Subgroups — C₂³×#88126 #16769477

Order 3328.

14334 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88126	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#6101	C₂	1	2	yes	maximal, complemented
4	C₂³×#6103	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
11	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×D₁₃	C₄	1	32	yes	complemented
13	C₂³×C₄×D₁₃	C₂²	1	28	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	28	yes	complemented
15	C₂²×#6103	C₂²	1	56	yes	—
16	C₂²×#6101	C₂²	1	56	yes	complemented
17	C₂²×C₄×Dic₁₃	C₂²	1	28	yes	complemented
18	C₂×#88126	C₂²	1	448	yes	direct product factor
19	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
20	C₂⁴×D₁₃	—	2	16	no	—
21	C₂⁴×D₁₃	C₂×C₄	1	16	yes	—
22	C₂⁴×D₁₃	C₂³	1	14	yes	—
23	C₂³×C₄×C₁₃	—	2	16	no	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	28	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
27	C₂³×Dic₁₃	—	2	16	no	—
28	C₂³×Dic₁₃	C₂×C₄	1	16	yes	—
29	C₂³×Dic₁₃	C₂³	1	28	yes	—
30	C₂³×Dic₁₃	C₂³	1	14	yes	—
31	C₂²×C₄×D₁₃	C₂×C₄	1	448	yes	complemented
32	C₂²×C₄×D₁₃	C₂³	1	112	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	112	yes	complemented
34	C₂×#6103	C₂³	1	224	yes	—
35	C₂×#6101	C₂³	1	224	yes	complemented
36	C₂×C₄×Dic₁₃	C₂³	1	112	yes	complemented
37	#88126	C₂³	1	512	yes	direct product factor
38	C₂³×#117	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
40	C₂⁴×C₁₃	#41	1	16	yes	—
41	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
42	C₂³×D₁₃	—	2	128	no	—
43	C₂³×D₁₃	—	2	112	no	—
44	C₂³×D₁₃	C₂²×C₄	1	112	yes	—
45	C₂³×D₁₃	C₂⁴	1	28	yes	—
46	C₂²×C₄×C₁₃	—	2	112	no	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	112	yes	—
48	C₂²×C₄×C₁₃	C₂⁴	1	56	yes	—
49	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
50	C₂²×Dic₁₃	—	2	112	no	—
51	C₂²×Dic₁₃	C₂²×C₄	1	112	yes	—
52	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
53	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
54	C₂×C₄×D₁₃	C₂²×C₄	1	896	yes	complemented
55	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
56	C₄²×C₁₃	C₂⁴	1	64	yes	complemented
57	#6103	C₂⁴	1	128	yes	—
58	#6101	C₂⁴	1	128	yes	complemented
59	C₄×Dic₁₃	C₂⁴	1	64	yes	complemented
60	C₂⁵×C₄	—	13	1	no	—
61	C₂²×#117	—	13	56	no	—
62	C₂³×C₄⋊C₄	—	13	2	no	—
63	C₂³×C₂²⋊C₄	—	13	2	no	—
64	C₂³×C₄²	—	13	1	no	—
65	C₂³×C₄²	D₁₃	1	1	yes	characteristic, complemented
66	C₂³×C₁₃	#117	1	64	yes	complemented
67	C₂³×C₁₃	C₂³×C₄	1	28	yes	—
68	C₂³×C₁₃	C₂×#41	1	56	yes	—
