Subgroups — C₂³×#88146 #16769478

Order 3328.

14334 conjugacy classes of subgroups.

The subgroup lattice has 200 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×#88146	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#6103	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₃×C₄⋊C₄	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×C₁₃×C₄⋊C₄	C₂²	1	28	yes	complemented
15	C₂²×#6103	C₂²	1	56	yes	—
16	C₂²×C₄⋊Dic₁₃	C₂²	1	28	yes	complemented
17	C₂²×C₄×Dic₁₃	C₂²	1	56	yes	complemented
18	C₂×#88146	C₂²	1	448	yes	direct product factor
19	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
20	C₂⁴×D₁₃	—	2	8	no	—
21	C₂⁴×D₁₃	—	2	8	no	—
22	C₂⁴×D₁₃	C₂×C₄	1	16	yes	—
23	C₂⁴×D₁₃	C₂³	1	14	yes	—
24	C₂³×C₄×C₁₃	—	2	16	no	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	28	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
28	C₂³×Dic₁₃	—	2	16	no	—
29	C₂³×Dic₁₃	C₂×C₄	1	16	yes	—
30	C₂³×Dic₁₃	C₂³	1	28	yes	—
31	C₂³×Dic₁₃	C₂³	1	14	yes	—
32	C₂²×C₄×D₁₃	C₂×C₄	1	448	yes	complemented
33	C₂²×C₄×D₁₃	C₂³	1	112	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂³	1	112	yes	complemented
35	C₂×#6103	C₂³	1	224	yes	—
36	C₂×C₄⋊Dic₁₃	C₂³	1	112	yes	complemented
37	C₂×C₄×Dic₁₃	C₂³	1	224	yes	complemented
38	#88146	C₂³	1	512	yes	direct product factor
39	C₂³×#117	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
41	C₂⁴×C₁₃	#41	1	8	yes	—
42	C₂⁴×C₁₃	#41	1	8	yes	—
43	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
44	C₂³×D₁₃	—	2	128	no	—
45	C₂³×D₁₃	—	2	56	no	—
46	C₂³×D₁₃	—	2	56	no	—
47	C₂³×D₁₃	C₂²×C₄	1	112	yes	—
48	C₂³×D₁₃	C₂⁴	1	28	yes	—
49	C₂²×C₄×C₁₃	—	2	112	no	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	112	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	56	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
53	C₂²×Dic₁₃	—	2	112	no	—
54	C₂²×Dic₁₃	C₂²×C₄	1	112	yes	—
55	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
56	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
57	C₂×C₄×D₁₃	C₂²×C₄	1	896	yes	complemented
58	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
59	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	complemented
60	#6103	C₂⁴	1	128	yes	—
61	C₄⋊Dic₁₃	C₂⁴	1	64	yes	complemented
62	C₄×Dic₁₃	C₂⁴	1	128	yes	complemented
63	C₂⁵×C₄	—	13	1	no	—
64	C₂²×#117	—	13	56	no	—
65	C₂³×C₄⋊C₄	—	13	1	no	—
66	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
67	C₂³×C₂²⋊C₄	—	13	2	no	—
68	C₂³×C₄²	—	13	2	no	—
69	C₂³×C₁₃	#117	1	64	yes	complemented
70	C₂³×C₁₃	C₂³×C₄	1	28	yes	—
71	C₂³×C₁₃	C₂×#41	1	28	yes	—
72	C₂³×C₁₃	C₂×#41	1	28	yes	—
73	C₂³×C₁₃	C₂⁵	1	7	yes	—
74	C₂²×D₁₃	—	2	448	no	—
75	C₂²×D₁₃	—	2	56	no	—
