Subgroups — C₄×C₅₃×#362 #27220157

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#362	C₂	1	2	yes	maximal
3	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1387	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
8	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₅₃	—	2	1	no	—
10	C₄×Q₁₆×C₅₃	—	2	1	no	—
11	C₄×Q₁₆×C₅₃	C₄	1	2	yes	—
12	C₄×Q₁₆×C₅₃	C₄	1	2	yes	—
13	C₂×#108×C₅₃	C₂²	1	2	yes	—
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#103×C₅₃	—	2	1	no	—
16	C₄×M₁₆×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#362	C₄	1	8	yes	direct product factor
19	C₅₃×#346	—	2	1	no	—
20	C₄×C₁₆×C₅₃	—	2	1	no	—
21	C₅₃×#341	—	2	1	no	—
22	C₂²×Q₈×C₅₃	—	2	1	no	—
23	C₂×Q₁₆×C₅₃	—	2	1	no	—
24	C₂×Q₁₆×C₅₃	—	2	2	no	—
25	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₄×Q₈×C₅₃	—	4	1	no	—
33	C₄×Q₈×C₅₃	—	4	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
36	C₂×C₄²×C₅₃	—	4	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	M₃₂×C₅₃	—	2	2	no	—
39	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	#108×C₅₃	—	2	2	no	—
42	#108×C₅₃	C₂×C₄	1	4	yes	—
43	#107×C₅₃	—	2	1	no	—
44	#107×C₅₃	C₂×C₄	1	2	yes	—
45	#103×C₅₃	—	2	2	no	—
46	#103×C₅₃	—	2	2	no	—
47	#97×C₅₃	—	4	1	no	—
48	C₄×C₈×C₅₃	—	4	1	no	—
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂×Q₈×C₅₃	—	2	2	no	—
54	C₂×Q₈×C₅₃	—	2	4	no	—
55	C₂×Q₈×C₅₃	—	4	1	no	—
56	C₂×Q₈×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₅₃	—	2	2	no	—
63	Q₁₆×C₅₃	—	2	4	no	—
64	Q₁₆×C₅₃	C₄²	1	4	yes	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	2	yes	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₄⋊C₄×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₅₃	D₈	1	1	yes	characteristic
84	C₄²×C₅₃	D₈	1	1	yes	characteristic
85	C₁₆×C₅₃	—	2	4	no	—
86	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	Q₈×C₅₃	—	2	4	no	—
88	Q₈×C₅₃	—	4	2	no	—
89	Q₈×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	8	2	no	—
96	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#135	1	2	yes	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	4	4	no	—
107	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
108	C₈×C₅₃	#117	1	1	yes	characteristic
109	C₈×C₅₃	C₄×D₄	1	2	yes	—
110	C₄×#362	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
114	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	4	4	no	—
119	C₄×C₅₃	—	8	2	no	—
120	C₄×C₅₃	#362	1	2	yes	direct product factor
121	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₅₃	#416	1	1	yes	characteristic
123	C₄×C₅₃	C₄×D₈	1	2	yes	—
124	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
125	C₂×#362	C₂×C₅₃	1	2	yes	—
126	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
127	C₄×#108	C₂×C₅₃	1	1	yes	characteristic
128	#1387	C₂×C₅₃	1	2	yes	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
132	C₂×C₅₃	#2148	1	1	yes	characteristic
133	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×C₄×Q₈	—	2	1	no	—
135	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
136	C₄×Q₁₆	—	2	1	no	—
137	C₄×Q₁₆	—	2	1	no	—
138	C₄×Q₁₆	C₄×C₅₃	1	2	yes	—
139	C₄×Q₁₆	C₄×C₅₃	1	2	yes	—
140	C₂×#108	C₂²×C₅₃	1	2	yes	—
141	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×#103	—	2	1	no	—
143	C₄×M₁₆	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
145	#362	C₄×C₅₃	1	8	yes	direct product factor
146	#346	—	2	1	no	—
147	C₄×C₁₆	—	2	1	no	—
148	#341	—	2	1	no	—
149	C₅₃	C₄×#362	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×Q₈	—	2	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
154	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	4	1	no	—
161	C₄×Q₈	—	4	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	4	1	no	—
164	C₂×C₄²	—	4	1	no	—
165	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
166	M₃₂	—	2	2	no	—
167	M₃₂	C₂×C₄×C₅₃	1	4	yes	—
168	C₂×C₁₆	—	2	2	no	—
169	#108	—	2	2	no	—
170	#108	C₂×C₄×C₅₃	1	4	yes	—
171	#107	—	2	1	no	—
172	#107	C₂×C₄×C₅₃	1	2	yes	—
173	#103	—	2	2	no	—
174	#103	—	2	2	no	—
175	#97	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
180	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	1	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
190	Q₁₆	—	2	2	no	—
191	Q₁₆	—	2	4	no	—
192	Q₁₆	C₄²×C₅₃	1	4	yes	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂×C₈	#41×C₅₃	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄²	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
211	C₄²	D₈×C₅₃	1	1	yes	characteristic
212	C₄²	D₈×C₅₃	1	1	yes	characteristic
213	C₁₆	—	2	4	no	—
214	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
225	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
226	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
227	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
229	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
230	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
231	C₂×C₄	#135×C₅₃	1	2	yes	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	4	4	no	—
235	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
236	C₈	#117×C₅₃	1	1	yes	characteristic
237	C₈	C₄×D₄×C₅₃	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
241	C₂²	C₄×SD₁₆×C₅₃	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	4	no	—
246	C₄	—	8	2	no	—
247	C₄	C₅₃×#362	1	2	yes	direct product factor
248	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
249	C₄	C₅₃×#416	1	1	yes	characteristic
250	C₄	C₄×D₈×C₅₃	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×#102×C₅₃	1	1	yes	characteristic
253	C₂	C₂×C₅₃×#362	1	1	yes	characteristic
254	C₂	C₅₃×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#102
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
18. C₅₃×#362	C₄	direct (N × H)
110. C₄×#362	C₅₃	direct (N × H)
120. C₄×C₅₃	#362	direct (N × H)
145. #362	C₄×C₅₃	direct (N × H)
149. C₅₃	C₄×#362	direct (N × H)
247. C₄	C₅₃×#362	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×C₄×Q₁₆×C₅₃	2
C₂×C₅₃×#362	2
C₄×M₃₂×C₅₃	2
C₄×#108×C₅₃	2
C₅₃×#1387	2
C₄×#362	53

Subgroups — C4×C53×#362 #27220157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#362 #27220157