Subgroups — C₂³×#88136 #16769551

Order 3328.

42862 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₅₂	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#88136	C₂	1	56	yes	maximal, direct product factor
5	C₂³×#6103	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁵×D₁₃	—	2	16	no	—
9	C₂⁵×D₁₃	—	2	32	no	—
10	C₂⁵×D₁₃	C₂²	1	2	yes	—
11	C₂⁵×D₁₃	C₂²	1	28	yes	complemented
12	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic, complemented
13	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	complemented
14	C₂³×C₂²⋊D₁₃	—	2	16	no	—
15	C₂³×C₂²⋊D₁₃	C₂²	1	28	yes	complemented
16	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₅₂	—	2	32	no	—
18	C₂³×D₅₂	C₂²	1	56	yes	complemented
19	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	28	yes	complemented
20	C₂²×#6103	C₂²	1	56	yes	complemented
21	C₂×#88136	C₂²	1	448	yes	direct product factor
22	C₂⁵×C₁₃	—	2	16	no	—
23	C₂⁵×C₁₃	D₄	1	16	yes	complemented
24	C₂⁵×C₁₃	C₂³	1	14	yes	—
25	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
26	C₂⁴×D₁₃	—	2	512	no	—
27	C₂⁴×D₁₃	—	2	16	no	—
28	C₂⁴×D₁₃	—	2	16	no	—
29	C₂⁴×D₁₃	—	2	224	no	—
30	C₂⁴×D₁₃	—	2	448	no	—
31	C₂⁴×D₁₃	—	2	16	no	—
32	C₂⁴×D₁₃	—	2	8	no	—
33	C₂⁴×D₁₃	D₄	1	32	yes	complemented
34	C₂⁴×D₁₃	C₂³	1	28	yes	—
35	C₂⁴×D₁₃	C₂³	1	112	yes	complemented
36	C₂⁴×D₁₃	C₂³	1	14	yes	complemented
37	C₂³×C₄×C₁₃	—	2	16	no	—
38	C₂³×C₄×C₁₃	C₂³	1	28	yes	complemented
39	C₂²×C₂²⋊D₁₃	—	2	224	no	—
40	C₂²×C₂²⋊D₁₃	C₂³	1	112	yes	complemented
41	C₂³×Dic₁₃	—	2	8	no	—
42	C₂³×Dic₁₃	C₂³	1	14	yes	complemented
43	C₂²×D₅₂	—	2	448	no	—
44	C₂²×D₅₂	C₂³	1	224	yes	complemented
45	C₂×C₁₃×C₂²⋊C₄	C₂³	1	112	yes	complemented
46	C₂×#6103	C₂³	1	224	yes	complemented
47	#88136	C₂³	1	512	yes	direct product factor
48	C₂³×#120	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	—	2	128	no	—
50	C₂⁴×C₁₃	—	2	112	no	—
51	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
52	C₂⁴×C₁₃	C₂×D₄	1	16	yes	—
53	C₂⁴×C₁₃	C₂×D₄	1	112	yes	complemented
54	C₂⁴×C₁₃	C₂⁴	1	28	yes	—
55	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
56	C₂³×D₁₃	—	2	3584	no	—
57	C₂³×D₁₃	—	2	256	no	—
58	C₂³×D₁₃	—	2	112	no	—
59	C₂³×D₁₃	—	2	112	no	—
60	C₂³×D₁₃	—	2	448	no	—
61	C₂³×D₁₃	—	2	896	no	—
62	C₂³×D₁₃	—	2	112	no	—
63	C₂³×D₁₃	—	2	56	no	—
64	C₂³×D₁₃	—	4	64	no	—
65	C₂³×D₁₃	C₂×D₄	1	224	yes	complemented
