Subgroups — C₄²×#11719945 #27542293

Order 13568.

4086 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×#11719945	C₂	1	3	yes	maximal
2	C₄²×Dic₁₀₆	C₂	1	4	yes	maximal, complemented
3	C₄²×D₂₁₂	C₂	1	4	yes	maximal, complemented
4	C₄²×C₂²⋊D₅₃	C₂	1	8	yes	maximal, complemented
5	C₄×#13133174	C₂	1	6	yes	maximal, complemented
6	C₄×#13133176	C₂	1	3	yes	maximal, complemented
7	C₄³×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719945	C₂²	1	1	yes	characteristic
10	C₄×#11719945	C₄	1	48	yes	direct product factor
11	C₂×C₄×Dic₁₀₆	C₂²	1	6	yes	—
12	C₂×C₄×D₂₁₂	C₂²	1	6	yes	—
13	C₂×C₄×C₂²⋊D₅₃	C₂²	1	12	yes	—
14	C₂×#13133176	C₂²	1	3	yes	—
15	C₂×#13133174	C₂²	1	6	yes	—
16	C₂×C₄²×D₅₃	C₂²	1	8	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	6	yes	—
18	C₄×#11719919	C₂²	1	24	yes	—
19	C₄×C₄⋊Dic₅₃	C₂²	1	12	yes	—
20	C₄×#11719921	C₂²	1	24	yes	—
21	C₄×C₂²⋊Dic₅₃	C₂²	1	12	yes	—
22	#16810469	C₂²	1	6	yes	—
23	C₄²×Dic₅₃	C₂²	1	6	yes	—
24	C₄²×Dic₅₃	C₂²	1	8	yes	—
25	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
27	C₄³×C₅₃	C₂²	1	3	yes	—
28	C₄³×C₅₃	C₂²	1	4	yes	—
29	C₄³×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#11719945	C₂×C₄	1	24	yes	—
31	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
32	C₂²×D₂₁₂	C₂³	1	1	yes	characteristic
33	C₂²×C₂²⋊D₅₃	C₂³	1	2	yes	—
34	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
35	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
36	C₄×Dic₁₀₆	C₂×C₄	1	48	yes	complemented
37	C₄×D₂₁₂	C₂×C₄	1	48	yes	complemented
38	C₄×C₂²⋊D₅₃	C₂×C₄	1	96	yes	complemented
39	#13133176	C₂×C₄	1	24	yes	complemented
40	#13133174	C₂×C₄	1	48	yes	complemented
41	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
42	C₂×#11719919	C₂³	1	12	yes	—
43	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
44	C₂×#11719921	C₂³	1	12	yes	—
45	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
46	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
47	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
48	C₄²×D₅₃	—	2	32	no	—
49	C₄²×D₅₃	C₂×C₄	1	48	yes	complemented
50	#13133124	C₂³	1	24	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	—	2	16	no	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	complemented
55	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
60	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂³×D₅₃	C₂⁴	1	2	yes	—
62	C₂²×C₄×C₅₃	—	2	24	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
69	C₂×C₂²⋊D₅₃	C₂²×C₄	1	24	yes	—
70	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
71	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
72	#11719945	C₄²	1	64	yes	direct product factor
73	C₂×D₂₁₂	C₂²×C₄	1	12	yes	—
74	C₂×C₄×D₅₃	—	2	48	no	—
75	C₂×C₄×D₅₃	C₂²×C₄	1	48	yes	—
76	C₂×C₄×D₅₃	C₂²×C₄	1	24	yes	—
77	C₂×Dic₁₀₆	C₂²×C₄	1	12	yes	—
78	C₄²×C₅₃	—	2	24	no	—
79	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	24	yes	—
82	C₄²×C₅₃	#41	1	16	yes	complemented
83	C₂²⋊Dic₅₃	C₂²×C₄	1	24	yes	—
84	#11719921	C₂²×C₄	1	48	yes	—
85	C₄⋊Dic₅₃	C₂²×C₄	1	24	yes	—
86	#11719919	C₂²×C₄	1	48	yes	—
87	C₄×Dic₅₃	—	2	48	no	—
88	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
89	C₄×Dic₅₃	C₂²×C₄	1	48	yes	—
90	C₂³×C₅₃	—	2	4	no	—
91	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
92	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
