Subgroups — C₄×#16810661 #27382142

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	2	yes	maximal
2	C₄×#13133176	C₂	1	2	yes	maximal
3	C₄×#13133151	C₂	1	4	yes	maximal, complemented
4	C₄×#13133149	C₂	1	4	yes	maximal, complemented
5	#20274432	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133176	C₂²	1	4	yes	—
8	#16810661	C₄	1	16	yes	direct product factor
9	C₂×#13133151	C₂²	1	4	yes	—
10	C₂×#13133149	C₂²	1	4	yes	—
11	C₄×C₄⋊Dic₅₃	C₂²	1	8	yes	—
12	C₄×C₂²⋊Dic₅₃	—	2	4	no	—
13	#16810469	—	2	2	no	—
14	C₄²×Dic₅₃	—	2	2	no	—
15	#16810386	C₂²	1	8	yes	—
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	#13133176	—	2	8	no	—
23	#13133176	C₂×C₄	1	16	yes	—
24	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
25	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
26	C₂×C₄×Dic₅₃	—	2	4	no	—
27	C₂×C₄×Dic₅₃	—	2	8	no	—
28	#13133151	C₂×C₄	1	16	yes	complemented
29	#13133149	C₂×C₄	1	16	yes	complemented
30	#13133124	—	2	8	no	—
31	C₂³×C₄×C₅₃	D₄	1	2	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	—	2	4	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	8	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₅₃	#41	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	8	no	—
59	C₂×C₈×C₅₃	—	2	8	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	C₄²	1	16	yes	complemented
62	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
70	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
71	C₂²⋊Dic₅₃	—	2	16	no	—
72	C₄⋊Dic₅₃	—	2	8	no	—
73	C₄⋊Dic₅₃	C₂²×C₄	1	16	yes	—
74	C₄×Dic₅₃	—	2	16	no	—
75	C₄×Dic₅₃	—	4	16	no	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₅₃	C₄×D₄	1	4	yes	—
79	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	16	no	—
82	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₅₃	#117	1	4	yes	—
87	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₅₃	C₄×Q₈	1	8	yes	—
89	C₂×C₄×C₅₃	#135	1	8	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
94	C₂×Dic₅₃	—	2	16	no	—
95	C₂×Dic₅₃	—	4	16	no	—
96	C₈×C₅₃	—	2	4	no	—
97	C₈×C₅₃	C₂×C₄²	1	4	yes	—
98	C₈×C₅₃	C₂×C₄²	1	4	yes	—
99	C₄×#427	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	4	no	—
101	C₂²×C₅₃	—	2	8	no	—
102	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
103	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×C₄×D₄	1	2	yes	—
105	C₂²×C₅₃	C₂×#135	1	4	yes	—
106	C₄×C₅₃	—	2	4	no	—
107	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
108	C₄×C₅₃	#427	1	8	yes	complemented
109	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
111	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
112	Dic₅₃	—	4	16	no	—
113	C₂×#427	—	53	2	no	—
114	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
115	C₄×#117	—	53	2	no	—
116	C₄×#107	—	53	4	no	—
117	C₄×#106	—	53	4	no	—
118	#1735	—	53	2	no	—
119	C₂×C₅₃	—	2	4	no	—
120	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₅₃	C₂×#427	1	2	yes	—
122	C₂×C₅₃	C₄×#135	1	4	yes	—
123	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#117	—	53	4	no	—
126	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
127	#427	—	53	16	no	—
128	C₂×#107	—	53	4	no	—
129	C₂×#106	—	53	4	no	—
130	C₂×C₄×C₈	Dic₅₃	1	8	yes	complemented
131	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
133	C₄×C₄⋊C₄	—	53	8	no	—
134	C₄×C₂²⋊C₄	—	106	4	no	—
135	#376	—	106	2	no	—
136	C₄³	—	106	2	no	—
137	#340	—	53	8	no	—
138	C₅₃	C₄×#427	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	D₂₁₂	1	2	yes	—
140	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₄×D₅₃	1	8	yes	—
143	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
144	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
145	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
146	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
148	#117	—	53	16	no	—
149	#117	—	106	8	no	—
150	C₂×C₄⋊C₄	—	53	8	no	—
151	C₂×C₂²⋊C₄	—	106	4	no	—
152	C₂×C₄²	—	106	4	no	—
153	C₂×C₄²	—	106	8	no	—
154	C₂×C₄²	Dic₁₀₆	1	4	yes	—
155	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
156	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
158	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
159	#107	—	53	16	no	—
160	#106	—	53	16	no	—
161	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
162	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
163	#95	—	106	8	no	—
164	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	106	8	no	—
167	C₂²×C₄	C₄⋊Dic₅₃	1	8	yes	—
168	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
170	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
171	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
173	C₂²×C₄	#11719945	1	4	yes	—
174	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
175	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄×Dic₅₃	1	16	yes	complemented
179	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
180	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
181	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
182	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
183	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
184	C₄⋊C₄	—	53	16	no	—
185	C₄⋊C₄	—	106	8	no	—
186	C₂²⋊C₄	—	106	16	no	—
187	C₄²	—	106	16	no	—
188	C₄²	—	212	16	no	—
189	C₄²	C₄⋊Dic₅₃	1	4	yes	—
190	C₄²	C₄⋊Dic₅₃	1	4	yes	—
191	C₄²	C₂×Dic₁₀₆	1	4	yes	—
192	C₄²	C₂²×Dic₅₃	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
195	C₂³	C₄×D₂₁₂	1	4	yes	—
196	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	106	16	no	—
200	C₂×C₄	—	212	16	no	—
201	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
205	C₂×C₄	#13133176	1	4	yes	—
206	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
207	C₂×C₄	C₄×Dic₁₀₆	1	8	yes	—
208	C₂×C₄	#13133239	1	8	yes	—
209	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
210	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
212	C₂×C₄	C₂×#11719945	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
215	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	4	no	—
218	C₂²	C₄×C₄⋊Dic₅₃	1	4	yes	—
219	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
220	C₂²	C₂×C₄×D₂₁₂	1	2	yes	—
221	C₂²	C₂×#13133239	1	4	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	212	16	no	—
224	C₄	C₄×C₄⋊Dic₅₃	1	4	yes	—
225	C₄	#16810661	1	8	yes	direct product factor
226	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
227	C₄	C₂×#13133176	1	1	yes	characteristic
228	C₄	C₂×C₄×Dic₁₀₆	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
231	C₂	C₂×#16810661	1	2	yes	—
232	C₂	C₄×#13133239	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₄×#427
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#13133151	C₂	semidirect
4. C₄×#13133149	C₂	semidirect
8. #16810661	C₄	direct (N × H)
19. C₂×C₄×C₈×C₅₃	C₄	semidirect
28. #13133151	C₂×C₄	semidirect
29. #13133149	C₂×C₄	semidirect
61. C₂×C₈×C₅₃	C₄²	semidirect
108. C₄×C₅₃	#427	semidirect
130. C₂×C₄×C₈	Dic₅₃	semidirect
138. C₅₃	C₄×#427	semidirect
178. C₂×C₈	C₄×Dic₅₃	semidirect
225. C₄	#16810661	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×#16810661	2
C₄×#13133176	2
C₄×#13133151	2
C₄×#13133149	2
#20274432	2
C₂²×C₄×C₈×C₅₃	2
C₄×#427	53

Subgroups — C4×#16810661 #27382142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810661 #27382142