Subgroups — C₂³×#88061 #16726428

Order 3328.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88061	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₁₃	—	2	1	no	—
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂⁴×Dic₁₃	—	2	1	no	—
8	C₂³×D₅₂	—	2	7	no	—
9	C₂³×D₅₂	—	2	8	no	—
10	C₂³×D₅₂	C₄	1	16	yes	complemented
11	C₂³×D₅₂	C₂²	1	14	yes	—
12	C₂³×C₈×C₁₃	—	2	8	no	—
13	C₂³×C₈×C₁₃	C₂²	1	14	yes	complemented
14	C₂²×C₄⋊Dic₁₃	—	2	7	no	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	14	yes	complemented
16	C₂×#88061	C₂²	1	112	yes	direct product factor
17	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
18	C₂⁴×D₁₃	—	2	16	no	—
19	C₂⁴×D₁₃	—	2	14	no	—
20	C₂⁴×D₁₃	—	4	8	no	—
21	C₂⁴×D₁₃	—	4	8	no	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₁₃	D₄	1	7	yes	—
24	C₂³×C₄×C₁₃	D₄	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
26	C₂³×Dic₁₃	—	2	14	no	—
27	C₂³×Dic₁₃	—	4	8	no	—
28	C₂²×D₅₂	—	2	112	no	—
29	C₂²×D₅₂	—	2	42	no	—
30	C₂²×D₅₂	—	2	56	no	—
31	C₂²×D₅₂	C₂×C₄	1	112	yes	complemented
32	C₂²×D₅₂	C₂³	1	28	yes	—
33	C₂²×C₈×C₁₃	—	2	56	no	—
34	C₂²×C₈×C₁₃	C₂³	1	28	yes	complemented
35	C₂×C₄⋊Dic₁₃	—	2	42	no	—
36	C₂×C₄⋊Dic₁₃	C₂³	1	28	yes	complemented
37	#88061	C₂³	1	64	yes	direct product factor
38	C₂³×#102	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	C₂²⋊C₄	1	8	yes	—
40	C₂⁴×C₁₃	D₈	1	8	yes	—
41	C₂⁴×C₁₃	SD₁₆	1	8	yes	—
42	C₂⁴×C₁₃	C₂×D₄	1	7	yes	—
43	C₂³×D₁₃	—	2	112	no	—
44	C₂³×D₁₃	—	2	28	no	—
45	C₂³×D₁₃	—	4	128	no	—
46	C₂³×D₁₃	—	4	56	no	—
47	C₂³×D₁₃	—	4	56	no	—
48	C₂²×C₄×C₁₃	C₂²⋊C₄	1	56	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	28	yes	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	28	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
53	C₂²×Dic₁₃	—	2	28	no	—
54	C₂²×Dic₁₃	—	4	56	no	—
55	C₂×D₅₂	—	2	336	no	—
56	C₂×D₅₂	—	2	28	no	—
57	C₂×D₅₂	—	2	56	no	—
58	C₂×D₅₂	C₂²×C₄	1	112	yes	complemented
59	C₂×D₅₂	C₂⁴	1	8	yes	—
60	C₂×C₈×C₁₃	—	2	56	no	—
61	C₂×C₈×C₁₃	C₂⁴	1	8	yes	complemented
62	C₄⋊Dic₁₃	—	2	28	no	—
63	C₄⋊Dic₁₃	C₂⁴	1	8	yes	complemented
64	C₂⁴×D₄	—	13	1	no	—
65	C₂⁴×C₈	D₁₃	1	1	yes	characteristic, complemented
66	C₂³×C₄⋊C₄	—	13	1	no	—
67	C₂²×#102	—	13	28	no	—
68	C₂³×C₁₃	#102	1	64	yes	complemented
69	C₂³×C₁₃	C₂×C₂²⋊C₄	1	28	yes	—
70	C₂³×C₁₃	C₂×D₈	1	28	yes	—
71	C₂³×C₁₃	C₂×SD₁₆	1	28	yes	—
72	C₂³×C₁₃	C₂²×D₄	1	7	yes	—
73	C₂²×D₁₃	—	2	112	no	—
74	C₂²×D₁₃	—	2	8	no	—
75	C₂²×D₁₃	—	4	56	no	—
76	C₂²×D₁₃	—	4	56	no	—
77	C₂²×D₁₃	—	4	448	no	—
78	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	84	yes	—
79	C₂×C₄×C₁₃	C₂³×C₄	1	14	yes	—
