Subgroups — C₂×C₅₃×#2858 #27244139

Order 13568.

4462 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#124×C₅₃	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2858	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#390	C₂	1	8	yes	maximal, complemented
6	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
9	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₅₃	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
12	C₂×#127×C₅₃	—	2	8	no	—
13	C₂×#127×C₅₃	C₂²	1	8	yes	—
14	C₂×#124×C₅₃	—	2	32	no	—
15	C₂×#124×C₅₃	C₂²	1	32	yes	complemented
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
17	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	16	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#390	C₂²	1	64	yes	complemented
22	C₄³×C₅₃	C₂²	1	8	yes	complemented
23	C₂⁵×C₅₃	C₂³	1	4	yes	—
24	C₂²×Q₈×C₅₃	—	2	16	no	—
25	C₂²×Q₈×C₅₃	—	2	8	no	—
26	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
27	C₂²×D₄×C₅₃	—	2	16	no	—
28	C₂²×D₄×C₅₃	—	2	16	no	—
29	C₂²×D₄×C₅₃	—	2	8	no	—
30	C₂²×D₄×C₅₃	—	2	32	no	—
31	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
32	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₅₃	—	2	32	no	—
38	#127×C₅₃	—	2	32	no	—
39	#124×C₅₃	—	2	128	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
42	C₂×C₂²⋊C₄×C₅₃	—	2	64	no	—
43	C₂×C₂²⋊C₄×C₅₃	C₂³	1	64	yes	—
44	C₂×C₄²×C₅₃	—	2	16	no	—
45	C₂×C₄²×C₅₃	D₄	1	8	yes	—
46	C₂×C₄²×C₅₃	D₄	1	16	yes	complemented
47	C₂×C₄²×C₅₃	C₂³	1	16	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
50	C₂⁴×C₅₃	—	2	16	no	—
51	C₂⁴×C₅₃	—	2	8	no	—
52	C₂⁴×C₅₃	—	2	32	no	—
53	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
54	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂×Q₈×C₅₃	—	2	32	no	—
56	C₂×Q₈×C₅₃	—	2	16	no	—
57	C₂×Q₈×C₅₃	—	4	32	no	—
58	C₂×D₄×C₅₃	—	2	32	no	—
59	C₂×D₄×C₅₃	—	2	32	no	—
60	C₂×D₄×C₅₃	—	2	16	no	—
61	C₂×D₄×C₅₃	—	2	64	no	—
62	C₂×D₄×C₅₃	—	4	64	no	—
63	C₂×D₄×C₅₃	—	4	32	no	—
64	C₂²×C₄×C₅₃	—	2	16	no	—
65	C₂²×C₄×C₅₃	—	2	16	no	—
66	C₂²×C₄×C₅₃	—	2	8	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
71	C₂²×C₄×C₅₃	#41	1	16	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
76	C₄⋊C₄×C₅₃	—	2	32	no	—
77	C₂²⋊C₄×C₅₃	—	2	128	no	—
78	C₄²×C₅₃	—	2	32	no	—
79	C₄²×C₅₃	C₂×D₄	1	16	yes	—
80	C₄²×C₅₃	C₂×D₄	1	32	yes	complemented
81	C₂³×C₅₃	—	2	8	no	—
82	C₂³×C₅₃	—	2	16	no	—
83	C₂³×C₅₃	—	2	32	no	—
84	C₂³×C₅₃	—	4	32	no	—
85	C₂³×C₅₃	—	4	16	no	—
86	C₂³×C₅₃	—	4	64	no	—
87	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
88	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
89	C₂³×C₅₃	C₂×#41	1	8	yes	—
90	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
91	Q₈×C₅₃	—	4	32	no	—
92	D₄×C₅₃	—	4	64	no	—
93	D₄×C₅₃	—	4	32	no	—
94	C₂×C₄×C₅₃	—	2	32	no	—
95	C₂×C₄×C₅₃	—	2	16	no	—
96	C₂×C₄×C₅₃	—	2	16	no	—
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	—	4	16	no	—
99	C₂×C₄×C₅₃	#124	1	32	yes	complemented
100	C₂×C₄×C₅₃	#127	1	16	yes	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
102	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
104	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
105	C₂×C₄×C₅₃	C₂×#41	1	16	yes	—
106	C₂×#2858	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₅₃	—	4	16	no	—
