Subgroups — C₂³×#88180 #16769479

Order 3328.

14334 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88180	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₁₃	C₂	1	2	yes	maximal
4	C₂³×C₄×Dic₁₃	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₃	C₄	1	32	yes	complemented
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	28	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₁₃	C₂²	1	4	yes	—
12	C₂²×C₂²⋊Dic₁₃	C₂²	1	56	yes	—
13	C₂×#88180	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₁₃	C₂²	1	56	yes	complemented
15	C₂²×C₄×Dic₁₃	C₂²	1	56	yes	complemented
16	C₂⁵×C₁₃	—	2	16	no	—
17	C₂⁵×C₁₃	C₂×C₄	1	16	yes	—
18	C₂⁵×C₁₃	C₂³	1	14	yes	—
19	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	448	yes	complemented
23	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	112	yes	—
26	C₂³×Dic₁₃	—	2	32	no	—
27	C₂³×Dic₁₃	C₂³	1	56	yes	—
28	C₂×C₂²⋊Dic₁₃	C₂³	1	224	yes	—
29	#88180	C₂³	1	512	yes	direct product factor
30	C₂×C₄⋊Dic₁₃	C₂³	1	224	yes	complemented
31	C₂×C₄×Dic₁₃	C₂³	1	224	yes	complemented
32	C₂³×#117	—	13	1	no	maximal, Sylow
33	C₂⁴×C₁₃	—	2	128	no	—
34	C₂⁴×C₁₃	—	2	112	no	—
35	C₂⁴×C₁₃	C₂²×C₄	1	112	yes	—
36	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
37	C₂⁴×C₁₃	#41	1	16	yes	—
38	C₂⁴×C₁₃	C₂⁴	1	28	yes	—
39	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	112	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	112	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	896	yes	complemented
43	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
44	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
45	C₂²×C₄×C₁₃	C₂⁴	1	64	yes	—
46	C₂²×Dic₁₃	—	2	224	no	—
47	C₂²×Dic₁₃	C₂⁴	1	112	yes	—
48	C₂²⋊Dic₁₃	C₂⁴	1	128	yes	—
49	C₄⋊Dic₁₃	C₂⁴	1	128	yes	complemented
50	C₄×Dic₁₃	C₂⁴	1	128	yes	complemented
51	C₂⁵×C₄	D₁₃	1	1	yes	characteristic
52	C₂²×#117	—	13	56	no	—
53	C₂³×C₄⋊C₄	—	13	2	no	—
54	C₂³×C₂²⋊C₄	—	13	2	no	—
55	C₂³×C₄²	—	13	2	no	—
56	C₂³×C₁₃	—	2	448	no	—
57	C₂³×C₁₃	—	2	112	no	—
58	C₂³×C₁₃	#117	1	64	yes	complemented
59	C₂³×C₁₃	C₂³×C₄	1	112	yes	—
60	C₂³×C₁₃	C₂³×C₄	1	28	yes	—
61	C₂³×C₁₃	C₂×#41	1	56	yes	—
62	C₂³×C₁₃	C₂⁵	1	8	yes	—
63	C₂³×C₁₃	C₂⁵	1	7	yes	—
64	C₂×C₄×C₁₃	C₂³×C₄	1	112	yes	—
65	C₂×C₄×C₁₃	C₂³×C₄	1	112	yes	—
66	C₂×C₄×C₁₃	C₂³×C₄	1	256	yes	complemented
67	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
68	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
69	C₂×Dic₁₃	—	2	224	no	—
70	C₂×Dic₁₃	C₂⁵	1	32	yes	—
71	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
72	C₂⁴×C₄	—	13	4	no	—
73	C₂⁴×C₄	Dic₁₃	1	32	yes	complemented
74	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
75	C₂⁴×C₄	C₂×D₁₃	1	28	yes	—
76	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
77	C₂×#117	—	13	448	no	—
78	C₂²×C₄⋊C₄	—	13	56	no	—
79	C₂²×C₂²⋊C₄	—	13	56	no	—
80	C₂²×C₄²	—	13	56	no	—
81	C₂²×C₁₃	—	2	16	no	—
82	C₂²×C₁₃	—	2	224	no	—
83	C₂²×C₁₃	C₂×#117	1	112	yes	complemented
84	C₂²×C₁₃	C₂⁴×C₄	1	16	yes	—
85	C₂²×C₁₃	C₂⁴×C₄	1	14	yes	—
86	C₂²×C₁₃	C₂²×#41	1	28	yes	—
87	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
88	C₄×C₁₃	C₂⁴×C₄	1	16	yes	—
89	C₄×C₁₃	C₂⁴×C₄	1	16	yes	—
90	Dic₁₃	—	2	32	no	—
91	C₂⁵	—	2	16	no	—
92	C₂⁵	C₂×Dic₁₃	1	16	yes	—
93	C₂⁵	C₂²×D₁₃	1	14	yes	—
94	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
95	C₂³×C₄	—	13	56	no	—
96	C₂³×C₄	—	26	32	no	—
97	C₂³×C₄	C₂×Dic₁₃	1	16	yes	—
98	C₂³×C₄	C₂×Dic₁₃	1	16	yes	—
