Subgroups — C₂×#20274471 #27381596

Order 13568.

930 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810581	C₂	1	3	yes	maximal
2	#20274471	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810380	C₂	1	3	yes	maximal
4	C₂×C₅₃×#338	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	1	yes	characteristic
6	C₂×#13133176	C₂²	1	3	yes	—
7	#16810581	C₄	1	12	yes	complemented
8	#16810581	C₂²	1	12	yes	—
9	C₂×#13133127	C₄	1	6	yes	complemented
10	C₂×C₈×Dic₅₃	C₄	1	6	yes	complemented
11	#16810380	C₂²	1	12	yes	—
12	C₂²×M₁₆×C₅₃	C₂²	1	3	yes	—
13	C₅₃×#338	C₂²	1	4	yes	—
14	C₂²×#88322	C₂×C₄	1	3	yes	—
15	C₂²×#88322	C₂×C₄	1	3	yes	—
16	C₂²×#88322	C₂×C₄	1	6	yes	—
17	C₂²×#88322	C₂³	1	2	yes	—
18	#13133176	—	2	3	no	—
19	#13133176	—	2	3	no	—
20	#13133176	C₂×C₄	1	6	yes	—
21	#13133176	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₅₃	—	2	3	no	—
23	C₂×C₂²⋊Dic₅₃	—	2	3	no	—
24	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
25	#13133127	C₂×C₄	1	24	yes	complemented
26	C₈×Dic₅₃	C₂×C₄	1	24	yes	complemented
27	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	—	2	6	no	—
29	C₂×M₁₆×C₅₃	—	2	6	no	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	6	yes	—
31	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
33	C₂⁴×C₅₃	#41	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
36	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	#41	1	3	yes	—
38	C₂²×C₄×C₅₃	#41	1	3	yes	—
39	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	2	6	no	—
41	M₁₆×C₅₃	—	2	12	no	—
42	M₁₆×C₅₃	—	2	12	no	—
43	C₂×C₈×C₅₃	—	2	6	no	—
44	C₂×C₈×C₅₃	—	2	6	no	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
46	C₂²⋊Dic₅₃	—	2	6	no	—
47	C₂²⋊Dic₅₃	—	2	6	no	—
48	C₄⋊Dic₅₃	—	2	6	no	—
49	C₄⋊Dic₅₃	—	2	6	no	—
50	C₄×Dic₅₃	—	2	6	no	—
51	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
52	C₂×#88322	C₄²	1	6	yes	—
53	C₂×#88322	C₄²	1	6	yes	—
54	C₂×#88322	C₄²	1	1	yes	characteristic
55	C₂×#88322	C₄²	1	3	yes	—
56	C₂×#88322	C₂²×C₄	1	6	yes	—
57	C₂×#88322	C₂²×C₄	1	6	yes	—
58	C₂³×C₅₃	—	4	2	no	—
59	C₂³×C₅₃	#117	1	3	yes	—
60	C₂³×C₅₃	#117	1	3	yes	—
61	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
63	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#117	1	3	yes	—
65	C₂×C₄×C₅₃	#117	1	3	yes	—
66	C₂×C₄×C₅₃	#117	1	6	yes	—
67	C₂×C₄×C₅₃	#117	1	3	yes	—
68	C₂×C₄×C₅₃	#117	1	3	yes	—
69	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
71	C₂×Dic₅₃	—	2	12	no	—
72	C₂×Dic₅₃	—	4	12	no	—
73	C₈×C₅₃	—	2	12	no	—
74	#88322	C₂×C₄²	1	2	yes	—
75	#88322	C₂×C₄²	1	6	yes	—
76	C₂×#1747	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	6	no	—
78	C₂²×C₅₃	—	4	3	no	—
79	C₂²×C₅₃	—	4	1	no	—
80	C₂²×C₅₃	#376	1	3	yes	—
81	C₂²×C₅₃	#376	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×#117	1	3	yes	—
83	C₄×C₅₃	#376	1	3	yes	—
84	C₄×C₅₃	#376	1	1	yes	characteristic
85	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
86	C₄×C₅₃	C₂×#117	1	3	yes	—
87	Dic₅₃	—	4	12	no	—
88	C₂×#405	—	53	3	no	—
89	#1747	—	53	8	no	—
90	C₂×#338	D₅₃	1	1	yes	characteristic
91	C₂×H(3,ℤ/4)	—	53	3	no	—
92	C₂×C₅₃	—	2	3	no	—
93	C₂×C₅₃	—	2	3	no	—
94	C₂×C₅₃	#1747	1	2	yes	complemented
95	C₂×C₅₃	C₂×#376	1	1	yes	characteristic
96	C₂²×M₁₆	C₂×D₅₃	1	3	yes	—