93	C₂²×D₅₃	—	2	8	no	—
94	C₂²×D₅₃	C₂³×C₄	1	12	yes	—
95	C₂×C₄×C₅₃	—	2	12	no	—
96	C₂×C₄×C₅₃	—	2	48	no	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	complemented
98	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
100	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
102	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
103	C₂²⋊D₅₃	C₂×C₄²	1	32	yes	complemented
104	C₂×Dic₅₃	—	2	24	no	—
105	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
106	C₂×Dic₅₃	C₂³×C₄	1	12	yes	—
107	D₂₁₂	C₂×C₄²	1	16	yes	complemented
108	C₄×D₅₃	—	2	96	no	—
109	C₄×D₅₃	C₂×C₄²	1	32	yes	complemented
110	Dic₁₀₆	C₂×C₄²	1	16	yes	complemented
111	C₄²×#41	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	12	no	—
113	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
114	C₂²×C₅₃	C₂⁴×C₄	1	3	yes	—
115	C₂²×C₅₃	C₂²×#41	1	4	yes	—
116	C₂×D₅₃	—	2	24	no	—
117	C₂×D₅₃	C₂²×C₄²	1	8	yes	—
118	C₄×C₅₃	—	2	12	no	—
119	C₄×C₅₃	C₂²×C₄²	1	4	yes	—
120	C₄×C₅₃	C₂²×C₄²	1	4	yes	—
121	C₄×C₅₃	C₄×#41	1	24	yes	complemented
122	Dic₅₃	—	2	24	no	—
123	Dic₅₃	C₂²×C₄²	1	8	yes	—
124	C₂×C₄×#41	—	53	3	no	—
125	C₄²×Q₈	—	53	4	no	—
126	C₄²×D₄	—	53	8	no	—
127	C₄²×D₄	—	53	4	no	—
128	C₄×#117	—	53	3	no	—
129	C₄×#117	—	53	6	no	—
130	C₂×C₄³	—	53	2	no	—
131	C₂×C₄³	D₅₃	1	1	yes	characteristic, complemented
132	C₂×C₅₃	—	2	4	no	—
133	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
134	C₂×C₅₃	C₂×C₄×#41	1	6	yes	—
135	D₅₃	—	2	8	no	—
136	C₂²×#41	—	53	1	no	—
137	C₄×#41	—	53	48	no	—
138	C₂×C₄×Q₈	—	53	6	no	—
139	C₂×C₄×D₄	—	53	12	no	—
140	C₂×C₄×D₄	—	53	6	no	—
141	C₂×#117	—	53	3	no	—
142	C₂×#117	—	53	6	no	—
143	C₂²×C₄²	—	53	6	no	—
144	C₂²×C₄²	—	53	8	no	—
145	C₂²×C₄²	C₂×D₅₃	1	4	yes	—
146	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
147	C₄×C₄⋊C₄	—	53	12	no	—
148	C₄×C₄⋊C₄	—	53	24	no	—
149	C₄×C₂²⋊C₄	—	53	12	no	—
150	C₄×C₂²⋊C₄	—	53	24	no	—
151	#376	—	53	6	no	—
152	C₄³	—	53	6	no	—
153	C₄³	—	53	8	no	—
154	C₄³	C₂×D₅₃	1	3	yes	—
155	C₄³	C₂×D₅₃	1	4	yes	—
156	C₄³	C₂×D₅₃	1	1	yes	characteristic
157	C₅₃	C₄²×#41	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	53	24	no	—
159	C₂²×Q₈	—	53	1	no	—
160	C₂²×D₄	—	53	2	no	—
161	C₂²×D₄	—	53	1	no	—
162	C₂³×C₄	—	53	2	no	—
163	C₂³×C₄	—	53	12	no	—
164	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
165	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
166	C₄×Q₈	—	53	48	no	—
167	C₄×D₄	—	53	96	no	—
168	C₄×D₄	—	53	48	no	—
169	#117	—	53	24	no	—
170	#117	—	53	48	no	—
171	C₂×C₄⋊C₄	—	53	12	no	—
172	C₂×C₄⋊C₄	—	53	6	no	—
173	C₂×C₂²⋊C₄	—	53	6	no	—
174	C₂×C₂²⋊C₄	—	53	12	no	—
175	C₂×C₄²	—	2	16	no	—
176	C₂×C₄²	—	53	6	no	—
177	C₂×C₄²	—	53	12	no	—
178	C₂×C₄²	—	53	48	no	—
179	C₂×C₄²	—	53	24	no	—
180	C₂×C₄²	—	106	32	no	—
181	C₂×C₄²	C₄×D₅₃	1	24	yes	complemented
182	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
183	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
184	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
185	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
186	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
187	#95	—	53	24	no	—
188	C₂⁴	—	53	2	no	—
189	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
190	#41	—	53	64	no	—
191	C₂×Q₈	—	53	12	no	—
192	C₂×D₄	—	53	24	no	—
193	C₂×D₄	—	53	12	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	—	53	2	no	—