76	C₂²×D₁₃	—	2	56	no	—
77	C₂²×D₁₃	C₂³×C₄	1	112	yes	—
78	C₂²×D₁₃	C₂⁵	1	8	yes	—
79	C₂×C₄×C₁₃	—	2	112	no	—
80	C₂×C₄×C₁₃	C₂³×C₄	1	112	yes	—
81	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
82	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
83	C₂×Dic₁₃	—	2	112	no	—
84	C₂×Dic₁₃	C₂³×C₄	1	112	yes	—
85	C₂×Dic₁₃	C₂⁵	1	16	yes	—
86	C₂×Dic₁₃	C₂⁵	1	8	yes	—
87	C₄×D₁₃	C₂³×C₄	1	256	yes	complemented
88	C₂⁶	—	13	1	no	—
89	C₂⁴×C₄	—	13	1	no	—
90	C₂⁴×C₄	—	13	32	no	—
91	C₂⁴×C₄	—	13	28	no	—
92	C₂⁴×C₄	—	13	2	no	—
93	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
94	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
95	C₂×#117	—	13	448	no	—
96	C₂²×C₄⋊C₄	—	13	28	no	—
97	C₂²×C₄⋊C₄	C₂×D₁₃	1	28	yes	complemented
98	C₂²×C₂²⋊C₄	—	13	56	no	—
99	C₂²×C₄²	—	13	56	no	—
100	C₂²×C₁₃	C₂×#117	1	112	yes	complemented
101	C₂²×C₁₃	C₂⁴×C₄	1	14	yes	—
102	C₂²×C₁₃	C₂²×#41	1	14	yes	—
103	C₂²×C₁₃	C₂²×#41	1	14	yes	—
104	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
105	C₂×D₁₃	—	2	8	no	—
106	C₂×D₁₃	—	2	8	no	—
107	C₂×D₁₃	—	2	224	no	—
108	C₂×D₁₃	C₂⁴×C₄	1	16	yes	—
109	C₄×C₁₃	—	2	16	no	—
110	C₄×C₁₃	C₂⁴×C₄	1	16	yes	—
111	Dic₁₃	—	2	16	no	—
112	Dic₁₃	C₂⁴×C₄	1	16	yes	—
113	C₂⁵	—	13	16	no	—
114	C₂⁵	—	13	14	no	—
115	C₂⁵	—	26	8	no	—
116	C₂⁵	—	26	8	no	—
117	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
118	C₂³×C₄	—	2	16	no	—
119	C₂³×C₄	—	13	16	no	—
120	C₂³×C₄	—	13	14	no	—
121	C₂³×C₄	—	13	448	no	—
122	C₂³×C₄	—	13	112	no	—
123	C₂³×C₄	—	13	28	no	—
124	C₂³×C₄	—	26	16	no	—
125	C₂³×C₄	C₄×D₁₃	1	16	yes	—
126	C₂³×C₄	C₂²×D₁₃	1	28	yes	—
127	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
128	#117	—	13	512	no	—
129	C₂×C₄⋊C₄	—	13	112	no	—
130	C₂×C₄⋊C₄	C₂²×D₁₃	1	112	yes	complemented
131	C₂×C₂²⋊C₄	—	13	224	no	—
132	C₂×C₄²	—	13	224	no	—
133	C₂×C₁₃	C₂²×#117	1	28	yes	complemented
134	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
135	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
136	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
137	D₁₃	—	2	16	no	—
138	C₂⁴	—	13	112	no	—
139	C₂⁴	—	13	28	no	—
140	C₂⁴	—	26	128	no	—
141	C₂⁴	—	26	56	no	—
142	C₂⁴	—	26	56	no	—
143	C₂⁴	C₂×C₄×D₁₃	1	8	yes	—
144	C₂⁴	#6129	1	8	yes	—
145	C₂⁴	#6131	1	8	yes	—
146	C₂⁴	C₂³×D₁₃	1	7	yes	—
147	C₂²×C₄	—	2	112	no	—
148	C₂²×C₄	—	13	112	no	—
149	C₂²×C₄	—	13	28	no	—
150	C₂²×C₄	—	13	896	no	—
151	C₂²×C₄	—	13	64	no	—
152	C₂²×C₄	—	13	56	no	—
153	C₂²×C₄	—	26	112	no	—
154	C₂²×C₄	C₂×C₄×D₁₃	1	112	yes	—
155	C₂²×C₄	C₂³×D₁₃	1	56	yes	—
156	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
157	C₄⋊C₄	—	13	64	no	—
158	C₄⋊C₄	C₂³×D₁₃	1	64	yes	complemented
159	C₂²⋊C₄	—	13	128	no	—