97	C₂³×C₈	—	53	1	no	—
98	C₂×#117	—	53	3	no	—
99	#405	—	53	12	no	—
100	#405	—	53	12	no	—
101	C₂×#97	—	53	6	no	—
102	C₂×C₄×C₈	—	53	6	no	—
103	#338	C₂×D₅₃	1	4	yes	—
104	H(3,ℤ/4)	—	53	12	no	—
105	C₅₃	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
106	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×M₁₆	—	2	6	no	—
108	C₂×M₁₆	—	2	6	no	—
109	C₂×M₁₆	C₄×D₅₃	1	6	yes	—
110	C₂×M₁₆	C₂²×D₅₃	1	6	yes	—
111	C₂²×C₈	—	53	3	no	—
112	C₂²×C₈	—	53	3	no	—
113	C₂²×C₈	—	53	6	no	—
114	C₂²×C₈	—	53	2	no	—
115	C₂²×C₈	C₄×D₅₃	1	6	yes	—
116	#117	—	53	6	no	—
117	#117	—	53	6	no	—
118	#117	—	106	3	no	—
119	#117	—	106	3	no	—
120	C₂×C₄⋊C₄	—	106	3	no	—
121	C₂×C₂²⋊C₄	—	106	3	no	—
122	C₂×C₄²	—	53	6	no	—
123	#97	—	53	24	no	—
124	C₄×C₈	—	53	24	no	—
125	C₂⁴	#11719949	1	1	yes	characteristic
126	C₂²×C₄	—	106	6	no	—
127	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
128	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
129	C₂²×C₄	#11719947	1	3	yes	—
130	C₂²×C₄	#11719947	1	3	yes	—
131	C₂²×C₄	#11719949	1	1	yes	characteristic
132	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
133	M₁₆	—	2	12	no	—
134	M₁₆	—	2	12	no	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	53	6	no	—
138	C₂×C₈	—	53	6	no	—
139	C₂×C₈	—	53	1	no	—
140	C₂×C₈	—	53	3	no	—
141	C₂×C₈	—	53	6	no	—
142	C₂×C₈	—	53	6	no	—
143	C₂×C₈	C₂×C₄×D₅₃	1	12	yes	—
144	C₄⋊C₄	—	106	6	no	—
145	C₄⋊C₄	—	106	6	no	—
146	C₂²⋊C₄	—	106	6	no	—
147	C₂²⋊C₄	—	106	6	no	—
148	C₄²	—	53	12	no	—
149	C₄²	—	106	6	no	—
150	C₂³	—	4	2	no	—
151	C₂³	#13133175	1	3	yes	—
152	C₂³	#13133175	1	3	yes	—
153	C₂³	C₂×#11719949	1	1	yes	characteristic
154	C₂×C₄	—	106	12	no	—
155	C₂×C₄	—	212	12	no	—
156	C₂×C₄	C₄²×D₅₃	1	3	yes	—
157	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
158	C₂×C₄	#13133173	1	3	yes	—
159	C₂×C₄	#13133173	1	3	yes	—
160	C₂×C₄	#13133173	1	6	yes	—
161	C₂×C₄	#13133175	1	3	yes	—
162	C₂×C₄	#13133175	1	3	yes	—
163	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
164	C₂×C₄	C₂×#11719947	1	3	yes	—
165	C₈	—	2	12	no	—
166	C₈	—	53	2	no	—
167	C₈	—	53	6	no	—
168	C₂²	—	2	6	no	—
169	C₂²	—	4	3	no	—
170	C₂²	—	4	1	no	—
171	C₂²	#16810468	1	3	yes	—
172	C₂²	#16810468	1	1	yes	characteristic
173	C₂²	C₂×#13133175	1	3	yes	—
174	C₄	—	212	12	no	—
175	C₄	#16810468	1	3	yes	—
176	C₄	#16810468	1	1	yes	characteristic
177	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
178	C₄	C₂×#13133173	1	3	yes	—
179	C₂	—	2	3	no	—
180	C₂	—	2	3	no	—
181	C₂	#20274471	1	2	yes	direct product factor
182	C₂	C₂×#16810468	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#338	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#338	D₅₃
hypocenter	C₅₃	C₂×#1747
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
2. #20274471	C₂	direct (N × H)
7. #16810581	C₄	semidirect
9. C₂×#13133127	C₄	semidirect
10. C₂×C₈×Dic₅₃	C₄	semidirect
25. #13133127	C₂×C₄	semidirect
26. C₈×Dic₅₃	C₂×C₄	semidirect
94. C₂×C₅₃	#1747	semidirect
105. C₅₃	C₂×#1747	semidirect
181. C₂	#20274471	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×#16810581	2
#20274471	2
C₂×#16810380	2
C₂×C₅₃×#338	2
C₂×#1747	53

Subgroups — C2×#20274471 #27381596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274471 #27381596