	no	—
137	#427	—	106	1	no	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	#358	—	53	16	no	—
140	#346	C₂×D₅₃	1	2	yes	—
141	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
142	C₅₃	#33705	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₈	—	106	2	no	—
144	C₂×Q₁₆	—	53	4	no	—
145	C₂×Q₁₆	—	106	2	no	—
146	C₂×Q₁₆	—	106	2	no	—
147	C₂×Q₁₆	—	106	2	no	—
148	C₂×Q₁₆	—	106	8	no	—
149	C₂×Q₁₆	—	212	4	no	—
150	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	106	2	no	—
153	C₄⋊Q₈	—	106	2	no	—
154	C₄⋊Q₈	—	106	2	no	—
155	C₄⋊Q₈	—	212	2	no	—
156	#125	—	106	2	no	—
157	C₂²⋊Q₈	—	212	2	no	—
158	C₄×Q₈	—	212	2	no	—
159	#117	—	106	2	no	—
160	C₂×C₄⋊C₄	—	212	2	no	—
161	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
162	M₃₂	—	2	4	no	—
163	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
164	#107	—	53	4	no	—
165	#107	—	106	2	no	—
166	#106	—	106	2	no	—
167	#106	—	106	2	no	—
168	C₄×C₈	D₂₁₂	1	1	yes	characteristic
169	C₄×C₈	D₂₁₂	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×Q₈	—	106	4	no	—
173	C₂×Q₈	—	212	4	no	—
174	C₂×Q₈	—	212	4	no	—
175	C₂×Q₈	—	212	2	no	—
176	C₂×Q₈	—	212	2	no	—
177	C₂²×C₄	—	212	2	no	—
178	C₂²×C₄	D₄₂₄	1	2	yes	—
179	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
180	Q₁₆	—	106	8	no	—
181	Q₁₆	—	212	4	no	—
182	Q₁₆	—	212	4	no	—
183	Q₁₆	—	212	4	no	—
184	C₂×C₈	#11719913	1	4	yes	—
185	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
186	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
187	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
188	C₂×C₈	#11719945	1	4	yes	—
189	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
190	C₄⋊C₄	—	106	4	no	—
191	C₄⋊C₄	—	212	4	no	—
192	C₄⋊C₄	—	212	4	no	—
193	C₄⋊C₄	—	212	4	no	—
194	C₄⋊C₄	—	212	2	no	—
195	C₂²⋊C₄	—	212	2	no	—
196	C₄²	—	212	2	no	—
197	C₄²	D₄₂₄	1	2	yes	—
198	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
199	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
200	C₁₆	—	2	4	no	—
201	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
202	Q₈	—	424	4	no	—
203	Q₈	—	424	4	no	—
204	Q₈	—	212	4	no	—
205	Q₈	—	212	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	424	4	no	—
208	C₂×C₄	—	212	4	no	—
209	C₂×C₄	—	212	4	no	—
210	C₂×C₄	#13133205	1	2	yes	—
211	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
212	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
213	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719913	1	2	yes	—
215	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
216	C₈	#13133205	1	2	yes	—
217	C₈	C₂×#11719913	1	2	yes	—
218	C₈	C₂×#11719913	1	2	yes	—
219	C₈	C₂×#11719945	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#16810970	1	2	yes	—
222	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	424	4	no	—
225	C₄	—	424	4	no	—
226	C₄	#16810970	1	2	yes	—
227	C₄	#16811066	1	4	yes	—
228	C₄	C₂×#13133205	1	1	yes	characteristic
229	C₄	C₂²×#11719913	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#16810970	1	1	yes	characteristic
232	C₂	C₂×#16811066	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33705
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20276961	C₂	semidirect
4. #20276943	C₂	semidirect
142. C₅₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
C₂×#16811003	2
#20276961	2
#20276943	2
#20276436	2
C₄×M₃₂×C₅₃	2
#33705	53