Subgroups — C₂×C₅₃×#2225 #27241135

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2225	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#367	C₂	1	2	yes	maximal
5	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#358	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
9	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
10	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#500	—	2	2	no	—
12	C₅₃×#500	C₂²	1	4	yes	—
13	C₄×Q₁₆×C₅₃	—	2	2	no	—
14	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
15	C₂×#107×C₅₃	—	2	1	no	—
16	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#103×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#367	C₂²	1	8	yes	—
21	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	—
22	C₅₃×#358	C₂²	1	8	yes	—
23	C₂²×Q₈×C₅₃	—	2	1	no	—
24	C₂²×Q₈×C₅₃	—	4	1	no	—
25	C₂×Q₁₆×C₅₃	—	2	1	no	—
26	C₂×Q₁₆×C₅₃	—	2	2	no	—
27	C₂×Q₁₆×C₅₃	Q₈	1	4	yes	complemented
28	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₃	—	2	4	no	—
32	C₄⋊Q₈×C₅₃	—	4	2	no	—
33	C₄×Q₈×C₅₃	—	2	4	no	—
34	C₄×Q₈×C₅₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	—	4	1	no	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	—	2	4	no	—
42	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
43	#107×C₅₃	—	2	2	no	—
44	#107×C₅₃	—	2	4	no	—
45	#107×C₅₃	—	2	1	no	—
46	#107×C₅₃	C₂³	1	4	yes	—
47	#107×C₅₃	C₂³	1	2	yes	—
48	#103×C₅₃	—	2	4	no	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂×Q₈×C₅₃	—	2	4	no	—
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂×Q₈×C₅₃	—	4	4	no	—
55	C₂×Q₈×C₅₃	—	4	2	no	—
56	C₂×Q₈×C₅₃	—	8	4	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₅₃	—	2	4	no	—
64	Q₁₆×C₅₃	—	2	2	no	—
65	Q₁₆×C₅₃	C₂×Q₈	1	4	yes	complemented
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₅₃	#41	1	2	yes	—
71	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	—	4	1	no	—
75	C₄⋊C₄×C₅₃	—	4	2	no	—
76	C₄⋊C₄×C₅₃	—	4	4	no	—
77	C₄⋊C₄×C₅₃	—	4	4	no	—
78	C₄²×C₅₃	—	4	4	no	—
79	C₄²×C₅₃	D₈	1	2	yes	—
80	C₄²×C₅₃	C₂×D₄	1	2	yes	—
81	C₁₆×C₅₃	—	2	4	no	—
82	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	4	no	—
84	Q₈×C₅₃	—	4	2	no	—
85	Q₈×C₅₃	—	4	4	no	—
86	Q₈×C₅₃	—	8	8	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	8	4	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
95	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₅₃	#137	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×#2225	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	#474	1	2	yes	—
107	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
108	C₂²×C₅₃	#510	1	2	yes	—
109	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	4	4	no	—
112	C₄×C₅₃	—	8	4	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	#474	1	2	yes	—
115	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
116	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
118	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
119	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
120	#2225	C₂×C₅₃	1	8	yes	direct product factor
121	C₂×#367	C₂×C₅₃	1	2	yes	—
122	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
123	C₂×#358	C₂×C₅₃	1	2	yes	—
124	C₂×C₅₃	#2225	1	4	yes	direct product factor
125	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
126	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
128	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×C₄⋊Q₈	—	2	1	no	—
130	C₂×C₄×Q₈	—	2	1	no	—
131	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
132	#500	—	2	2	no	—
133	#500	C₂²×C₅₃	1	4	yes	—
134	C₄×Q₁₆	—	2	2	no	—
135	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	—
136	C₂×#107	—	2	1	no	—
137	C₂×#107	C₂²×C₅₃	1	2	yes	—
138	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
139	C₂×#103	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
141	#367	C₂²×C₅₃	1	8	yes	—
142	C₄⋊C₁₆	C₂²×C₅₃	1	4	yes	—
143	#358	C₂²×C₅₃	1	8	yes	—
144	C₅₃	C₂×#2225	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	—	4	1	no	—
147	C₂×Q₁₆	—	2	1	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	Q₈×C₅₃	1	4	yes	complemented
150	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
151	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	4	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄×Q₈	—	2	4	no	—
156	C₄×Q₈	—	4	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	4	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄²	—	4	1	no	—
162	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
163	C₂×C₁₆	—	2	4	no	—
164	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
165	#107	—	2	4	no	—
166	#107	—	2	1	no	—
167	#107	—	2	2	no	—
168	#107	C₂³×C₅₃	1	4	yes	—
169	#107	C₂³×C₅₃	1	2	yes	—
170	#103	—	2	4	no	—
171	C₄×C₈	D₄×C₅₃	1	2	yes	—
172	C₄×C₈	C₂³×C₅₃	1	2	yes	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	8	4	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
183	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	2	2	no	—
187	Q₁₆	C₂×Q₈×C₅₃	1	4	yes	complemented
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
190	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
191	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
192	C₂×C₈	#41×C₅₃	1	2	yes	—
193	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄²	—	4	4	no	—
201	C₄²	D₈×C₅₃	1	2	yes	—
202	C₄²	C₂×D₄×C₅₃	1	2	yes	—
203	C₁₆	—	2	4	no	—
204	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	4	no	—
208	Q₈	—	8	8	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	SD₃₂×C₅₃	1	4	yes	—
217	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
218	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
219	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
220	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
221	C₂×C₄	#137×C₅₃	1	2	yes	—
222	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
225	C₈	C₂²×Q₈×C₅₃	1	1	yes	characteristic
226	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
227	C₂²	C₅₃×#474	1	2	yes	—
228	C₂²	C₂×SD₃₂×C₅₃	1	2	yes	—
229	C₂²	C₅₃×#510	1	2	yes	—
230	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	4	no	—
233	C₄	—	8	4	no	—
234	C₄	—	8	2	no	—
235	C₄	C₅₃×#474	1	2	yes	—
236	C₄	C₂×SD₃₂×C₅₃	1	4	yes	—
237	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
238	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
239	C₂	C₅₃×#2225	1	4	yes	direct product factor
240	C₂	C₂×C₅₃×#474	1	1	yes	characteristic
241	C₂	C₂²×SD₃₂×C₅₃	1	1	yes	characteristic
242	C₂	C₂×C₅₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2225	C₂	direct (N × H)
27. C₂×Q₁₆×C₅₃	Q₈	semidirect
65. Q₁₆×C₅₃	C₂×Q₈	semidirect
105. C₂×#2225	C₅₃	direct (N × H)
120. #2225	C₂×C₅₃	direct (N × H)
124. C₂×C₅₃	#2225	direct (N × H)
144. C₅₃	C₂×#2225	direct (N × H)
149. C₂×Q₁₆	Q₈×C₅₃	semidirect
187. Q₁₆	C₂×Q₈×C₅₃	semidirect
239. C₂	C₅₃×#2225	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×#500	2
C₂×C₄×Q₁₆×C₅₃	2
C₅₃×#2225	2
C₂×C₅₃×#367	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×C₅₃×#358	2
C₂×#2225	53

Subgroups — C2×C53×#2225 #27241135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2225 #27241135