Subgroups — C₂×#20274867 #27384471

Order 13568.

1470 conjugacy classes of subgroups.

The subgroup lattice has 156 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	6	yes	maximal
2	#20274867	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₂²	1	4	yes	—
5	C₂×C₄⋊#88322	—	2	24	no	—
6	C₂×C₄⋊#88322	C₂²	1	24	yes	—
7	C₂²×C₄²×C₅₃	C₄	1	3	yes	—
8	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
10	C₄³×C₅₃	C₄	1	4	yes	—
11	C₄³×C₅₃	C₂²	1	4	yes	—
12	C₂²×#88322	—	2	24	no	—
13	C₂²×#88322	C₂³	1	8	yes	—
14	C₄⋊#88322	—	2	48	no	—
15	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
16	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
17	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	—	2	12	no	—
19	C₂×C₄²×C₅₃	C₈	1	8	yes	complemented
20	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅₃	D₄	1	12	yes	—
24	C₂×C₄²×C₅₃	Q₈	1	12	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
26	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	—	2	12	no	—
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	12	yes	—
31	C₂²×C₄×C₅₃	M₁₆	1	12	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
36	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
37	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
38	C₄²×C₅₃	—	2	24	no	—
39	C₄²×C₅₃	C₄⋊C₄	1	24	yes	—
40	C₄²×C₅₃	C₂×C₈	1	16	yes	complemented
41	C₄²×C₅₃	C₂×D₄	1	12	yes	—
42	C₄²×C₅₃	C₂×Q₈	1	12	yes	—
43	C₂×#88322	—	2	24	no	—
44	C₂×#88322	—	4	16	no	—
45	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
46	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
47	C₂³×C₅₃	C₂×M₁₆	1	6	yes	—
48	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	12	no	—
50	C₂×C₄×C₅₃	—	2	24	no	—
51	C₂×C₄×C₅₃	C₄⋊C₈	1	48	yes	complemented
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
55	C₂×C₄×C₅₃	#127	1	4	yes	—
56	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
57	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
59	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
60	C₂×C₄×C₅₃	C₂²×Q₈	1	3	yes	—
61	#88322	—	4	8	no	—
62	C₂×#1842	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	#384	1	4	yes	—
64	C₂²×C₅₃	C₂×C₄⋊C₈	1	12	yes	—
65	C₂²×C₅₃	#423	1	12	yes	—
66	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
67	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
68	C₂²×C₅₃	C₂²×M₁₆	1	3	yes	—
69	C₄×C₅₃	—	2	12	no	—
70	C₄×C₅₃	#384	1	4	yes	—
71	C₄×C₅₃	C₂×C₄⋊C₈	1	24	yes	complemented
72	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
73	C₄×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
74	C₂²×C₄⋊C₈	—	53	6	no	—
75	C₂×C₄³	D₅₃	1	1	yes	characteristic
76	#1842	—	53	8	no	—
77	C₂×C₅₃	#1842	1	8	yes	complemented
78	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
79	C₂×C₅₃	C₂²×C₄⋊C₈	1	3	yes	—
80	C₂×C₅₃	C₂×#423	1	3	yes	—
81	C₂³×C₈	—	53	4	no	—
82	C₂²×C₄²	Dic₅₃	1	3	yes	—
83	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
84	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
85	C₂×C₄⋊C₈	—	53	24	no	—
86	C₂×C₄⋊C₈	—	106	24	no	—
87	C₄³	Dic₅₃	1	4	yes	—
88	C₄³	C₂×D₅₃	1	4	yes	—
89	C₅₃	C₂×#1842	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
91	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
92	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
93	C₂²×C₈	—	53	8	no	—
94	C₂²×C₈	—	106	24	no	—
95	C₂×C₄²	—	2	12	no	—
96	C₂×C₄²	#88322	1	8	yes	complemented
97	C₂×C₄²	Dic₁₀₆	1	12	yes	—
98	C₂×C₄²	D₂₁₂	1	12	yes	—
99	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
100	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
101	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
102	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
103	C₄⋊C₈	—	106	48	no	—
104	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	C₂×#88322	1	12	yes	—
107	C₂²×C₄	#11719917	1	12	yes	—
108	C₂²×C₄	C₄⋊Dic₅₃	1	12	yes	—
109	C₂²×C₄	C₄⋊Dic₅₃	1	12	yes	—
110	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
111	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
112	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
113	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
115	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
116	C₂×C₈	—	106	24	no	—
117	C₂×C₈	—	212	16	no	—
118	C₄²	—	2	24	no	—
119	C₄²	C₂×#88322	1	16	yes	complemented
120	C₄²	C₄⋊Dic₅₃	1	24	yes	—
121	C₄²	C₂×Dic₁₀₆	1	12	yes	—
122	C₄²	C₂×D₂₁₂	1	12	yes	—
123	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
124	C₂³	C₂²×#88322	1	2	yes	—
125	C₂³	C₂×#11719917	1	6	yes	—
126	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	24	no	—
129	C₂×C₄	C₄⋊#88322	1	48	yes	complemented
130	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
131	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
133	C₂×C₄	#13133214	1	4	yes	—
134	C₂×C₄	#13133217	1	12	yes	—
135	C₂×C₄	C₂²×#88322	1	12	yes	—
136	C₂×C₄	C₂×#11719917	1	12	yes	—
137	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
138	C₂×C₄	C₂²×Dic₁₀₆	1	3	yes	—
139	C₈	—	212	8	no	—
140	C₂²	#16810493	1	4	yes	—
141	C₂²	C₂×C₄⋊#88322	1	12	yes	—
142	C₂²	#16810647	1	12	yes	—
143	C₂²	C₂²×C₄⋊Dic₅₃	1	3	yes	—
144	C₂²	C₂³×#88322	1	1	yes	characteristic
145	C₂²	C₂²×#11719917	1	3	yes	—
146	C₄	—	2	12	no	—
147	C₄	#16810493	1	4	yes	—
148	C₄	C₂×C₄⋊#88322	1	24	yes	complemented
149	C₄	C₂×#13133214	1	1	yes	characteristic
150	C₄	C₂×#13133217	1	3	yes	—
151	C₂	#20274867	1	8	yes	direct product factor
152	C₂	C₂×#16810493	1	1	yes	characteristic
153	C₂	C₂²×C₄⋊#88322	1	3	yes	—
154	C₂	C₂×#16810647	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×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×#1842
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. #20274867	C₂	direct (N × H)
19. C₂×C₄²×C₅₃	C₈	semidirect
40. C₄²×C₅₃	C₂×C₈	semidirect
51. C₂×C₄×C₅₃	C₄⋊C₈	semidirect
71. C₄×C₅₃	C₂×C₄⋊C₈	semidirect
77. C₂×C₅₃	#1842	semidirect
89. C₅₃	C₂×#1842	semidirect
96. C₂×C₄²	#88322	semidirect
119. C₄²	C₂×#88322	semidirect
129. C₂×C₄	C₄⋊#88322	semidirect
148. C₄	C₂×C₄⋊#88322	semidirect
151. C₂	#20274867	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
#20274867	2
C₂×C₄³×C₅₃	2
C₂×#1842	53

Subgroups — C2×#20274867 #27384471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274867 #27384471