196	C₂²×C₄	—	53	24	no	—
197	C₂²×C₄	—	53	12	no	—
198	C₂²×C₄	—	53	48	no	—
199	C₂²×C₄	—	106	48	no	—
200	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
201	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
202	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
203	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
206	C₄⋊C₄	—	53	48	no	—
207	C₄⋊C₄	—	53	24	no	—
208	C₂²⋊C₄	—	53	24	no	—
209	C₂²⋊C₄	—	53	48	no	—
210	C₄²	—	2	24	no	—
211	C₄²	—	53	24	no	—
212	C₄²	—	53	48	no	—
213	C₄²	—	106	48	no	—
214	C₄²	C₂×C₄×D₅₃	1	12	yes	—
215	C₄²	C₂×C₄×D₅₃	1	12	yes	—
216	C₄²	C₂×C₄×D₅₃	1	24	yes	—
217	C₄²	#11719945	1	16	yes	direct product factor
218	C₂³	—	2	4	no	—
219	C₂³	—	53	12	no	—
220	C₂³	—	106	8	no	—
221	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
222	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
223	Q₈	—	53	16	no	—
224	D₄	—	53	32	no	—
225	D₄	—	53	16	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	48	no	—
228	C₂×C₄	—	53	12	no	—
229	C₂×C₄	—	53	32	no	—
230	C₂×C₄	—	53	24	no	—
231	C₂×C₄	—	106	24	no	—
232	C₂×C₄	—	106	96	no	—
233	C₂×C₄	C₄²×D₅₃	1	16	yes	complemented
234	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
235	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
236	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
237	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
238	C₂×C₄	C₂×#11719945	1	24	yes	—
239	C₂²	—	2	12	no	—
240	C₂²	—	53	8	no	—
241	C₂²	—	106	24	no	—
242	C₂²	C₂×C₄²×D₅₃	1	4	yes	—
243	C₂²	C₂³×C₄×D₅₃	1	3	yes	—
244	C₂²	C₂²×#11719945	1	4	yes	—
245	C₄	—	2	12	no	—
246	C₄	—	53	8	no	—
247	C₄	—	106	24	no	—
248	C₄	C₂×C₄²×D₅₃	1	4	yes	—
249	C₄	C₂×C₄²×D₅₃	1	4	yes	—
250	C₄	C₄×#11719945	1	24	yes	direct product factor
251	C₂	—	2	4	no	—
252	C₂	—	106	8	no	—
253	C₂	C₂²×C₄²×D₅₃	1	1	yes	characteristic
254	C₂	C₂×C₄×#11719945	1	6	yes	—

	subgroup	G/H
center	C₄³	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×#11719945	C₂²
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₄²×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄³ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₄²×Dic₁₀₆	C₂	semidirect
3. C₄²×D₂₁₂	C₂	semidirect
4. C₄²×C₂²⋊D₅₃	C₂	semidirect
5. C₄×#13133174	C₂	semidirect
6. C₄×#13133176	C₂	semidirect
7. C₄³×D₅₃	C₂	semidirect
8. C₂×C₄³×C₅₃	C₂	semidirect
10. C₄×#11719945	C₄	direct (N × H)
36. C₄×Dic₁₀₆	C₂×C₄	semidirect
37. C₄×D₂₁₂	C₂×C₄	semidirect
38. C₄×C₂²⋊D₅₃	C₂×C₄	semidirect
39. #13133176	C₂×C₄	semidirect
40. #13133174	C₂×C₄	semidirect
49. C₄²×D₅₃	C₂×C₄	semidirect
54. C₂×C₄²×C₅₃	C₂×C₄	semidirect
72. #11719945	C₄²	direct (N × H)
82. C₄²×C₅₃	#41	semidirect
97. C₂×C₄×C₅₃	C₂×C₄²	semidirect
103. C₂²⋊D₅₃	C₂×C₄²	semidirect
107. D₂₁₂	C₂×C₄²	semidirect
109. C₄×D₅₃	C₂×C₄²	semidirect
110. Dic₁₀₆	C₂×C₄²	semidirect
121. C₄×C₅₃	C₄×#41	semidirect
131. C₂×C₄³	D₅₃	semidirect
157. C₅₃	C₄²×#41	semidirect
181. C₂×C₄²	C₄×D₅₃	semidirect
217. C₄²	#11719945	direct (N × H)
233. C₂×C₄	C₄²×D₅₃	semidirect
250. C₄	C₄×#11719945	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×#11719945	2
C₄²×Dic₁₀₆	2
C₄²×D₂₁₂	2
C₄²×C₂²⋊D₅₃	2
C₄×#13133174	2
C₄×#13133176	2
C₄³×D₅₃	2
C₂×C₄³×C₅₃	2
C₄²×#41	53

Subgroups — C42×#11719945 #27542293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#11719945 #27542293