Subgroups — C₂×#20276471 #27537071

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	2	yes	maximal
2	#20276471	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	8	no	—
6	C₂×#13133157	C₂²	1	8	yes	—
7	C₂²×#11719908	C₂²	1	4	yes	—
8	C₄⋊#11719908	C₂²	1	16	yes	complemented
9	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
16	#13133157	—	2	16	no	—
17	C₂×#11719908	—	2	8	no	—
18	C₂×#11719908	C₂³	1	8	yes	—
19	C₂³×C₄×C₅₃	C₈	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	—	2	4	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	D₄	1	4	yes	—
27	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₈	1	4	yes	—
32	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₅₃	C₂³	1	4	yes	—
38	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	—	2	8	no	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	—	2	8	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×C₈	1	4	yes	—
62	C₄²×C₅₃	C₂×C₈	1	4	yes	—
63	C₄²×C₅₃	C₂×C₈	1	4	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
65	#11719908	—	2	8	no	—
66	C₂³×C₅₃	—	2	4	no	—
67	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
68	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
69	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	16	no	—
72	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
73	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
74	C₂×C₄×C₅₃	M₃₂	1	8	yes	complemented
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	C₈×C₅₃	—	2	4	no	—
85	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
89	C₂×#2150	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	4	no	—
91	C₂²×C₅₃	—	2	8	no	—
92	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
93	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
94	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
95	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₅₃	—	2	4	no	—
97	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
98	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
99	C₄×C₅₃	C₂×M₃₂	1	8	yes	complemented
100	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
102	C₂²×M₃₂	—	53	2	no	—
103	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
104	#2150	—	53	8	no	—
105	C₂×C₄⋊C₁₆	—	53	4	no	—
106	C₂×C₅₃	—	2	4	no	—
107	C₂×C₅₃	#2150	1	4	yes	complemented
108	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
110	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
111	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
112	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
113	C₂×M₃₂	—	53	8	no	—
114	C₂×M₃₂	—	106	8	no	—
115	C₂²×C₁₆	—	53	4	no	—
116	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
117	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
118	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
119	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
120	C₄⋊C₁₆	—	53	16	no	—
121	C₅₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	#88322	1	2	yes	—
123	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	Dic₁₀₆	1	4	yes	—
126	C₂²×C₈	D₂₁₂	1	4	yes	—
127	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
128	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
129	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
131	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
132	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
133	C₂×C₄²	#88322	1	1	yes	characteristic
134	C₂×C₄²	#88322	1	4	yes	—
135	C₂×C₄²	#88322	1	1	yes	characteristic
136	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
137	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
138	M₃₂	—	106	16	no	—
139	C₂×C₁₆	—	53	8	no	—
140	C₂×C₁₆	—	106	8	no	—
141	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
142	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
143	C₄×C₈	C₂²×D₅₃	1	4	yes	—
144	C₂⁴	C₂×#88322	1	1	yes	characteristic
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	C₂×#88322	1	2	yes	—
147	C₂²×C₄	C₂×#88322	1	8	yes	—
148	C₂²×C₄	C₂×#88322	1	2	yes	—
149	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
150	C₂²×C₄	C₂×#88322	1	2	yes	—
151	C₂²×C₄	#11719917	1	4	yes	—
152	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
153	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
158	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
159	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
160	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
161	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
162	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
163	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
164	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
165	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
167	C₄²	C₂×#88322	1	4	yes	—
168	C₄²	C₂×#88322	1	4	yes	—
169	C₄²	C₂×#88322	1	4	yes	—
170	C₄²	C₂²×Dic₅₃	1	4	yes	—
171	C₁₆	—	106	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	C₄⋊#88322	1	4	yes	—
174	C₂³	C₂²×#88322	1	2	yes	—
175	C₂³	C₂²×#88322	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	C₄⋊#88322	1	8	yes	—
179	C₂×C₄	C₄⋊#88322	1	4	yes	—
180	C₂×C₄	#13133157	1	8	yes	complemented
181	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
183	C₂×C₄	C₂²×#88322	1	4	yes	—
184	C₂×C₄	C₂²×#88322	1	4	yes	—
185	C₂×C₄	C₂²×#88322	1	2	yes	—
186	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
187	C₂×C₄	C₂×#11719917	1	4	yes	—
188	C₂×C₄	C₂×#11719917	1	2	yes	—
189	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
192	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
193	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
194	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	C₂×C₄⋊#88322	1	4	yes	—
198	C₂²	C₂×C₄⋊#88322	1	2	yes	—
199	C₂²	C₂×#13133157	1	4	yes	—
200	C₂²	C₂³×#88322	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	C₂×C₄⋊#88322	1	2	yes	—
203	C₄	C₂×C₄⋊#88322	1	4	yes	—
204	C₄	C₂×#13133157	1	8	yes	complemented
205	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₄	C₂²×#11719917	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#20276471	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
210	C₂	C₂²×#13133157	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#2150
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20276471	C₂	direct (N × H)
3. C₂×C₄⋊#11719908	C₂	semidirect
8. C₄⋊#11719908	C₂²	semidirect
74. C₂×C₄×C₅₃	M₃₂	semidirect
99. C₄×C₅₃	C₂×M₃₂	semidirect
107. C₂×C₅₃	#2150	semidirect
121. C₅₃	C₂×#2150	semidirect
180. C₂×C₄	#13133157	semidirect
204. C₄	C₂×#13133157	semidirect
208. C₂	#20276471	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×#13133157	2
#20276471	2
C₂×C₄⋊#11719908	2
C₂²×C₄×C₈×C₅₃	2
C₂×#2150	53

Subgroups — C2×#20276471 #27537071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276471 #27537071