80	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
81	C₂×C₄×C₁₃	C₂²×D₄	1	14	yes	—
82	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	8	no	—
84	C₂×Dic₁₃	—	4	56	no	—
85	D₅₂	—	2	112	no	—
86	D₅₂	—	2	8	no	—
87	D₅₂	C₂³×C₄	1	16	yes	complemented
88	C₈×C₁₃	—	2	8	no	—
89	C₂⁶	—	26	1	no	—
90	C₂³×D₄	—	13	16	no	—
91	C₂³×D₄	—	13	14	no	—
92	C₂³×D₄	—	26	7	no	—
93	C₂³×D₄	—	26	8	no	—
94	C₂⁴×C₄	—	26	1	no	—
95	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
96	C₂³×C₈	—	2	8	no	—
97	C₂³×C₈	C₂×D₁₃	1	14	yes	complemented
98	C₂²×C₄⋊C₄	—	13	14	no	—
99	C₂²×C₄⋊C₄	—	26	7	no	—
100	C₂×#102	—	13	112	no	—
101	C₂²×C₁₃	C₂×#102	1	112	yes	complemented
102	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	14	yes	—
103	C₂²×C₁₃	C₂²×D₈	1	14	yes	—
104	C₂²×C₁₃	C₂²×SD₁₆	1	14	yes	—
105	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
106	C₂×D₁₃	—	2	16	no	—
107	C₂×D₁₃	—	4	224	no	—
108	C₂×D₁₃	—	4	8	no	—
109	C₂×D₁₃	—	4	8	no	—
110	C₄×C₁₃	C₂²×C₂²⋊C₄	1	14	yes	—
111	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
112	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
113	Dic₁₃	—	4	8	no	—
114	C₂⁵	—	26	16	no	—
115	C₂⁵	—	26	14	no	—
116	C₂⁵	—	52	8	no	—
117	C₂⁵	—	52	8	no	—
118	C₂⁵	D₅₂	1	1	yes	characteristic
119	C₂²×D₄	—	13	112	no	—
120	C₂²×D₄	—	13	28	no	—
121	C₂²×D₄	—	26	112	no	—
122	C₂²×D₄	—	26	42	no	—
123	C₂²×D₄	—	26	56	no	—
124	C₂³×C₄	—	26	14	no	—
125	C₂³×C₄	—	52	8	no	—
126	C₂³×C₄	C₄×D₁₃	1	8	yes	—
127	C₂³×C₄	D₅₂	1	7	yes	—
128	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
129	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
130	C₂²×C₈	—	2	56	no	—
131	C₂²×C₈	C₂²×D₁₃	1	28	yes	complemented
132	C₂×C₄⋊C₄	—	13	28	no	—
133	C₂×C₄⋊C₄	—	26	42	no	—
134	#102	—	13	64	no	—
135	C₂×C₁₃	C₂²×#102	1	28	yes	complemented
136	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
138	C₂×C₁₃	C₂³×SD₁₆	1	1	yes	characteristic
139	D₁₃	—	4	16	no	—
140	C₂⁴	—	26	112	no	—
141	C₂⁴	—	26	28	no	—
142	C₂⁴	—	52	56	no	—
143	C₂⁴	—	52	56	no	—
144	C₂⁴	—	52	128	no	—
145	C₂⁴	#6095	1	8	yes	—
146	C₂⁴	D₁₀₄	1	8	yes	—
147	C₂⁴	#6103	1	8	yes	—
148	C₂⁴	C₂×D₅₂	1	7	yes	—
149	C₂×D₄	—	13	112	no	—
150	C₂×D₄	—	13	8	no	—
151	C₂×D₄	—	26	336	no	—
152	C₂×D₄	—	26	28	no	—
153	C₂×D₄	—	26	56	no	—
154	C₂²×C₄	—	26	28	no	—
155	C₂²×C₄	—	52	56	no	—
156	C₂²×C₄	#6103	1	56	yes	—
157	C₂²×C₄	C₂×C₄×D₁₃	1	28	yes	—
158	C₂²×C₄	C₂×D₅₂	1	21	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₁₃	1	28	yes	—
160	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
161	C₂×C₈	—	2	56	no	—
