Subgroups — C₂×#20274461 #27381554

Order 13568.

1438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88322	C₂	1	2	yes	maximal
2	#20274461	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×C₄×#88322	C₄	1	16	yes	complemented
7	C₂×C₄×#88322	C₂²	1	8	yes	—
8	#16810378	C₂²	1	16	yes	—
9	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄³×C₅₃	C₄	1	4	yes	—
14	C₄³×C₅₃	C₂²	1	4	yes	—
15	C₂²×#88322	—	2	16	no	—
16	C₂²×#88322	C₂×C₄	1	16	yes	—
17	C₂²×#88322	C₂³	1	8	yes	—
18	C₄×#88322	C₂×C₄	1	32	yes	complemented
19	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	—	2	8	no	—
24	C₂×C₄²×C₅₃	—	2	8	no	—
25	C₂×C₄²×C₅₃	C₈	1	16	yes	complemented
26	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
33	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	8	no	—
35	C₂²×C₄×C₅₃	—	2	8	no	—
36	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
37	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	—
39	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
40	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	#41	1	8	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₄²×C₅₃	—	2	8	no	—
49	C₄²×C₅₃	—	2	16	no	—
50	C₄²×C₅₃	—	2	8	no	—
51	C₄²×C₅₃	C₄²	1	8	yes	—
52	C₄²×C₅₃	C₂×C₈	1	32	yes	complemented
53	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
54	C₂×#88322	—	2	32	no	—
55	C₂×#88322	—	2	16	no	—
56	C₂×#88322	C₂²×C₄	1	16	yes	—
57	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
58	C₂³×C₅₃	#117	1	4	yes	—
59	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
60	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
61	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	8	no	—
63	C₂×C₄×C₅₃	—	2	8	no	—
64	C₂×C₄×C₅₃	—	2	32	no	—
65	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	—
66	C₂×C₄×C₅₃	#97	1	16	yes	—
67	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
70	C₂×C₄×C₅₃	#117	1	16	yes	—
71	C₂×C₄×C₅₃	#117	1	4	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	16	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
77	#88322	—	2	16	no	—
78	C₂×#1744	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	#376	1	4	yes	—
80	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
81	C₂²×C₅₃	C₂×#97	1	4	yes	—
82	C₂²×C₅₃	#431	1	16	yes	—
83	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×#117	1	2	yes	—
85	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
86	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
87	C₄×C₅₃	—	2	16	no	—
88	C₄×C₅₃	#376	1	4	yes	—
89	C₄×C₅₃	C₂×C₄×C₈	1	8	yes	—
90	C₄×C₅₃	C₂×#97	1	8	yes	—
91	C₄×C₅₃	C₂×#117	1	4	yes	—
92	C₂²×C₄×C₈	—	53	2	no	—
93	C₂×C₄³	D₅₃	1	1	yes	characteristic
94	#1744	—	53	8	no	—
95	C₂×#336	—	53	4	no	—
96	C₂×C₅₃	#1744	1	8	yes	complemented
97	C₂×C₅₃	C₂×#376	1	1	yes	characteristic
98	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
99	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
100	C₂×C₅₃	C₂×#431	1	4	yes	—
101	C₂³×C₈	—	53	4	no	—
102	C₂²×C₄²	Dic₅₃	1	2	yes	—
103	C₂²×C₄²	Dic₅₃	1	2	yes	—
104	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
105	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
106	C₂×C₄×C₈	—	53	16	no	—
107	C₂×C₄×C₈	—	53	8	no	—
108	C₄³	Dic₅₃	1	4	yes	—
109	C₄³	C₂×D₅₃	1	4	yes	—
110	#336	—	53	16	no	—
111	C₅₃	C₂×#1744	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
113	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
115	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
116	C₂²×C₈	—	53	16	no	—
117	C₂²×C₈	—	53	8	no	—
118	C₂²×C₈	—	106	16	no	—
119	C₂×C₄²	—	2	8	no	—
120	C₂×C₄²	—	2	8	no	—
121	C₂×C₄²	#88322	1	16	yes	complemented
122	C₂×C₄²	C₄×D₅₃	1	8	yes	—
123	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
124	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
125	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
126	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
128	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
129	C₄×C₈	—	53	32	no	—
130	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	C₂×#88322	1	16	yes	—
134	C₂²×C₄	C₂×#88322	1	8	yes	—
135	C₂²×C₄	#11719917	1	8	yes	—
136	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
137	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
138	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
139	C₂²×C₄	#11719945	1	8	yes	—
140	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
143	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
144	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
145	C₂×C₈	—	53	16	no	—
146	C₂×C₈	—	106	32	no	—
147	C₂×C₈	—	106	16	no	—
148	C₄²	—	2	8	no	—
149	C₄²	—	2	16	no	—
150	C₄²	—	2	8	no	—
151	C₄²	C₂×#88322	1	32	yes	complemented
152	C₄²	C₄×Dic₅₃	1	8	yes	—
153	C₄²	C₂×C₄×D₅₃	1	8	yes	—
154	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
155	C₂³	#13133176	1	4	yes	—
156	C₂³	C₂²×#88322	1	4	yes	—
157	C₂³	C₂×#11719917	1	4	yes	—
158	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	32	no	—
162	C₂×C₄	C₄×#88322	1	16	yes	—
163	C₂×C₄	#13133128	1	16	yes	—
164	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
165	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
166	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
167	C₂×C₄	#13133174	1	16	yes	—
168	C₂×C₄	#13133176	1	4	yes	—
169	C₂×C₄	C₂²×#88322	1	16	yes	—
170	C₂×C₄	C₂²×#88322	1	8	yes	—
171	C₂×C₄	C₂×#11719917	1	8	yes	—
172	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
173	C₂×C₄	C₂×#11719945	1	4	yes	—
174	C₈	—	106	16	no	—
175	C₂²	#16810469	1	4	yes	—
176	C₂²	C₂×C₄×#88322	1	4	yes	—
177	C₂²	C₂×#13133128	1	4	yes	—
178	C₂²	#16810676	1	16	yes	—
179	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
180	C₂²	C₂×#13133176	1	2	yes	—
181	C₂²	C₂³×#88322	1	2	yes	—
182	C₂²	C₂²×#11719917	1	2	yes	—
183	C₄	—	2	16	no	—
184	C₄	#16810469	1	4	yes	—
185	C₄	C₂×C₄×#88322	1	8	yes	—
186	C₄	C₂×#13133128	1	8	yes	—
187	C₄	C₂×#13133174	1	4	yes	—
188	C₂	#20274461	1	8	yes	direct product factor
189	C₂	C₂×#16810469	1	1	yes	characteristic
190	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
191	C₂	C₂²×#13133128	1	1	yes	characteristic
192	C₂	C₂×#16810676	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
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×#1744
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20274461	C₂	direct (N × H)
6. C₂×C₄×#88322	C₄	semidirect
18. C₄×#88322	C₂×C₄	semidirect
25. C₂×C₄²×C₅₃	C₈	semidirect
52. C₄²×C₅₃	C₂×C₈	semidirect
96. C₂×C₅₃	#1744	semidirect
111. C₅₃	C₂×#1744	semidirect
121. C₂×C₄²	#88322	semidirect
151. C₄²	C₂×#88322	semidirect
188. C₂	#20274461	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×C₄×#88322	2
#20274461	2
C₂×#16810378	2
C₂×C₄³×C₅₃	2
C₂×#1744	53

Subgroups — C2×#20274461 #27381554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274461 #27381554