108	C₂²×C₅₃	—	4	8	no	—
109	C₂²×C₅₃	—	4	32	no	—
110	C₂²×C₅₃	—	8	32	no	—
111	C₂²×C₅₃	C₂×#124	1	8	yes	—
112	C₂²×C₅₃	C₂×#127	1	4	yes	—
113	C₂²×C₅₃	#532	1	16	yes	—
114	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
116	C₂²×C₅₃	C₂²×#41	1	4	yes	—
117	C₄×C₅₃	—	2	16	no	—
118	C₄×C₅₃	—	4	8	no	—
119	C₄×C₅₃	C₂×#124	1	16	yes	complemented
120	C₄×C₅₃	C₂×#127	1	8	yes	—
121	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
122	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
123	C₂²×#124	C₂×C₅₃	1	4	yes	complemented
124	#2858	C₂×C₅₃	1	16	yes	direct product factor
125	C₂×#390	C₂×C₅₃	1	8	yes	complemented
126	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic, complemented
127	C₂×C₅₃	—	8	8	no	—
128	C₂×C₅₃	#2858	1	8	yes	direct product factor
129	C₂×C₅₃	C₂²×#124	1	2	yes	—
130	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#532	1	4	yes	—
132	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
133	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
134	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
135	C₂×C₄⋊Q₈	—	2	8	no	—
136	C₂×C₄⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
137	C₂×#127	—	2	8	no	—
138	C₂×#127	C₂²×C₅₃	1	8	yes	—
139	C₂×#124	—	2	32	no	—
140	C₂×#124	C₂²×C₅₃	1	32	yes	complemented
141	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
142	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	16	yes	—
143	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
144	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
145	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
146	#390	C₂²×C₅₃	1	64	yes	complemented
147	C₄³	C₂²×C₅₃	1	8	yes	complemented
148	C₅₃	C₂×#2858	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁵	C₂³×C₅₃	1	4	yes	—
150	C₂²×Q₈	—	2	16	no	—
151	C₂²×Q₈	—	2	8	no	—
152	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
153	C₂²×D₄	—	2	16	no	—
154	C₂²×D₄	—	2	16	no	—
155	C₂²×D₄	—	2	8	no	—
156	C₂²×D₄	—	2	32	no	—
157	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
158	C₂²×D₄	C₂³×C₅₃	1	16	yes	—
159	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
160	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
161	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
163	C₄⋊Q₈	—	2	32	no	—
164	#127	—	2	32	no	—
165	#124	—	2	128	no	—
166	C₂×C₄⋊C₄	—	2	16	no	—
167	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
168	C₂×C₂²⋊C₄	—	2	64	no	—
169	C₂×C₂²⋊C₄	C₂³×C₅₃	1	64	yes	—
170	C₂×C₄²	—	2	16	no	—
171	C₂×C₄²	D₄×C₅₃	1	8	yes	—
172	C₂×C₄²	D₄×C₅₃	1	16	yes	complemented
173	C₂×C₄²	C₂³×C₅₃	1	16	yes	—
174	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
175	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
176	C₂⁴	—	2	16	no	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	—	2	32	no	—
179	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
180	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
181	C₂×Q₈	—	2	32	no	—
182	C₂×Q₈	—	2	16	no	—
183	C₂×Q₈	—	4	32	no	—
184	C₂×D₄	—	2	32	no	—
185	C₂×D₄	—	2	32	no	—
186	C₂×D₄	—	2	16	no	—
187	C₂×D₄	—	2	64	no	—
188	C₂×D₄	—	4	64	no	—
189	C₂×D₄	—	4	32	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	16	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	—
194	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
