Subgroups — C₂×C₅₃×#1537 #27225173

Order 13568.

954 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	C₂×C₅₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#432	C₂	1	2	yes	maximal
3	C₅₃×#1537	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#330	C₂	1	4	yes	maximal, complemented
5	C₅₃×#533	—	2	4	no	—
6	C₅₃×#533	C₄	1	4	yes	—
7	C₅₃×#533	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
9	C₂×#125×C₅₃	C₂²	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
11	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
12	C₂×#117×C₅₃	C₄	1	2	yes	—
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#432	—	2	4	no	—
15	C₅₃×#432	C₂²	1	8	yes	—
16	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
17	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
18	C₂×#97×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#330	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	2	yes	—
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₄⋊Q₈×C₅₃	—	2	4	no	—
25	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
26	#125×C₅₃	—	2	4	no	—
27	#125×C₅₃	C₂×C₄	1	4	yes	—
28	#125×C₅₃	C₂³	1	4	yes	—
29	C₂²⋊Q₈×C₅₃	—	2	8	no	—
30	C₂²⋊Q₈×C₅₃	—	2	8	no	—
31	C₄×Q₈×C₅₃	—	2	8	no	—
32	C₄×Q₈×C₅₃	—	2	8	no	—
33	#117×C₅₃	—	2	4	no	—
34	#117×C₅₃	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
46	C₄⋊C₈×C₅₃	—	2	8	no	—
47	C₂²⋊C₈×C₅₃	—	2	8	no	—
48	#97×C₅₃	—	2	4	no	—
49	#97×C₅₃	C₂³	1	8	yes	—
50	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₅₃	—	2	8	no	—
52	C₂×Q₈×C₅₃	—	4	8	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	4	4	no	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₅₃	—	2	8	no	—
65	C₂×C₈×C₅₃	—	4	8	no	—
66	C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₄⋊C₄×C₅₃	—	2	8	no	—
68	C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₄⋊C₄×C₅₃	—	2	8	no	—
70	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
71	C₂²⋊C₄×C₅₃	—	2	8	no	—
72	C₄²×C₅₃	—	2	8	no	—
73	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₄²×C₅₃	C₂×D₄	1	4	yes	—
79	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	—	2	4	no	—
82	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₅₃	—	4	16	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	8	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	4	8	no	—
94	C₂×C₄×C₅₃	#101	1	4	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
101	C₈×C₅₃	—	4	8	no	—
102	C₂×#1537	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	4	no	—
105	C₂²×C₅₃	—	4	4	no	—
106	C₂²×C₅₃	C₂×#101	1	2	yes	—
107	C₂²×C₅₃	#421	1	4	yes	—
108	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	2	4	no	—
114	C₄×C₅₃	—	4	8	no	—
115	C₄×C₅₃	C₂×#101	1	4	yes	—
116	C₂×#533	C₂×C₅₃	1	1	yes	characteristic
117	C₂×#432	C₂×C₅₃	1	2	yes	—
118	#1537	C₂×C₅₃	1	8	yes	direct product factor
119	C₂×#330	C₂×C₅₃	1	4	yes	complemented
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	#1537	1	4	yes	direct product factor
122	C₂×C₅₃	C₂²×#101	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#421	1	2	yes	—
124	#533	—	2	4	no	—
125	#533	C₄×C₅₃	1	4	yes	—
126	#533	C₂²×C₅₃	1	4	yes	—
127	C₂×C₄⋊Q₈	C₄×C₅₃	1	2	yes	—
128	C₂×#125	C₂²×C₅₃	1	2	yes	—
129	C₂×C₂²⋊Q₈	—	2	2	no	—
130	C₂×C₄×Q₈	—	2	2	no	—
131	C₂×#117	C₄×C₅₃	1	2	yes	—
132	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
133	#432	—	2	4	no	—
134	#432	C₂²×C₅₃	1	8	yes	—
135	C₂×C₄⋊C₈	—	2	2	no	—
136	C₂×C₂²⋊C₈	—	2	2	no	—
137	C₂×#97	C₂²×C₅₃	1	4	yes	—
138	#330	C₂²×C₅₃	1	16	yes	complemented
139	C₅₃	C₂×#1537	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×Q₈	—	2	2	no	—
141	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
142	C₂³×C₄	D₄×C₅₃	1	2	yes	—
143	C₂²×C₈	—	2	4	no	—
144	C₄⋊Q₈	—	2	4	no	—
145	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
146	#125	—	2	4	no	—
147	#125	C₂×C₄×C₅₃	1	4	yes	—
148	#125	C₂³×C₅₃	1	4	yes	—
149	C₂²⋊Q₈	—	2	8	no	—
150	C₂²⋊Q₈	—	2	8	no	—
151	C₄×Q₈	—	2	8	no	—
152	C₄×Q₈	—	2	8	no	—
153	#117	—	2	4	no	—
154	#117	C₂×C₄×C₅₃	1	8	yes	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
158	C₂×C₂²⋊C₄	—	2	2	no	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
161	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
162	C₂×C₄²	D₄×C₅₃	1	2	yes	—
163	C₂×C₄²	D₄×C₅₃	1	4	yes	—
164	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
166	C₄⋊C₈	—	2	8	no	—
167	C₂²⋊C₈	—	2	8	no	—
168	#97	—	2	4	no	—
169	#97	C₂³×C₅₃	1	8	yes	—
170	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	—	4	8	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
177	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
179	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	4	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	8	no	—
190	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
191	C₂²⋊C₄	—	2	8	no	—
192	C₄²	—	2	8	no	—
193	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
194	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
195	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
196	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
197	C₄²	C₂×D₄×C₅₃	1	2	yes	—
198	C₄²	C₂×D₄×C₅₃	1	4	yes	—
199	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
203	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
204	Q₈	—	4	16	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	#101×C₅₃	1	4	yes	—
215	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
217	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
219	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
221	C₈	—	4	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×#101×C₅₃	1	2	yes	—
226	C₂²	C₅₃×#421	1	4	yes	—
227	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	4	no	—
233	C₄	—	4	8	no	—
234	C₄	C₂×#101×C₅₃	1	4	yes	—
235	C₂	—	4	2	no	—
236	C₂	C₅₃×#1537	1	4	yes	direct product factor
237	C₂	C₂²×#101×C₅₃	1	1	yes	characteristic
238	C₂	C₂×C₅₃×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1537	C₂	direct (N × H)
4. C₂×C₅₃×#330	C₂	semidirect
19. C₅₃×#330	C₂²	semidirect
102. C₂×#1537	C₅₃	direct (N × H)
118. #1537	C₂×C₅₃	direct (N × H)
119. C₂×#330	C₂×C₅₃	semidirect
121. C₂×C₅₃	#1537	direct (N × H)
138. #330	C₂²×C₅₃	semidirect
139. C₅₃	C₂×#1537	direct (N × H)
236. C₂	C₅₃×#1537	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#533	2
C₂×C₅₃×#432	2
C₅₃×#1537	2
C₂×C₅₃×#330	2
C₂×#1537	53

Subgroups — C2×C53×#1537 #27225173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1537 #27225173