Subgroups — C₂×C₄×#13133128 #27381416

Order 13568.

1790 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	4	yes	maximal
2	C₂²×C₄×#88322	C₂	1	2	yes	maximal
3	C₄×#13133128	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	4	yes	—
6	C₂×#13133128	C₄	1	32	yes	direct product factor
7	C₂×#13133128	C₂²	1	16	yes	—
8	C₂×C₄×#88322	C₄	1	16	yes	complemented
9	C₂×C₄×#88322	C₂²	1	8	yes	—
10	C₂²×C₄²×C₅₃	C₄	1	4	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	C₂×C₄	1	16	yes	—
16	C₂²×#88322	C₂×C₄	1	32	yes	—
17	C₂²×#88322	C₂³	1	8	yes	—
18	#13133128	C₂×C₄	1	64	yes	direct product factor
19	C₄×#88322	C₂×C₄	1	32	yes	complemented
20	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	—	2	16	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
31	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	—	2	16	no	—
33	C₂²×C₄×C₅₃	C₄²	1	16	yes	—
34	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
35	C₂²×C₄×C₅₃	M₁₆	1	16	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	C₂²×C₄	1	8	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
42	C₄²×C₅₃	—	2	32	no	—
43	C₄²×C₅₃	C₄²	1	16	yes	—
44	C₄²×C₅₃	C₄²	1	8	yes	—
45	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
46	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
47	C₂×#88322	C₄²	1	64	yes	complemented
48	C₂×#88322	C₂²×C₄	1	16	yes	—
49	C₂×#88322	C₂²×C₄	1	32	yes	—
50	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
51	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
52	C₂³×C₅₃	C₂×M₁₆	1	8	yes	—
53	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	16	no	—
55	C₂×C₄×C₅₃	—	2	32	no	—
56	C₂×C₄×C₅₃	#97	1	32	yes	complemented
57	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
58	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
59	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	16	yes	—
63	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
65	#88322	C₂×C₄²	1	32	yes	complemented
66	C₂×C₄×#97	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	C₄³	1	4	yes	—
68	C₂²×C₅₃	C₂×#97	1	8	yes	—
69	C₂²×C₅₃	C₄×M₁₆	1	16	yes	—
70	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
71	C₂²×C₅₃	C₂²×C₄²	1	2	yes	—
72	C₂²×C₅₃	C₂²×M₁₆	1	4	yes	—
73	C₄×C₅₃	—	2	16	no	—
74	C₄×C₅₃	C₄³	1	4	yes	—
75	C₄×C₅₃	C₂×#97	1	16	yes	complemented
76	C₄×C₅₃	C₂²×C₄²	1	4	yes	—
77	C₂²×#97	—	53	4	no	—
78	C₂²×C₄×C₈	—	53	2	no	—
79	C₂×C₄³	D₅₃	1	1	yes	characteristic
80	C₄×#97	—	53	8	no	—
81	C₂×C₅₃	C₄×#97	1	8	yes	complemented
82	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
83	C₂×C₅₃	C₂²×#97	1	2	yes	—
84	C₂×C₅₃	C₂×C₄×M₁₆	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	32	no	—
90	C₂×#97	—	53	16	no	—
91	C₂×C₄×C₈	—	53	16	no	—
92	C₂×C₄×C₈	—	53	8	no	—
93	C₄³	Dic₅₃	1	4	yes	—
94	C₄³	C₂×D₅₃	1	4	yes	—
95	C₅₃	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
97	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
98	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂²×C₈	—	53	16	no	—
100	C₂²×C₈	—	53	32	no	—
101	C₂²×C₈	—	53	8	no	—
102	C₂×C₄²	—	2	16	no	—
103	C₂×C₄²	C₄×D₅₃	1	8	yes	—
104	C₂×C₄²	C₄×D₅₃	1	16	yes	—
105	C₂×C₄²	C₂×Dic₅₃	1	16	yes	—
106	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
107	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
108	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
109	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
110	#97	—	53	64	no	—
111	C₄×C₈	—	53	32	no	—
112	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
113	C₂²×C₄	—	2	16	no	—
114	C₂²×C₄	#11719917	1	16	yes	—
115	C₂²×C₄	C₄×Dic₅₃	1	16	yes	—
116	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
117	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
118	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
119	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
120	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
122	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
123	C₂×C₈	—	53	64	no	—
124	C₂×C₈	—	53	16	no	—
125	C₂×C₈	—	53	32	no	—
126	C₄²	—	2	32	no	—
127	C₄²	C₄×Dic₅₃	1	16	yes	—
128	C₄²	C₄×Dic₅₃	1	8	yes	—
129	C₄²	C₂×C₄×D₅₃	1	8	yes	—
130	C₄²	C₂×C₄×D₅₃	1	16	yes	—
131	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
132	C₂³	C₂×C₄×Dic₅₃	1	4	yes	—
133	C₂³	C₂×#11719917	1	8	yes	—
134	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
135	C₂×C₄	—	2	32	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	#13133128	1	32	yes	direct product factor
138	C₂×C₄	C₄²×D₅₃	1	16	yes	—
139	C₂×C₄	C₂×C₄×Dic₅₃	1	16	yes	—
140	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
141	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
142	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
143	C₂×C₄	C₂×#11719917	1	16	yes	—
144	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
145	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
146	C₈	—	53	32	no	—
147	C₂²	C₄²×Dic₅₃	1	4	yes	—
148	C₂²	C₂×#13133128	1	8	yes	—
149	C₂²	C₄×#11719917	1	16	yes	—
150	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
151	C₂²	C₂²×C₄×Dic₅₃	1	2	yes	—
152	C₂²	C₂²×#11719917	1	4	yes	—
153	C₄	—	2	16	no	—
154	C₄	C₄²×Dic₅₃	1	4	yes	—
155	C₄	C₂×#13133128	1	16	yes	direct product factor
156	C₄	C₂×C₄²×D₅₃	1	4	yes	—
157	C₂	C₄×#13133128	1	8	yes	direct product factor
158	C₂	C₂×C₄²×Dic₅₃	1	1	yes	characteristic
159	C₂	C₂²×#13133128	1	2	yes	—
160	C₂	C₂×C₄×#11719917	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄³
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×C₄×#97
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
3. C₄×#13133128	C₂	direct (N × H)
6. C₂×#13133128	C₄	direct (N × H)
8. C₂×C₄×#88322	C₄	semidirect
18. #13133128	C₂×C₄	direct (N × H)
19. C₄×#88322	C₂×C₄	semidirect
47. C₂×#88322	C₄²	semidirect
56. C₂×C₄×C₅₃	#97	semidirect
65. #88322	C₂×C₄²	semidirect
75. C₄×C₅₃	C₂×#97	semidirect
81. C₂×C₅₃	C₄×#97	semidirect
95. C₅₃	C₂×C₄×#97	semidirect
137. C₂×C₄	#13133128	direct (N × H)
155. C₄	C₂×#13133128	direct (N × H)
157. C₂	C₄×#13133128	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
C₂²×C₄×#88322	2
C₄×#13133128	2
C₂×C₄³×C₅₃	2
C₂×C₄×#97	53

Subgroups — C2×C4×#13133128 #27381416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133128 #27381416