99	C₂³×C₄	C₂×Dic₁₃	1	448	yes	complemented
100	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
101	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
102	C₂³×C₄	C₂²×D₁₃	1	112	yes	—
103	#117	—	13	512	no	—
104	C₂×C₄⋊C₄	—	13	224	no	—
105	C₂×C₂²⋊C₄	—	13	224	no	—
106	C₂×C₄²	—	13	224	no	—
107	C₂×C₁₃	—	2	16	no	—
108	C₂×C₁₃	C₂²×#117	1	28	yes	complemented
109	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
110	C₂×C₁₃	C₂³×#41	1	2	yes	—
111	C₂⁴	—	2	128	no	—
112	C₂⁴	—	2	112	no	—
113	C₂⁴	#6127	1	16	yes	—
114	C₂⁴	C₂²×Dic₁₃	1	112	yes	—
115	C₂⁴	C₂²×Dic₁₃	1	8	yes	—
116	C₂⁴	C₂³×D₁₃	1	28	yes	—
117	C₂⁴	C₂³×D₁₃	1	7	yes	—
118	C₂²×C₄	—	13	112	no	—
119	C₂²×C₄	—	26	224	no	—
120	C₂²×C₄	C₂²×Dic₁₃	1	112	yes	—
121	C₂²×C₄	C₂²×Dic₁₃	1	112	yes	—
122	C₂²×C₄	C₂²×Dic₁₃	1	896	yes	complemented
123	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
124	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
125	C₂²×C₄	C₂³×D₁₃	1	64	yes	—
126	C₄⋊C₄	—	13	128	no	—
127	C₂²⋊C₄	—	13	128	no	—
128	C₄²	—	13	128	no	—
129	C₁₃	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
130	C₂³	—	2	448	no	—
131	C₂³	—	2	112	no	—
132	C₂³	#88180	1	64	yes	direct product factor
133	C₂³	C₂×#6127	1	56	yes	—
134	C₂³	C₂³×Dic₁₃	1	112	yes	—
135	C₂³	C₂³×Dic₁₃	1	28	yes	—
136	C₂³	C₂⁴×D₁₃	1	8	yes	—
137	C₂³	C₂⁴×D₁₃	1	7	yes	—
138	C₂×C₄	—	13	32	no	—
139	C₂×C₄	—	26	224	no	—
140	C₂×C₄	C₂³×Dic₁₃	1	112	yes	—
141	C₂×C₄	C₂³×Dic₁₃	1	112	yes	—
142	C₂×C₄	C₂³×Dic₁₃	1	256	yes	complemented
143	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
144	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
145	C₂²	—	2	224	no	—
146	C₂²	—	2	16	no	—
147	C₂²	C₂×#88180	1	112	yes	direct product factor
148	C₂²	C₂²×#6127	1	28	yes	—
149	C₂²	C₂⁴×Dic₁₃	1	16	yes	—
150	C₂²	C₂⁴×Dic₁₃	1	14	yes	—
151	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
152	C₄	—	26	32	no	—
153	C₄	C₂⁴×Dic₁₃	1	16	yes	—
154	C₄	C₂⁴×Dic₁₃	1	16	yes	—
155	C₂	—	2	16	no	—
156	C₂	C₂²×#88180	1	28	yes	direct product factor
157	C₂	C₂⁵×Dic₁₃	1	1	yes	characteristic
158	C₂	C₂³×#6127	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂⁵×C₄×C₁₃	C₂
socle	C₂⁵×C₁₃	C₂³
prime radical	C₂⁶×C₁₃	C₂²
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂⁵×C₄	D₁₃
hypocenter	C₁₃	C₂³×#117
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88180	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₁₃	C₂	semidirect
4. C₂³×C₄×Dic₁₃	C₂	semidirect
7. C₂⁴×C₄×C₁₃	C₄	semidirect
13. C₂×#88180	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
15. C₂²×C₄×Dic₁₃	C₂²	semidirect
22. C₂³×C₄×C₁₃	C₂×C₄	semidirect
29. #88180	C₂³	direct (N × H)
30. C₂×C₄⋊Dic₁₃	C₂³	semidirect
31. C₂×C₄×Dic₁₃	C₂³	semidirect
42. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
49. C₄⋊Dic₁₃	C₂⁴	semidirect
50. C₄×Dic₁₃	C₂⁴	semidirect
58. C₂³×C₁₃	#117	semidirect
66. C₂×C₄×C₁₃	C₂³×C₄	semidirect
73. C₂⁴×C₄	Dic₁₃	semidirect
83. C₂²×C₁₃	C₂×#117	semidirect
99. C₂³×C₄	C₂×Dic₁₃	semidirect
108. C₂×C₁₃	C₂²×#117	semidirect
122. C₂²×C₄	C₂²×Dic₁₃	semidirect
129. C₁₃	C₂³×#117	semidirect
132. C₂³	#88180	direct (N × H)
142. C₂×C₄	C₂³×Dic₁₃	semidirect
147. C₂²	C₂×#88180	direct (N × H)
156. C₂	C₂²×#88180	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×#88180	2
C₂³×C₄⋊Dic₁₃	2
C₂³×C₂²⋊Dic₁₃	2
C₂³×C₄×Dic₁₃	2
C₂⁵×C₄×C₁₃	2
C₂³×#117	13

Subgroups — C23×#88180 #16769479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88180 #16769479