Subgroups — C₄²×#11719949 #27542292

Order 13568.

4086 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×C₄×#11719949	C₂	1	3	yes	maximal
2	C₄×Q₈×Dic₅₃	C₂	1	3	yes	maximal, complemented
3	C₄²×D₂₁₂	C₂	1	3	yes	maximal, complemented
4	C₄×#13133179	C₂	1	9	yes	maximal, complemented
5	C₄×#13133175	C₂	1	9	yes	maximal, complemented
6	C₄³×D₅₃	C₂	1	3	yes	maximal, complemented
7	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719949	C₂²	1	1	yes	characteristic
9	C₄×#11719949	C₄	1	48	yes	direct product factor
10	C₂×Q₈×Dic₅₃	C₂²	1	3	yes	—
11	C₂×C₄×D₂₁₂	C₂²	1	9	yes	—
12	C₂×#13133179	C₂²	1	9	yes	—
13	C₂×#13133175	C₂²	1	9	yes	—
14	C₂×C₄²×D₅₃	C₂²	1	9	yes	—
15	C₂×C₄²×D₅₃	C₂²	1	3	yes	—
16	C₂×C₄²×D₅₃	C₂²	1	9	yes	—
17	C₄⋊C₄×Dic₅₃	C₂²	1	18	yes	—
18	C₄×C₄⋊Dic₅₃	C₂²	1	9	yes	—
19	C₄×#11719921	C₂²	1	18	yes	—
20	#16810470	C₂²	1	18	yes	—
21	#16810468	C₂²	1	6	yes	—
22	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄²×Dic₅₃	C₂²	1	9	yes	—
24	C₄²×Dic₅₃	C₂²	1	9	yes	—
25	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	9	yes	—
27	C₄³×C₅₃	C₂²	1	3	yes	—
28	C₂×#11719949	C₂×C₄	1	24	yes	—
29	C₂²×D₂₁₂	C₂³	1	3	yes	—
30	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
31	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
32	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
33	Q₈×Dic₅₃	C₂×C₄	1	24	yes	complemented
34	C₄×D₂₁₂	C₂×C₄	1	72	yes	complemented
35	#13133179	C₂×C₄	1	72	yes	complemented
36	#13133175	C₂×C₄	1	72	yes	complemented
37	C₂×C₄⋊Dic₅₃	C₂³	1	9	yes	—
38	C₂×#11719921	C₂³	1	18	yes	—
39	C₂×C₄×Dic₅₃	C₂³	1	18	yes	—
40	C₂×C₄×Dic₅₃	C₂³	1	9	yes	—
41	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
42	C₂×C₄×Dic₅₃	C₂³	1	9	yes	—
43	C₂×C₄×Dic₅₃	C₂³	1	9	yes	—
44	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
45	C₄²×D₅₃	—	2	12	no	—
46	C₄²×D₅₃	—	2	36	no	—
47	C₄²×D₅₃	C₂×C₄	1	72	yes	complemented
48	#13133124	C₂³	1	18	yes	—
49	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₅₃	C₂×C₄	1	24	yes	complemented
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	9	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	9	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	9	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#95×C₅₃	C₂³	1	6	yes	—
56	C₂³×D₅₃	C₂⁴	1	3	yes	—
57	C₂×Q₈×C₅₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
61	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
62	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
64	#11719949	C₄²	1	64	yes	direct product factor
65	C₂×D₂₁₂	C₂²×C₄	1	36	yes	—
66	C₂×C₄×D₅₃	—	2	18	no	—
67	C₂×C₄×D₅₃	—	2	18	no	—
68	C₂×C₄×D₅₃	—	2	36	no	—
69	C₂×C₄×D₅₃	C₂²×C₄	1	36	yes	—
70	C₂×C₄×D₅₃	C₂²×C₄	1	36	yes	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	36	yes	—
72	C₄⋊C₄×C₅₃	C₂²×C₄	1	36	yes	—
73	C₄²×C₅₃	—	2	18	no	—
74	C₄²×C₅₃	C₂²×C₄	1	36	yes	—
75	C₄²×C₅₃	#41	1	4	yes	complemented
76	#11719921	C₂²×C₄	1	72	yes	—
77	C₄⋊Dic₅₃	C₂²×C₄	1	36	yes	—
78	C₄×Dic₅₃	—	2	18	no	—
79	C₄×Dic₅₃	—	2	36	no	—
80	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
81	C₄×Dic₅₃	C₂²×C₄	1	36	yes	—
82	C₄×Dic₅₃	C₂²×C₄	1	36	yes	—
