Subgroups — C₂×#20274464 #27381558

Order 13568.

1310 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133128	C₂	1	2	yes	maximal
2	#20274464	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810378	C₂	1	4	yes	maximal
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	4	yes	—
6	C₂×#13133128	C₄	1	16	yes	complemented
7	C₂×#13133128	C₂²	1	8	yes	—
8	#16810378	C₂²	1	16	yes	—
9	C₂²×C₄²×C₅₃	C₄	1	4	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄³×C₅₃	C₄	1	4	yes	—
13	C₄³×C₅₃	C₂²	1	4	yes	—
14	C₂²×#88322	—	2	16	no	—
15	C₂²×#88322	C₂×C₄	1	16	yes	—
16	C₂²×#88322	C₂³	1	8	yes	—
17	#13133128	C₂×C₄	1	32	yes	complemented
18	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	—	2	16	no	—
22	C₂×C₄²×C₅₃	—	2	8	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
29	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	—	2	8	no	—
31	C₂²×C₄×C₅₃	—	2	8	no	—
32	C₂²×C₄×C₅₃	—	2	8	no	—
33	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
34	C₂²×C₄×C₅₃	M₁₆	1	16	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	#41	1	8	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₄²×C₅₃	—	2	8	no	—
42	C₄²×C₅₃	—	2	32	no	—
43	C₄²×C₅₃	—	2	8	no	—
44	C₄²×C₅₃	C₄²	1	8	yes	—
45	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
46	C₂×#88322	—	2	32	no	—
47	C₂×#88322	—	2	16	no	—
48	C₂×#88322	C₂²×C₄	1	16	yes	—
49	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
50	C₂³×C₅₃	#117	1	4	yes	—
51	C₂³×C₅₃	C₂×M₁₆	1	8	yes	—
52	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	8	no	—
54	C₂×C₄×C₅₃	—	2	8	no	—
55	C₂×C₄×C₅₃	—	2	8	no	—
56	C₂×C₄×C₅₃	—	2	16	no	—
57	C₂×C₄×C₅₃	—	2	32	no	—
58	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
59	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₅₃	#117	1	16	yes	—
61	C₂×C₄×C₅₃	#117	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	16	yes	—
63	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
65	#88322	—	2	16	no	—
66	C₂×#1745	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	#376	1	4	yes	—
68	C₂²×C₅₃	C₄×M₁₆	1	8	yes	—
69	C₂²×C₅₃	#432	1	16	yes	—
70	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
71	C₂²×C₅₃	C₂×#117	1	2	yes	—
72	C₂²×C₅₃	C₂²×M₁₆	1	4	yes	—
73	C₄×C₅₃	—	2	8	no	—
74	C₄×C₅₃	—	2	16	no	—
75	C₄×C₅₃	#376	1	4	yes	—
76	C₄×C₅₃	C₂×#117	1	4	yes	—
77	C₂²×#97	—	53	2	no	—
78	C₂×C₄³	D₅₃	1	1	yes	characteristic
79	#1745	—	53	8	no	—
80	C₂×#336	—	53	4	no	—
81	C₂×C₅₃	#1745	1	8	yes	complemented
82	C₂×C₅₃	C₂×#376	1	1	yes	characteristic
83	C₂×C₅₃	C₂×C₄×M₁₆	1	2	yes	—
84	C₂×C₅₃	C₂×#432	1	4	yes	—
85	C₂³×C₈	—	53	4	no	—
86	C₂²×C₄²	Dic₅₃	1	4	yes	—
87	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
88	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
89	C₂×#97	—	53	16	no	—
90	C₂×#97	—	53	8	no	—
91	C₄³	Dic₅₃	1	4	yes	—
92	C₄³	C₂×D₅₃	1	4	yes	—
93	#336	—	53	16	no	—
94	C₅₃	C₂×#1745	1	1	yes	characteristic, complemented, Sylow
95	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
96	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
97	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	C₂²×C₈	—	53	16	no	—
99	C₂²×C₈	—	53	8	no	—
100	C₂²×C₈	—	106	16	no	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	—	2	16	no	—
103	C₂×C₄²	C₄×D₅₃	1	8	yes	—
104	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
105	C₂×C₄²	C₂×Dic₅₃	1	16	yes	—
106	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
107	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
108	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
109	#97	—	53	32	no	—
110	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	#11719917	1	16	yes	—
115	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
116	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
117	C₂²×C₄	#11719945	1	8	yes	—
118	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
119	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
120	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
122	C₂×C₈	—	53	16	no	—
123	C₂×C₈	—	106	32	no	—
124	C₂×C₈	—	106	16	no	—
125	C₄²	—	2	8	no	—
126	C₄²	—	2	32	no	—
127	C₄²	—	2	8	no	—
128	C₄²	C₄×Dic₅₃	1	8	yes	—
129	C₄²	C₂×C₄×D₅₃	1	8	yes	—
130	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
131	C₂³	#13133176	1	4	yes	—
132	C₂³	C₂×#11719917	1	8	yes	—
133	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	32	no	—
139	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
140	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
141	C₂×C₄	#13133174	1	16	yes	—
142	C₂×C₄	#13133176	1	4	yes	—
143	C₂×C₄	C₂×#11719917	1	16	yes	—
144	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
145	C₂×C₄	C₂×#11719945	1	4	yes	—
146	C₈	—	106	16	no	—
147	C₂²	#16810469	1	4	yes	—
148	C₂²	C₄×#11719917	1	8	yes	—
149	C₂²	#16810680	1	16	yes	—
150	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
151	C₂²	C₂×#13133176	1	2	yes	—
152	C₂²	C₂²×#11719917	1	4	yes	—
153	C₄	—	2	16	no	—
154	C₄	—	2	8	no	—
155	C₄	#16810469	1	4	yes	—
156	C₄	C₂×#13133174	1	4	yes	—
157	C₂	#20274464	1	8	yes	direct product factor
158	C₂	C₂×#16810469	1	1	yes	characteristic
159	C₂	C₂×C₄×#11719917	1	2	yes	—
160	C₂	C₂×#16810680	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₂×#1745
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20274464	C₂	direct (N × H)
6. C₂×#13133128	C₄	semidirect
17. #13133128	C₂×C₄	semidirect
81. C₂×C₅₃	#1745	semidirect
94. C₅₃	C₂×#1745	semidirect
157. C₂	#20274464	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×#13133128	2
#20274464	2
C₂×#16810378	2
C₂×C₄³×C₅₃	2
C₂×#1745	53

Subgroups — C2×#20274464 #27381558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274464 #27381558