Subgroups — C₂×#20274625 #27383196

Order 13568.

6110 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊#88322	C₂	1	6	yes	maximal, complemented
2	#20274625	C₂	1	8	yes	maximal, direct product factor
3	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₂²	1	4	yes	complemented
5	C₂×C₂²⋊#88322	—	2	24	no	—
6	C₂×C₂²⋊#88322	C₂²	1	24	yes	complemented
7	C₂⁶×C₅₃	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₅₃	—	2	24	no	—
9	C₂⁴×C₄×C₅₃	C₄	1	3	yes	—
10	C₂⁴×C₄×C₅₃	C₄	1	4	yes	—
11	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	—
12	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
13	C₂²×#88322	—	2	24	no	—
14	C₂²×#88322	C₂³	1	8	yes	complemented
15	C₂²⋊#88322	—	2	48	no	—
16	C₂⁵×C₅₃	—	2	12	no	—
17	C₂⁵×C₅₃	—	2	12	no	—
18	C₂⁵×C₅₃	C₈	1	8	yes	complemented
19	C₂⁵×C₅₃	C₂×C₄	1	4	yes	—
20	C₂⁵×C₅₃	C₂×C₄	1	3	yes	—
21	C₂³×C₄×C₅₃	—	2	128	no	—
22	C₂³×C₄×C₅₃	—	2	12	no	—
23	C₂³×C₄×C₅₃	—	2	24	no	—
24	C₂³×C₄×C₅₃	—	2	48	no	—
25	C₂³×C₄×C₅₃	—	2	48	no	—
26	C₂³×C₄×C₅₃	—	2	24	no	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
29	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
30	C₂³×C₄×C₅₃	D₄	1	24	yes	—
31	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂⁴×C₅₃	—	2	32	no	—
34	C₂⁴×C₅₃	—	2	96	no	—
35	C₂⁴×C₅₃	—	2	96	no	—
36	C₂⁴×C₅₃	—	2	24	no	—
37	C₂⁴×C₅₃	—	2	12	no	—
38	C₂⁴×C₅₃	—	2	24	no	—
39	C₂⁴×C₅₃	—	2	12	no	—
40	C₂⁴×C₅₃	C₂²⋊C₄	1	12	yes	—
41	C₂⁴×C₅₃	C₂×C₈	1	12	yes	—
42	C₂⁴×C₅₃	C₂×C₈	1	16	yes	complemented
43	C₂⁴×C₅₃	M₁₆	1	12	yes	—
44	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
45	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	96	no	—
47	C₂²×C₄×C₅₃	—	2	128	no	—
48	C₂²×C₄×C₅₃	—	2	128	no	—
49	C₂²×C₄×C₅₃	—	2	24	no	—
50	C₂²×C₄×C₅₃	—	2	12	no	—
51	C₂²×C₄×C₅₃	—	2	48	no	—
52	C₂²×C₄×C₅₃	—	2	24	no	—
53	C₂²×C₄×C₅₃	—	2	96	no	—
54	C₂²×C₄×C₅₃	—	2	24	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	24	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂×#88322	—	2	24	no	—
63	C₂×#88322	—	4	16	no	—
64	C₂³×C₅₃	—	2	32	no	—
65	C₂³×C₅₃	—	2	96	no	—
66	C₂³×C₅₃	—	2	256	no	—
67	C₂³×C₅₃	—	2	12	no	—
68	C₂³×C₅₃	—	2	12	no	—
69	C₂³×C₅₃	—	2	24	no	—
70	C₂³×C₅₃	—	2	48	no	—
71	C₂³×C₅₃	—	2	96	no	—
72	C₂³×C₅₃	—	2	24	no	—
73	C₂³×C₅₃	—	2	48	no	—
74	C₂³×C₅₃	C₂²⋊C₈	1	48	yes	complemented
75	C₂³×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
76	C₂³×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
77	C₂³×C₅₃	C₂²×C₈	1	12	yes	—
78	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
79	C₂³×C₅₃	C₂×M₁₆	1	12	yes	—
80	C₂³×C₅₃	C₂×M₁₆	1	6	yes	—
81	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	2	96	no	—
83	C₂×C₄×C₅₃	—	2	12	no	—
84	C₂×C₄×C₅₃	—	2	24	no	—