83	C₄×Dic₅₃	#41	1	12	yes	complemented
84	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
85	C₂²×D₅₃	—	2	9	no	—
86	C₂²×D₅₃	—	2	3	no	—
87	C₂²×D₅₃	C₂³×C₄	1	18	yes	—
88	Q₈×C₅₃	C₂×C₄²	1	16	yes	complemented
89	C₂×C₄×C₅₃	—	2	18	no	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	18	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
92	C₂×C₄×C₅₃	C₂³×C₄	1	18	yes	—
93	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
94	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
95	C₂×Dic₅₃	—	2	9	no	—
96	C₂×Dic₅₃	—	2	9	no	—
97	C₂×Dic₅₃	C₂³×C₄	1	18	yes	—
98	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
99	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
100	C₂×Dic₅₃	C₂×#41	1	12	yes	—
101	D₂₁₂	C₂×C₄²	1	48	yes	complemented
102	C₄×D₅₃	—	2	72	no	—
103	C₄×D₅₃	—	2	72	no	—
104	C₄×D₅₃	C₂×C₄²	1	48	yes	complemented
105	C₄²×#41	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	C₂⁴×C₄	1	3	yes	—
107	C₂²×C₅₃	C₂²×#41	1	3	yes	—
108	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
109	C₂×D₅₃	—	2	18	no	—
110	C₂×D₅₃	—	2	18	no	—
111	C₂×D₅₃	C₂²×C₄²	1	12	yes	—
112	C₄×C₅₃	—	2	18	no	—
113	C₄×C₅₃	C₂²×C₄²	1	12	yes	—
114	C₄×C₅₃	C₄×#41	1	12	yes	complemented
115	Dic₅₃	—	2	18	no	—
116	Dic₅₃	C₂²×C₄²	1	4	yes	—
117	Dic₅₃	C₄×#41	1	12	yes	complemented
118	C₂×C₄×#41	—	53	3	no	—
119	C₄²×Q₈	—	53	3	no	—
120	C₄²×Q₈	D₅₃	1	1	yes	characteristic, complemented
121	C₄²×D₄	—	53	9	no	—
122	C₄²×D₄	—	53	3	no	—
123	C₄×#117	—	53	9	no	—
124	C₂×C₄³	—	53	3	no	—
125	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
126	C₂×C₅₃	C₂×C₄×#41	1	3	yes	—
127	C₂×C₅₃	C₂×C₄×#41	1	3	yes	—
128	D₅₃	—	2	12	no	—
129	C₂²×#41	—	53	1	no	—
130	C₄×#41	—	53	48	no	—
131	C₂×C₄×Q₈	—	53	3	no	—
132	C₂×C₄×Q₈	C₂×D₅₃	1	3	yes	—
133	C₂×C₄×D₄	—	53	9	no	—
134	C₂×C₄×D₄	—	53	9	no	—
135	C₂×#117	—	53	9	no	—
136	C₂²×C₄²	—	53	9	no	—
137	C₂²×C₄²	—	53	9	no	—
138	C₂²×C₄²	—	53	3	no	—
139	C₄×C₄⋊C₄	—	53	18	no	—
140	C₄×C₄⋊C₄	—	53	9	no	—
141	C₄×C₄⋊C₄	C₂×D₅₃	1	9	yes	—
142	C₄×C₂²⋊C₄	—	53	18	no	—
143	C₄×C₂²⋊C₄	—	53	18	no	—
144	#376	—	53	6	no	—
145	C₄³	—	53	9	no	—
146	C₄³	—	53	9	no	—
147	C₄³	—	53	1	no	—
148	C₄³	C₂×D₅₃	1	3	yes	—
149	C₅₃	C₄²×#41	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	53	24	no	—
151	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
152	C₂²×D₄	—	53	3	no	—
153	C₂³×C₄	—	53	3	no	—
154	C₂³×C₄	—	53	9	no	—
155	C₂³×C₄	—	53	9	no	—
156	C₄×Q₈	—	53	24	no	—
157	C₄×Q₈	C₄×D₅₃	1	24	yes	complemented
158	C₄×D₄	—	53	72	no	—
159	C₄×D₄	—	53	72	no	—
160	#117	—	53	72	no	—
161	C₂×C₄⋊C₄	—	53	9	no	—
162	C₂×C₄⋊C₄	C₂²×D₅₃	1	9	yes	—
163	C₂×C₂²⋊C₄	—	53	18	no	—
164	C₂×C₄²	—	53	9	no	—
165	C₂×C₄²	—	53	9	no	—
166	C₂×C₄²	—	53	3	no	—
167	C₂×C₄²	—	53	3	no	—
168	C₂×C₄²	—	53	72	no	—
169	C₂×C₄²	—	53	18	no	—
170	C₂×C₄²	—	53	9	no	—
171	C₂×C₄²	—	106	36	no	—
172	C₂×C₄²	—	106	12	no	—
173	C₂×C₄²	C₂²×D₅₃	1	9	yes	—
174	C₂×C₄²	C₂²×D₅₃	1	9	yes	—
175	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	#95	—	53	18	no	—
177	#95	C₂²×D₅₃	1	6	yes	—
178	C₂⁴	—	53	3	no	—
179	#41	—	53	64	no	—
180	C₂×Q₈	C₂×C₄×D₅₃	1	12	yes	—
181	C₂×D₄	—	53	36	no	—