162	C₂×C₈	C₂³×D₁₃	1	8	yes	complemented
163	C₄⋊C₄	—	13	8	no	—
164	C₄⋊C₄	—	26	28	no	—
165	C₁₃	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	26	112	no	—
167	C₂³	—	26	8	no	—
168	C₂³	—	52	56	no	—
169	C₂³	—	52	56	no	—
170	C₂³	—	52	448	no	—
171	C₂³	#88061	1	64	yes	direct product factor
172	C₂³	C₂×#6095	1	28	yes	—
173	C₂³	C₂×D₁₀₄	1	28	yes	—
174	C₂³	C₂×#6103	1	28	yes	—
175	C₂³	C₂²×D₅₂	1	7	yes	—
176	D₄	—	13	16	no	—
177	D₄	—	26	112	no	—
178	D₄	—	26	8	no	—
179	C₂×C₄	—	26	8	no	—
180	C₂×C₄	—	52	56	no	—
181	C₂×C₄	C₂×#6103	1	84	yes	—
182	C₂×C₄	C₂²×C₄×D₁₃	1	14	yes	—
183	C₂×C₄	C₂²×D₅₂	1	7	yes	—
184	C₂×C₄	C₂²×C₂²⋊D₁₃	1	14	yes	—
185	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
186	C₈	—	2	8	no	—
187	C₂²	—	26	16	no	—
188	C₂²	—	52	8	no	—
189	C₂²	—	52	8	no	—
190	C₂²	—	52	224	no	—
191	C₂²	C₂×#88061	1	112	yes	direct product factor
192	C₂²	C₂²×#6095	1	14	yes	—
193	C₂²	C₂²×D₁₀₄	1	14	yes	—
194	C₂²	C₂²×#6103	1	14	yes	—
195	C₂²	C₂³×D₅₂	1	1	yes	characteristic
196	C₄	—	52	8	no	—
197	C₄	C₂²×#6103	1	14	yes	—
198	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
199	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
200	C₂	—	52	16	no	—
201	C₂	C₂²×#88061	1	28	yes	direct product factor
202	C₂	C₂³×#6103	1	1	yes	characteristic
203	C₂	C₂³×D₁₀₄	1	1	yes	characteristic
204	C₂	C₂³×#6095	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂⁵×C₁₃	D₄
prime radical	C₂⁴×D₅₂	C₂
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂³×#102
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊Dic₁₃	C₂	semidirect
3. C₂²×#88061	C₂	direct (N × H)
4. C₂⁴×C₈×C₁₃	C₂	semidirect
10. C₂³×D₅₂	C₄	semidirect
13. C₂³×C₈×C₁₃	C₂²	semidirect
15. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
16. C₂×#88061	C₂²	direct (N × H)
31. C₂²×D₅₂	C₂×C₄	semidirect
34. C₂²×C₈×C₁₃	C₂³	semidirect
36. C₂×C₄⋊Dic₁₃	C₂³	semidirect
37. #88061	C₂³	direct (N × H)
58. C₂×D₅₂	C₂²×C₄	semidirect
61. C₂×C₈×C₁₃	C₂⁴	semidirect
63. C₄⋊Dic₁₃	C₂⁴	semidirect
65. C₂⁴×C₈	D₁₃	semidirect
68. C₂³×C₁₃	#102	semidirect
87. D₅₂	C₂³×C₄	semidirect
97. C₂³×C₈	C₂×D₁₃	semidirect
101. C₂²×C₁₃	C₂×#102	semidirect
131. C₂²×C₈	C₂²×D₁₃	semidirect
135. C₂×C₁₃	C₂²×#102	semidirect
162. C₂×C₈	C₂³×D₁₃	semidirect
165. C₁₃	C₂³×#102	semidirect
171. C₂³	#88061	direct (N × H)
191. C₂²	C₂×#88061	direct (N × H)
201. C₂	C₂²×#88061	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×D₅₂	2
C₂³×C₄⋊Dic₁₃	2
C₂²×#88061	2
C₂⁴×C₈×C₁₃	2
C₂³×#102	13

Subgroups — C23×#88061 #16726428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88061 #16726428