85	C₂×C₄×C₅₃	—	2	48	no	—
86	C₂×C₄×C₅₃	—	2	48	no	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂×C₄×C₅₃	#120	1	16	yes	—
90	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
91	#88322	—	4	8	no	—
92	C₂×#1779	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	128	no	—
94	C₂²×C₅₃	—	2	24	no	—
95	C₂²×C₅₃	—	2	12	no	—
96	C₂²×C₅₃	—	2	24	no	—
97	C₂²×C₅₃	—	2	12	no	—
98	C₂²×C₅₃	—	2	96	no	—
99	C₂²×C₅₃	#379	1	4	yes	—
100	C₂²×C₅₃	C₂×C₂²⋊C₈	1	12	yes	—
101	C₂²×C₅₃	C₂×C₂²⋊C₈	1	24	yes	complemented
102	C₂²×C₅₃	#407	1	12	yes	—
103	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	3	yes	—
104	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
105	C₂²×C₅₃	C₂²×M₁₆	1	3	yes	—
106	C₄×C₅₃	—	2	12	no	—
107	C₄×C₅₃	#379	1	4	yes	—
108	C₄×C₅₃	C₂×#120	1	4	yes	—
109	C₂⁵×C₄	D₅₃	1	1	yes	characteristic
110	C₂²×C₂²⋊C₈	—	53	6	no	—
111	#1779	—	53	8	no	—
112	C₂×C₅₃	—	2	24	no	—
113	C₂×C₅₃	#1779	1	8	yes	complemented
114	C₂×C₅₃	C₂×#379	1	1	yes	characteristic
115	C₂×C₅₃	C₂²×C₂²⋊C₈	1	3	yes	—
116	C₂×C₅₃	C₂×#407	1	3	yes	—
117	C₂⁶	Dic₅₃	1	1	yes	characteristic
118	C₂⁴×C₄	—	2	24	no	—
119	C₂⁴×C₄	Dic₅₃	1	3	yes	—
120	C₂⁴×C₄	Dic₅₃	1	4	yes	—
121	C₂⁴×C₄	C₂×D₅₃	1	4	yes	—
122	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
123	C₂³×C₈	—	53	4	no	—
124	C₂×C₂²⋊C₈	—	53	24	no	—
125	C₂×C₂²⋊C₈	—	106	24	no	—
126	C₅₃	C₂×#1779	1	1	yes	characteristic, complemented, Sylow
127	C₂⁵	—	2	12	no	—
128	C₂⁵	—	2	12	no	—
129	C₂⁵	#88322	1	8	yes	complemented
130	C₂⁵	C₂×Dic₅₃	1	4	yes	—
131	C₂⁵	C₂×Dic₅₃	1	3	yes	—
132	C₂³×C₄	—	2	128	no	—
133	C₂³×C₄	—	2	12	no	—
134	C₂³×C₄	—	2	48	no	—
135	C₂³×C₄	—	2	24	no	—
136	C₂³×C₄	—	2	24	no	—
137	C₂³×C₄	—	2	48	no	—
138	C₂³×C₄	C₂×Dic₅₃	1	12	yes	—
139	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
140	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
141	C₂³×C₄	C₂²⋊D₅₃	1	24	yes	—
142	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
143	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
144	C₂²×C₈	—	53	8	no	—
145	C₂²×C₈	—	106	24	no	—
146	C₂²⋊C₈	—	106	48	no	—
147	C₂⁴	—	2	32	no	—
148	C₂⁴	—	2	96	no	—
149	C₂⁴	—	2	96	no	—
150	C₂⁴	—	2	24	no	—
151	C₂⁴	—	2	12	no	—
152	C₂⁴	—	2	24	no	—
153	C₂⁴	—	2	12	no	—
154	C₂⁴	C₂×#88322	1	12	yes	—
155	C₂⁴	C₂×#88322	1	16	yes	complemented
156	C₂⁴	#11719917	1	12	yes	—
157	C₂⁴	C₂²⋊Dic₅₃	1	12	yes	—
158	C₂⁴	C₂²×Dic₅₃	1	6	yes	—
159	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	96	no	—
161	C₂²×C₄	—	2	128	no	—
162	C₂²×C₄	—	2	128	no	—
163	C₂²×C₄	—	2	24	no	—
164	C₂²×C₄	—	2	12	no	—
165	C₂²×C₄	—	2	48	no	—
166	C₂²×C₄	—	2	24	no	—
167	C₂²×C₄	—	2	96	no	—
168	C₂²×C₄	—	2	24	no	—
169	C₂²×C₄	C₂²⋊Dic₅₃	1	12	yes	—