69	C₂³×C₁₃	C₂⁵	1	7	yes	—
70	C₂²×D₁₃	—	2	448	no	—
71	C₂²×D₁₃	—	2	112	no	—
72	C₂²×D₁₃	C₂³×C₄	1	112	yes	—
73	C₂²×D₁₃	C₂⁵	1	8	yes	—
74	C₂×C₄×C₁₃	—	2	112	no	—
75	C₂×C₄×C₁₃	C₂³×C₄	1	112	yes	—
76	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
77	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
78	C₂×Dic₁₃	—	2	112	no	—
79	C₂×Dic₁₃	C₂³×C₄	1	112	yes	—
80	C₂×Dic₁₃	C₂⁵	1	16	yes	—
81	C₂×Dic₁₃	C₂⁵	1	8	yes	—
82	C₄×D₁₃	C₂³×C₄	1	256	yes	complemented
83	C₂⁶	—	13	1	no	—
84	C₂⁴×C₄	—	13	1	no	—
85	C₂⁴×C₄	—	13	32	no	—
86	C₂⁴×C₄	—	13	28	no	—
87	C₂⁴×C₄	—	13	2	no	—
88	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
89	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
90	C₂×#117	—	13	448	no	—
91	C₂²×C₄⋊C₄	—	13	56	no	—
92	C₂²×C₂²⋊C₄	—	13	56	no	—
93	C₂²×C₄²	—	13	28	no	—
94	C₂²×C₄²	C₂×D₁₃	1	28	yes	complemented
95	C₂²×C₁₃	C₂×#117	1	112	yes	complemented
96	C₂²×C₁₃	C₂⁴×C₄	1	14	yes	—
97	C₂²×C₁₃	C₂²×#41	1	28	yes	—
98	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
99	C₂×D₁₃	—	2	16	no	—
100	C₂×D₁₃	—	2	224	no	—
101	C₂×D₁₃	C₂⁴×C₄	1	16	yes	—
102	C₄×C₁₃	—	2	16	no	—
103	C₄×C₁₃	C₂⁴×C₄	1	16	yes	—
104	Dic₁₃	—	2	16	no	—
105	Dic₁₃	C₂⁴×C₄	1	16	yes	—
106	C₂⁵	—	13	16	no	—
107	C₂⁵	—	13	14	no	—
108	C₂⁵	—	26	16	no	—
109	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
110	C₂³×C₄	—	2	16	no	—
111	C₂³×C₄	—	13	16	no	—
112	C₂³×C₄	—	13	14	no	—
113	C₂³×C₄	—	13	448	no	—
114	C₂³×C₄	—	13	112	no	—
115	C₂³×C₄	—	13	28	no	—
116	C₂³×C₄	—	26	16	no	—
117	C₂³×C₄	C₄×D₁₃	1	16	yes	—
118	C₂³×C₄	C₂²×D₁₃	1	28	yes	—
119	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
120	#117	—	13	512	no	—
121	C₂×C₄⋊C₄	—	13	224	no	—
122	C₂×C₂²⋊C₄	—	13	224	no	—
123	C₂×C₄²	—	13	112	no	—
124	C₂×C₄²	C₂²×D₁₃	1	112	yes	complemented
125	C₂×C₁₃	C₂²×#117	1	28	yes	complemented
126	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
127	C₂×C₁₃	C₂³×#41	1	2	yes	—
128	D₁₃	—	2	16	no	—
129	C₂⁴	—	13	112	no	—
130	C₂⁴	—	13	28	no	—
131	C₂⁴	—	26	128	no	—
132	C₂⁴	—	26	112	no	—
133	C₂⁴	C₂×C₄×D₁₃	1	8	yes	—
134	C₂⁴	#6127	1	16	yes	—
135	C₂⁴	C₂³×D₁₃	1	7	yes	—
136	C₂²×C₄	—	2	112	no	—
137	C₂²×C₄	—	13	112	no	—
138	C₂²×C₄	—	13	28	no	—
139	C₂²×C₄	—	13	896	no	—
140	C₂²×C₄	—	13	64	no	—
141	C₂²×C₄	—	13	56	no	—
142	C₂²×C₄	—	26	112	no	—
143	C₂²×C₄	C₂×C₄×D₁₃	1	112	yes	—
144	C₂²×C₄	C₂³×D₁₃	1	56	yes	—
145	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
146	C₄⋊C₄	—	13	128	no	—
147	C₂²⋊C₄	—	13	128	no	—
148	C₄²	—	13	64	no	—