195	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
196	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
197	C₂²×C₄	#41×C₅₃	1	16	yes	—
198	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
199	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
200	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
202	C₄⋊C₄	—	2	32	no	—
203	C₂²⋊C₄	—	2	128	no	—
204	C₄²	—	2	32	no	—
205	C₄²	C₂×D₄×C₅₃	1	16	yes	—
206	C₄²	C₂×D₄×C₅₃	1	32	yes	complemented
207	C₂³	—	2	16	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	32	no	—
210	C₂³	—	4	32	no	—
211	C₂³	—	4	16	no	—
212	C₂³	—	4	64	no	—
213	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
214	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
215	C₂³	C₂×#41×C₅₃	1	8	yes	—
216	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
217	Q₈	—	4	32	no	—
218	D₄	—	4	64	no	—
219	D₄	—	4	32	no	—
220	C₂×C₄	—	2	32	no	—
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	16	no	—
225	C₂×C₄	#124×C₅₃	1	32	yes	complemented
226	C₂×C₄	#127×C₅₃	1	16	yes	—
227	C₂×C₄	C₂²×D₄×C₅₃	1	16	yes	—
228	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
229	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
231	C₂×C₄	C₂×#41×C₅₃	1	16	yes	—
232	C₂²	—	4	16	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	32	no	—
235	C₂²	—	8	32	no	—
236	C₂²	C₂×#124×C₅₃	1	8	yes	—
237	C₂²	C₂×#127×C₅₃	1	4	yes	—
238	C₂²	C₅₃×#532	1	16	yes	—
239	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
240	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
241	C₂²	C₂²×#41×C₅₃	1	4	yes	—
242	C₄	—	2	16	no	—
243	C₄	—	4	8	no	—
244	C₄	C₂×#124×C₅₃	1	16	yes	complemented
245	C₄	C₂×#127×C₅₃	1	8	yes	—
246	C₂	—	8	8	no	—
247	C₂	C₅₃×#2858	1	8	yes	direct product factor
248	C₂	C₂²×#124×C₅₃	1	2	yes	—
249	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
250	C₂	C₂×C₅₃×#532	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂²×#127×C₅₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
3. C₂²×#124×C₅₃	C₂	semidirect
4. C₅₃×#2858	C₂	direct (N × H)
5. C₂×C₅₃×#390	C₂	semidirect
6. C₂×C₄³×C₅₃	C₂	semidirect
11. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
15. C₂×#124×C₅₃	C₂²	semidirect
21. C₅₃×#390	C₂²	semidirect
22. C₄³×C₅₃	C₂²	semidirect
46. C₂×C₄²×C₅₃	D₄	semidirect
80. C₄²×C₅₃	C₂×D₄	semidirect
99. C₂×C₄×C₅₃	#124	semidirect
106. C₂×#2858	C₅₃	direct (N × H)
119. C₄×C₅₃	C₂×#124	semidirect
121. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
123. C₂²×#124	C₂×C₅₃	semidirect
124. #2858	C₂×C₅₃	direct (N × H)
125. C₂×#390	C₂×C₅₃	semidirect
126. C₂×C₄³	C₂×C₅₃	semidirect
128. C₂×C₅₃	#2858	direct (N × H)
136. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
140. C₂×#124	C₂²×C₅₃	semidirect
146. #390	C₂²×C₅₃	semidirect
147. C₄³	C₂²×C₅₃	semidirect
148. C₅₃	C₂×#2858	direct (N × H)
172. C₂×C₄²	D₄×C₅₃	semidirect
206. C₄²	C₂×D₄×C₅₃	semidirect
225. C₂×C₄	#124×C₅₃	semidirect
244. C₄	C₂×#124×C₅₃	semidirect
247. C₂	C₅₃×#2858	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×C₄⋊Q₈×C₅₃	2
C₂²×#127×C₅₃	2
C₂²×#124×C₅₃	2
C₅₃×#2858	2
C₂×C₅₃×#390	2
C₂×C₄³×C₅₃	2
C₂×#2858	53

Subgroups — C2×C53×#2858 #27244139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2858 #27244139