170	C₂²×C₄	C₂²⋊Dic₅₃	1	24	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
172	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
174	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
175	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
176	C₂×C₈	—	106	24	no	—
177	C₂×C₈	—	212	16	no	—
178	C₂³	—	2	32	no	—
179	C₂³	—	2	96	no	—
180	C₂³	—	2	256	no	—
181	C₂³	—	2	12	no	—
182	C₂³	—	2	48	no	—
183	C₂³	—	2	96	no	—
184	C₂³	—	2	48	no	—
185	C₂³	—	2	24	no	—
186	C₂³	—	2	12	no	—
187	C₂³	—	2	24	no	—
188	C₂³	C₂²⋊#88322	1	48	yes	complemented
189	C₂³	C₂×C₂²⋊Dic₅₃	1	12	yes	—
190	C₂³	C₂×C₂²⋊Dic₅₃	1	6	yes	—
191	C₂³	C₂²×#88322	1	12	yes	—
192	C₂³	C₂²×#88322	1	2	yes	—
193	C₂³	C₂×#11719917	1	12	yes	—
194	C₂³	C₂×#11719917	1	6	yes	—
195	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
196	C₂×C₄	—	2	96	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	2	24	no	—
199	C₂×C₄	—	2	48	no	—
200	C₂×C₄	—	2	48	no	—
201	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	12	yes	—
202	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	6	yes	—
203	C₂×C₄	#13133186	1	16	yes	—
204	C₂×C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
205	C₈	—	212	8	no	—
206	C₂²	—	2	128	no	—
207	C₂²	—	2	24	no	—
208	C₂²	—	2	12	no	—
209	C₂²	—	2	24	no	—
210	C₂²	—	2	12	no	—
211	C₂²	—	2	96	no	—
212	C₂²	#16810479	1	4	yes	—
213	C₂²	C₂×C₂²⋊#88322	1	12	yes	—
214	C₂²	C₂×C₂²⋊#88322	1	24	yes	complemented
215	C₂²	#16810587	1	12	yes	—
216	C₂²	C₂²×C₂²⋊Dic₅₃	1	3	yes	—
217	C₂²	C₂³×#88322	1	1	yes	characteristic
218	C₂²	C₂²×#11719917	1	3	yes	—
219	C₄	—	2	12	no	—
220	C₄	#16810479	1	4	yes	—
221	C₄	C₂×#13133186	1	4	yes	—
222	C₂	—	2	24	no	—
223	C₂	#20274625	1	8	yes	direct product factor
224	C₂	C₂×#16810479	1	1	yes	characteristic
225	C₂	C₂²×C₂²⋊#88322	1	3	yes	—
226	C₂	C₂×#16810587	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁶×C₅₃	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂×#1779
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊#88322	C₂	semidirect
2. #20274625	C₂	direct (N × H)
4. C₂³×#88322	C₂²	semidirect
6. C₂×C₂²⋊#88322	C₂²	semidirect
14. C₂²×#88322	C₂³	semidirect
18. C₂⁵×C₅₃	C₈	semidirect
42. C₂⁴×C₅₃	C₂×C₈	semidirect
74. C₂³×C₅₃	C₂²⋊C₈	semidirect
101. C₂²×C₅₃	C₂×C₂²⋊C₈	semidirect
113. C₂×C₅₃	#1779	semidirect
126. C₅₃	C₂×#1779	semidirect
129. C₂⁵	#88322	semidirect
155. C₂⁴	C₂×#88322	semidirect
188. C₂³	C₂²⋊#88322	semidirect
214. C₂²	C₂×C₂²⋊#88322	semidirect
223. C₂	#20274625	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₂²⋊#88322	2
#20274625	2
C₂⁵×C₄×C₅₃	2
C₂×#1779	53

Subgroups — C2×#20274625 #27383196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274625 #27383196