149	C₄²	C₂³×D₁₃	1	64	yes	complemented
150	C₁₃	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
151	C₂³	—	13	112	no	—
152	C₂³	—	13	8	no	—
153	C₂³	—	26	448	no	—
154	C₂³	—	26	112	no	—
155	C₂³	#88126	1	64	yes	direct product factor
156	C₂³	C₂²×C₄×D₁₃	1	28	yes	—
157	C₂³	C₂×#6127	1	56	yes	—
158	C₂³	C₂⁴×D₁₃	1	7	yes	—
159	C₂×C₄	—	2	112	no	—
160	C₂×C₄	—	13	112	no	—
161	C₂×C₄	—	13	8	no	—
162	C₂×C₄	—	13	256	no	—
163	C₂×C₄	—	13	16	no	—
164	C₂×C₄	—	26	112	no	—
165	C₂×C₄	C₂²×C₄×D₁₃	1	112	yes	—
166	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
167	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
168	C₂²	—	13	16	no	—
169	C₂²	—	26	224	no	—
170	C₂²	—	26	16	no	—
171	C₂²	C₂×#88126	1	112	yes	direct product factor
172	C₂²	C₂³×C₄×D₁₃	1	14	yes	—
173	C₂²	C₂²×#6127	1	28	yes	—
174	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
175	C₄	—	2	16	no	—
176	C₄	—	13	16	no	—
177	C₄	—	26	16	no	—
178	C₄	C₂³×C₄×D₁₃	1	16	yes	—
179	C₂	—	26	16	no	—
180	C₂	C₂²×#88126	1	28	yes	direct product factor
181	C₂	C₂⁴×C₄×D₁₃	1	1	yes	characteristic
182	C₂	C₂³×#6127	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle	C₂⁵×C₁₃	C₂³
prime radical	C₂⁵×D₁₃	C₂²
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂³×#117
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×#88126	C₂	direct (N × H)
3. C₂³×#6101	C₂	semidirect
5. C₂³×C₄×Dic₁₃	C₂	semidirect
6. C₂³×C₄²×C₁₃	C₂	semidirect
12. C₂³×C₄×D₁₃	C₄	semidirect
14. C₂²×C₄²×C₁₃	C₂²	semidirect
16. C₂²×#6101	C₂²	semidirect
17. C₂²×C₄×Dic₁₃	C₂²	semidirect
18. C₂×#88126	C₂²	direct (N × H)
31. C₂²×C₄×D₁₃	C₂×C₄	semidirect
33. C₂×C₄²×C₁₃	C₂³	semidirect
35. C₂×#6101	C₂³	semidirect
36. C₂×C₄×Dic₁₃	C₂³	semidirect
37. #88126	C₂³	direct (N × H)
54. C₂×C₄×D₁₃	C₂²×C₄	semidirect
56. C₄²×C₁₃	C₂⁴	semidirect
58. #6101	C₂⁴	semidirect
59. C₄×Dic₁₃	C₂⁴	semidirect
65. C₂³×C₄²	D₁₃	semidirect
66. C₂³×C₁₃	#117	semidirect
82. C₄×D₁₃	C₂³×C₄	semidirect
94. C₂²×C₄²	C₂×D₁₃	semidirect
95. C₂²×C₁₃	C₂×#117	semidirect
124. C₂×C₄²	C₂²×D₁₃	semidirect
125. C₂×C₁₃	C₂²×#117	semidirect
149. C₄²	C₂³×D₁₃	semidirect
150. C₁₃	C₂³×#117	semidirect
155. C₂³	#88126	direct (N × H)
171. C₂²	C₂×#88126	direct (N × H)
180. C₂	C₂²×#88126	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₄×D₁₃	2
C₂²×#88126	2
C₂³×#6101	2
C₂³×#6103	2
C₂³×C₄×Dic₁₃	2
C₂³×C₄²×C₁₃	2
C₂³×#117	13

Subgroups — C23×#88126 #16769477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88126 #16769477