Subgroups — C₂×C₅₃×#1991 #27228829

Order 13568.

1246 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1991	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#340	C₂	1	4	yes	maximal
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×#125×C₅₃	—	2	4	no	—
8	C₂×#125×C₅₃	—	2	2	no	—
9	C₂×#125×C₅₃	C₄	1	8	yes	complemented
10	C₂×#125×C₅₃	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	—	2	4	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#382	—	2	2	no	—
18	C₅₃×#382	C₂²	1	4	yes	—
19	C₂×#95×C₅₃	—	2	2	no	—
20	C₅₃×#340	C₂²	1	16	yes	—
21	C₂³×C₄×C₅₃	—	2	2	no	—
22	C₂³×C₄×C₅₃	—	2	2	no	—
23	C₂³×C₄×C₅₃	D₄	1	2	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	—	2	4	no	—
26	C₂²×C₈×C₅₃	—	2	4	no	—
27	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
29	#125×C₅₃	—	2	16	no	—
30	#125×C₅₃	—	2	8	no	—
31	#125×C₅₃	C₂×C₄	1	16	yes	complemented
32	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅₃	D₄	1	4	yes	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	C₄×C₈×C₅₃	—	2	8	no	—
46	#95×C₅₃	—	2	8	no	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	8	no	—
49	C₂²×C₄×C₅₃	—	2	8	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	—	4	8	no	—
53	C₂²×C₄×C₅₃	—	4	4	no	—
54	C₂²×C₄×C₅₃	—	4	8	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₅₃	#41	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₅₃	—	2	16	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
67	C₄⋊C₄×C₅₃	—	2	32	no	—
68	C₄⋊C₄×C₅₃	—	2	8	no	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	—	2	8	no	—
71	C₄⋊C₄×C₅₃	—	4	4	no	—
72	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
73	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₅₃	C₂×D₄	1	4	yes	—
76	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂³×C₅₃	#124	1	2	yes	—
78	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
79	C₂³×C₅₃	#135	1	8	yes	—
80	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	—	2	16	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	4	8	no	—
85	C₂×C₄×C₅₃	—	4	16	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	4	8	no	—
88	C₂×C₄×C₅₃	—	8	8	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₄×Q₈	1	8	yes	—
92	C₂×C₄×C₅₃	C₂²⋊Q₈	1	8	yes	—
93	C₂×C₄×C₅₃	#124	1	2	yes	—
94	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
99	C₈×C₅₃	—	2	8	no	—
100	C₂×#1991	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₅₃	#391	1	4	yes	—
102	C₂²×C₅₃	#416	1	8	yes	—
103	C₂²×C₅₃	#488	1	8	yes	—
104	C₂²×C₅₃	#499	1	8	yes	—
105	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#135	1	4	yes	—
109	C₄×C₅₃	—	2	8	no	—
110	C₄×C₅₃	—	4	8	no	—
111	C₄×C₅₃	—	8	4	no	—
112	C₄×C₅₃	#391	1	4	yes	—
113	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
114	C₄×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
115	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic
116	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
117	C₂×#382	C₂×C₅₃	1	1	yes	characteristic
118	#1991	C₂×C₅₃	1	8	yes	direct product factor
119	C₂×#340	C₂×C₅₃	1	4	yes	—
120	C₂×C₅₃	#1991	1	8	yes	direct product factor
121	C₂×C₅₃	C₂×#391	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#416	1	2	yes	—
123	C₂×C₅₃	C₂×#488	1	2	yes	—
124	C₂×C₅₃	C₂×#499	1	2	yes	—
125	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
126	C₂×#125	—	2	4	no	—
127	C₂×#125	—	2	2	no	—
128	C₂×#125	C₄×C₅₃	1	8	yes	complemented
129	C₂×#125	C₂²×C₅₃	1	4	yes	—
130	C₂²×C₄⋊C₄	—	2	2	no	—
131	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
133	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×C₄×C₈	—	2	4	no	—
135	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
136	#382	—	2	2	no	—
137	#382	C₂²×C₅₃	1	4	yes	—
138	C₂×#95	—	2	2	no	—
139	#340	C₂²×C₅₃	1	16	yes	—
140	C₅₃	C₂×#1991	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	—	2	2	no	—
143	C₂³×C₄	D₄×C₅₃	1	2	yes	—
144	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	Q₈×C₅₃	1	8	yes	—
148	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
149	#125	—	2	8	no	—
150	#125	—	2	16	no	—
151	#125	C₂×C₄×C₅₃	1	16	yes	complemented
152	C₂×C₄⋊C₄	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	16	no	—
156	C₂×C₄⋊C₄	—	2	8	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
159	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
161	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
162	C₂×C₄²	D₄×C₅₃	1	4	yes	—
163	C₂×C₄²	D₄×C₅₃	1	2	yes	—
164	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
165	C₄×C₈	—	2	8	no	—
166	#95	—	2	8	no	—
167	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	—	4	8	no	—
175	C₂²×C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
176	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
178	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
180	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
181	C₂²×C₄	#41×C₅₃	1	4	yes	—
182	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
183	C₂×C₈	—	2	16	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₂×Q₈×C₅₃	1	8	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
193	C₄²	C₂²⋊C₄×C₅₃	1	8	yes	—
194	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
195	C₄²	C₂×D₄×C₅₃	1	4	yes	—
196	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
197	C₂³	#124×C₅₃	1	2	yes	—
198	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
199	C₂³	#135×C₅₃	1	8	yes	—
200	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	16	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	16	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
210	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
211	C₂×C₄	C₄×Q₈×C₅₃	1	8	yes	—
212	C₂×C₄	C₂²⋊Q₈×C₅₃	1	8	yes	—
213	C₂×C₄	#124×C₅₃	1	2	yes	—
214	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
215	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
218	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₂²	C₅₃×#391	1	4	yes	—
221	C₂²	C₅₃×#416	1	8	yes	—
222	C₂²	C₅₃×#488	1	8	yes	—
223	C₂²	C₅₃×#499	1	8	yes	—
224	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
225	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
226	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
227	C₂²	C₂×#135×C₅₃	1	4	yes	—
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	—	8	4	no	—
231	C₄	C₅₃×#391	1	4	yes	—
232	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
233	C₄	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
234	C₂	C₅₃×#1991	1	8	yes	direct product factor
235	C₂	C₂×C₅₃×#391	1	1	yes	characteristic
236	C₂	C₂×C₅₃×#416	1	2	yes	—
237	C₂	C₂×C₅₃×#488	1	2	yes	—
238	C₂	C₂×C₅₃×#499	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1991	C₂	direct (N × H)
9. C₂×#125×C₅₃	C₄	semidirect
31. #125×C₅₃	C₂×C₄	semidirect
100. C₂×#1991	C₅₃	direct (N × H)
118. #1991	C₂×C₅₃	direct (N × H)
120. C₂×C₅₃	#1991	direct (N × H)
128. C₂×#125	C₄×C₅₃	semidirect
140. C₅₃	C₂×#1991	direct (N × H)
151. #125	C₂×C₄×C₅₃	semidirect
234. C₂	C₅₃×#1991	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#382	2
C₅₃×#1991	2
C₂×C₅₃×#340	2
C₂×#1991	53

Subgroups — C2×C53×#1991 #27228829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1991 #27228829