182	C₂²×C₄	—	53	1	no	—
183	C₂²×C₄	—	53	3	no	—
184	C₂²×C₄	—	53	36	no	—
185	C₂²×C₄	—	53	9	no	—
186	C₂²×C₄	—	53	36	no	—
187	C₂²×C₄	—	53	36	no	—
188	C₂²×C₄	—	106	36	no	—
189	C₂²×C₄	—	106	18	no	—
190	C₂²×C₄	—	106	18	no	—
191	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
192	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
193	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
194	C₄⋊C₄	—	53	36	no	—
195	C₄⋊C₄	C₂×C₄×D₅₃	1	36	yes	—
196	C₂²⋊C₄	—	53	72	no	—
197	C₄²	—	2	18	no	—
198	C₄²	—	53	12	no	—
199	C₄²	—	53	36	no	—
200	C₄²	—	53	36	no	—
201	C₄²	—	53	12	no	—
202	C₄²	—	106	36	no	—
203	C₄²	—	106	18	no	—
204	C₄²	C₂×C₄×D₅₃	1	36	yes	—
205	C₄²	#11719949	1	4	yes	direct product factor
206	C₂³	—	53	18	no	—
207	C₂³	—	106	9	no	—
208	C₂³	—	106	3	no	—
209	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
210	Q₈	C₄²×D₅₃	1	16	yes	complemented
211	D₄	—	53	48	no	—
212	C₂×C₄	—	2	18	no	—
213	C₂×C₄	—	53	6	no	—
214	C₂×C₄	—	53	6	no	—
215	C₂×C₄	—	53	12	no	—
216	C₂×C₄	—	53	48	no	—
217	C₂×C₄	—	53	18	no	—
218	C₂×C₄	—	106	72	no	—
219	C₂×C₄	—	106	9	no	—
220	C₂×C₄	—	106	9	no	—
221	C₂×C₄	—	106	72	no	—
222	C₂×C₄	C₂²×C₄×D₅₃	1	18	yes	—
223	C₂×C₄	C₂²×C₄×D₅₃	1	18	yes	—
224	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
225	C₂×C₄	#41×D₅₃	1	6	yes	—
226	C₂×C₄	C₂×#11719949	1	6	yes	—
227	C₂²	—	53	12	no	—
228	C₂²	—	106	18	no	—
229	C₂²	—	106	18	no	—
230	C₂²	C₂³×C₄×D₅₃	1	3	yes	—
231	C₂²	C₂×#41×D₅₃	1	3	yes	—
232	C₂²	C₂²×#11719949	1	1	yes	characteristic
233	C₄	—	2	18	no	—
234	C₄	—	53	4	no	—
235	C₄	—	53	12	no	—
236	C₄	—	106	18	no	—
237	C₄	C₂×C₄²×D₅₃	1	12	yes	—
238	C₄	C₄×#11719949	1	12	yes	direct product factor
239	C₂	—	106	12	no	—
240	C₂	C₂²×C₄²×D₅₃	1	1	yes	characteristic
241	C₂	C₄×#41×D₅₃	1	3	yes	—
242	C₂	C₂×C₄×#11719949	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₄²×Q₈×C₅₃	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×#11719949	C₂²
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	C₄²×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×Dic₅₃	C₂	semidirect
3. C₄²×D₂₁₂	C₂	semidirect
4. C₄×#13133179	C₂	semidirect
5. C₄×#13133175	C₂	semidirect
6. C₄³×D₅₃	C₂	semidirect
7. C₄²×Q₈×C₅₃	C₂	semidirect
9. C₄×#11719949	C₄	direct (N × H)
33. Q₈×Dic₅₃	C₂×C₄	semidirect
34. C₄×D₂₁₂	C₂×C₄	semidirect
35. #13133179	C₂×C₄	semidirect
36. #13133175	C₂×C₄	semidirect
47. C₄²×D₅₃	C₂×C₄	semidirect
50. C₄×Q₈×C₅₃	C₂×C₄	semidirect
64. #11719949	C₄²	direct (N × H)
75. C₄²×C₅₃	#41	semidirect
83. C₄×Dic₅₃	#41	semidirect
88. Q₈×C₅₃	C₂×C₄²	semidirect
101. D₂₁₂	C₂×C₄²	semidirect
104. C₄×D₅₃	C₂×C₄²	semidirect
114. C₄×C₅₃	C₄×#41	semidirect
117. Dic₅₃	C₄×#41	semidirect
120. C₄²×Q₈	D₅₃	semidirect
149. C₅₃	C₄²×#41	semidirect
157. C₄×Q₈	C₄×D₅₃	semidirect
205. C₄²	#11719949	direct (N × H)
210. Q₈	C₄²×D₅₃	semidirect
238. C₄	C₄×#11719949	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×#11719949	2
C₄×Q₈×Dic₅₃	2
C₄²×D₂₁₂	2
C₄×#13133179	2
C₄×#13133175	2
C₄³×D₅₃	2
C₄²×Q₈×C₅₃	2
C₄²×#41	53

Subgroups — C42×#11719949 #27542292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#11719949 #27542292