160	C₄²	—	13	128	no	—
161	C₁₃	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
162	C₂³	—	13	112	no	—
163	C₂³	—	13	8	no	—
164	C₂³	—	26	448	no	—
165	C₂³	—	26	56	no	—
166	C₂³	—	26	56	no	—
167	C₂³	#88146	1	64	yes	direct product factor
168	C₂³	C₂²×C₄×D₁₃	1	28	yes	—
169	C₂³	C₂×#6129	1	28	yes	—
170	C₂³	C₂×#6131	1	28	yes	—
171	C₂³	C₂⁴×D₁₃	1	7	yes	—
172	C₂×C₄	—	2	112	no	—
173	C₂×C₄	—	13	112	no	—
174	C₂×C₄	—	13	8	no	—
175	C₂×C₄	—	13	256	no	—
176	C₂×C₄	—	13	16	no	—
177	C₂×C₄	—	26	112	no	—
178	C₂×C₄	C₂²×C₄×D₁₃	1	112	yes	—
179	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
180	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
181	C₂²	—	13	16	no	—
182	C₂²	—	26	224	no	—
183	C₂²	—	26	8	no	—
184	C₂²	—	26	8	no	—
185	C₂²	C₂×#88146	1	112	yes	direct product factor
186	C₂²	C₂³×C₄×D₁₃	1	14	yes	—
187	C₂²	C₂²×#6129	1	14	yes	—
188	C₂²	C₂²×#6131	1	14	yes	—
189	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
190	C₄	—	2	16	no	—
191	C₄	—	13	16	no	—
192	C₄	—	26	16	no	—
193	C₄	C₂³×C₄×D₁₃	1	16	yes	—
194	C₂	—	26	16	no	—
195	C₂	C₂²×#88146	1	28	yes	direct product factor
196	C₂	C₂⁴×C₄×D₁₃	1	1	yes	characteristic
197	C₂	C₂³×#6129	1	1	yes	characteristic
198	C₂	C₂³×#6131	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×#88146	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₁₃	C₂	semidirect
5. C₂³×C₄×Dic₁₃	C₂	semidirect
6. C₂³×C₁₃×C₄⋊C₄	C₂	semidirect
12. C₂³×C₄×D₁₃	C₄	semidirect
14. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
16. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
17. C₂²×C₄×Dic₁₃	C₂²	semidirect
18. C₂×#88146	C₂²	direct (N × H)
32. C₂²×C₄×D₁₃	C₂×C₄	semidirect
34. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
36. C₂×C₄⋊Dic₁₃	C₂³	semidirect
37. C₂×C₄×Dic₁₃	C₂³	semidirect
38. #88146	C₂³	direct (N × H)
57. C₂×C₄×D₁₃	C₂²×C₄	semidirect
59. C₁₃×C₄⋊C₄	C₂⁴	semidirect
61. C₄⋊Dic₁₃	C₂⁴	semidirect
62. C₄×Dic₁₃	C₂⁴	semidirect
66. C₂³×C₄⋊C₄	D₁₃	semidirect
69. C₂³×C₁₃	#117	semidirect
87. C₄×D₁₃	C₂³×C₄	semidirect
97. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
100. C₂²×C₁₃	C₂×#117	semidirect
130. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
133. C₂×C₁₃	C₂²×#117	semidirect
158. C₄⋊C₄	C₂³×D₁₃	semidirect
161. C₁₃	C₂³×#117	semidirect
167. C₂³	#88146	direct (N × H)
185. C₂²	C₂×#88146	direct (N × H)
195. C₂	C₂²×#88146	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×#88146	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6103	2
C₂³×C₄×Dic₁₃	2
C₂³×C₁₃×C₄⋊C₄	2
C₂³×#117	13

Subgroups — C23×#88146 #16769478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88146 #16769478