Subgroups — C₂×C₅₃×#1544 #27225211

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 246 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×#431	C₂	1	2	yes	maximal
3	C₅₃×#1544	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#332	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	C₂²	1	2	yes	—
9	C₂×#125×C₅₃	C₄	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×#431	—	2	4	no	—
15	C₅₃×#431	C₂²	1	8	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
18	C₂×C₄×C₈×C₅₃	—	2	2	no	—
19	C₅₃×#332	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	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
28	#125×C₅₃	—	2	4	no	—
29	#125×C₅₃	C₂×C₄	1	8	yes	—
30	C₂²⋊Q₈×C₅₃	—	2	8	no	—
31	C₂²⋊Q₈×C₅₃	—	2	8	no	—
32	C₄×Q₈×C₅₃	—	2	8	no	—
33	C₄×Q₈×C₅₃	—	2	8	no	—
34	#117×C₅₃	—	2	4	no	—
35	#117×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	4	yes	—
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
47	C₄⋊C₈×C₅₃	—	2	4	no	—
48	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
49	C₂²⋊C₈×C₅₃	—	2	8	no	—
50	C₄×C₈×C₅₃	—	2	8	no	—
51	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	—	2	4	no	—
53	C₂×Q₈×C₅₃	—	2	4	no	—
54	C₂×Q₈×C₅₃	—	4	8	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	8	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	—	2	8	no	—
67	C₂×C₈×C₅₃	—	4	8	no	—
68	C₄⋊C₄×C₅₃	—	2	16	no	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
73	C₂²⋊C₄×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	—	2	8	no	—
75	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	4	yes	—
80	C₄²×C₅₃	C₂×D₄	1	2	yes	—
81	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	4	8	no	—
87	Q₈×C₅₃	—	4	8	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	8	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	8	no	—
94	C₂×C₄×C₅₃	—	2	8	no	—
95	C₂×C₄×C₅₃	—	4	8	no	—
96	C₂×C₄×C₅₃	#101	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	#135	1	8	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
104	C₈×C₅₃	—	4	8	no	—
105	C₂×#1544	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	2	4	no	—
108	C₂²×C₅₃	—	4	4	no	—
109	C₂²×C₅₃	C₂×#101	1	2	yes	—
110	C₂²×C₅₃	#416	1	4	yes	—
111	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	—	4	8	no	—
116	C₄×C₅₃	C₂×#101	1	4	yes	—
117	C₄×C₅₃	#416	1	4	yes	—
118	C₄×C₅₃	C₂×#135	1	4	yes	—
119	C₂×#533	C₂×C₅₃	1	1	yes	characteristic
120	C₂×#431	C₂×C₅₃	1	2	yes	—
121	#1544	C₂×C₅₃	1	8	yes	direct product factor
122	C₂×#332	C₂×C₅₃	1	4	yes	complemented
123	C₂×C₅₃	—	4	2	no	—
124	C₂×C₅₃	#1544	1	4	yes	direct product factor
125	C₂×C₅₃	C₂²×#101	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#416	1	2	yes	—
127	#533	—	2	4	no	—
128	#533	C₄×C₅₃	1	4	yes	—
129	#533	C₂²×C₅₃	1	4	yes	—
130	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
131	C₂×#125	C₄×C₅₃	1	2	yes	—
132	C₂×C₂²⋊Q₈	—	2	2	no	—
133	C₂×C₄×Q₈	—	2	2	no	—
134	C₂×#117	C₄×C₅₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
136	#431	—	2	4	no	—
137	#431	C₂²×C₅₃	1	8	yes	—
138	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
139	C₂×C₂²⋊C₈	—	2	2	no	—
140	C₂×C₄×C₈	—	2	2	no	—
141	#332	C₂²×C₅₃	1	16	yes	complemented
142	C₅₃	C₂×#1544	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×Q₈	—	2	2	no	—
144	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
145	C₂³×C₄	D₄×C₅₃	1	2	yes	—
146	C₂²×C₈	—	2	4	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
150	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
151	#125	—	2	4	no	—
152	#125	C₂×C₄×C₅₃	1	8	yes	—
153	C₂²⋊Q₈	—	2	8	no	—
154	C₂²⋊Q₈	—	2	8	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₄×Q₈	—	2	8	no	—
157	#117	—	2	4	no	—
158	#117	C₂×C₄×C₅₃	1	8	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	2	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
166	C₂×C₄²	D₄×C₅₃	1	2	yes	—
167	C₂×C₄²	D₄×C₅₃	1	4	yes	—
168	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
172	C₂²⋊C₈	—	2	8	no	—
173	C₄×C₈	—	2	8	no	—
174	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
182	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
188	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
189	C₂×C₈	—	2	8	no	—
190	C₂×C₈	—	4	8	no	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₄²	—	2	8	no	—
198	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
199	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
200	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
201	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₅₃	1	4	yes	—
203	C₄²	C₂×D₄×C₅₃	1	2	yes	—
204	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
205	C₂³	—	2	2	no	—
206	C₂³	—	2	4	no	—
207	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
208	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
209	Q₈	—	4	8	no	—
210	Q₈	—	4	8	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	#101×C₅₃	1	4	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
224	C₂×C₄	#135×C₅₃	1	8	yes	—
225	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
227	C₈	—	4	8	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×#101×C₅₃	1	2	yes	—
232	C₂²	C₅₃×#416	1	4	yes	—
233	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	8	no	—
238	C₄	C₂×#101×C₅₃	1	4	yes	—
239	C₄	C₅₃×#416	1	4	yes	—
240	C₄	C₂×#135×C₅₃	1	4	yes	—
241	C₂	—	4	2	no	—
242	C₂	C₅₃×#1544	1	4	yes	direct product factor
243	C₂	C₂²×#101×C₅₃	1	1	yes	characteristic
244	C₂	C₂×C₅₃×#416	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1544	C₂	direct (N × H)
4. C₂×C₅₃×#332	C₂	semidirect
19. C₅₃×#332	C₂²	semidirect
105. C₂×#1544	C₅₃	direct (N × H)
121. #1544	C₂×C₅₃	direct (N × H)
122. C₂×#332	C₂×C₅₃	semidirect
124. C₂×C₅₃	#1544	direct (N × H)
141. #332	C₂²×C₅₃	semidirect
142. C₅₃	C₂×#1544	direct (N × H)
242. C₂	C₅₃×#1544	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×#431	2
C₅₃×#1544	2
C₂×C₅₃×#332	2
C₂×#1544	53

Subgroups — C2×C53×#1544 #27225211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1544 #27225211