66	C₂³×D₁₃	C₂⁴	1	56	yes	—
67	C₂³×D₁₃	C₂⁴	1	64	yes	complemented
68	C₂³×D₁₃	C₂⁴	1	28	yes	complemented
69	C₂²×C₄×C₁₃	—	2	112	no	—
70	C₂²×C₄×C₁₃	C₂⁴	1	56	yes	complemented
71	C₂×C₂²⋊D₁₃	—	2	448	no	—
72	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	complemented
73	C₂²×Dic₁₃	—	2	56	no	—
74	C₂²×Dic₁₃	C₂⁴	1	28	yes	complemented
75	C₂×D₅₂	—	2	896	no	—
76	C₂×D₅₂	C₂⁴	1	128	yes	complemented
77	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
78	#6103	C₂⁴	1	128	yes	complemented
79	C₂⁷	—	13	1	no	—
80	C₂⁴×D₄	—	13	1	no	—
81	C₂⁴×D₄	—	13	2	no	—
82	C₂²×#120	—	13	56	no	—
83	C₂³×C₂²⋊C₄	—	13	2	no	—
84	C₂³×C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
85	C₂³×C₁₃	—	2	448	no	—
86	C₂³×C₁₃	—	2	112	no	—
87	C₂³×C₁₃	#120	1	64	yes	complemented
88	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
89	C₂³×C₁₃	C₂²×D₄	1	56	yes	—
90	C₂³×C₁₃	C₂²×D₄	1	112	yes	complemented
91	C₂³×C₁₃	C₂⁵	1	8	yes	—
92	C₂³×C₁₃	C₂⁵	1	7	yes	—
93	C₂²×D₁₃	—	2	3584	no	—
94	C₂²×D₁₃	—	2	896	no	—
95	C₂²×D₁₃	—	2	112	no	—
96	C₂²×D₁₃	—	2	112	no	—
97	C₂²×D₁₃	—	2	128	no	—
98	C₂²×D₁₃	—	2	256	no	—
99	C₂²×D₁₃	—	2	112	no	—
100	C₂²×D₁₃	—	2	56	no	—
101	C₂²×D₁₃	—	4	224	no	—
102	C₂²×D₁₃	C₂²×D₄	1	224	yes	complemented
103	C₂²×D₁₃	C₂⁵	1	16	yes	—
104	C₂²×D₁₃	C₂⁵	1	8	yes	complemented
105	C₂×C₄×C₁₃	—	2	112	no	—
106	C₂×C₄×C₁₃	C₂⁵	1	16	yes	complemented
107	C₂²⋊D₁₃	—	2	128	no	—
108	C₂×Dic₁₃	—	2	56	no	—
109	C₂×Dic₁₃	C₂⁵	1	8	yes	complemented
110	D₅₂	—	2	256	no	—
111	C₂⁶	—	13	2	no	—
112	C₂⁶	—	13	28	no	—
113	C₂⁶	—	13	1	no	—
114	C₂⁶	—	26	32	no	—
115	C₂⁶	—	26	16	no	—
116	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
117	C₂³×D₄	—	13	28	no	—
118	C₂³×D₄	—	13	56	no	—
119	C₂³×D₄	—	26	32	no	—
120	C₂³×D₄	—	26	16	no	—
121	C₂⁴×C₄	—	13	1	no	—
122	C₂⁴×C₄	C₂×D₁₃	1	2	yes	complemented
123	C₂×#120	—	13	448	no	—
124	C₂²×C₂²⋊C₄	—	13	56	no	—
125	C₂²×C₂²⋊C₄	C₂×D₁₃	1	28	yes	complemented
126	C₂²×C₁₃	—	2	16	no	—
127	C₂²×C₁₃	—	2	224	no	—
128	C₂²×C₁₃	C₂×#120	1	112	yes	complemented
129	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
130	C₂²×C₁₃	C₂³×D₄	1	28	yes	—
131	C₂²×C₁₃	C₂³×D₄	1	16	yes	complemented
132	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
133	C₂×D₁₃	—	2	512	no	—
134	C₂×D₁₃	—	2	16	no	—
135	C₂×D₁₃	—	2	16	no	—
136	C₂×D₁₃	—	2	448	no	—
137	C₂×D₁₃	—	2	16	no	—
138	C₂×D₁₃	—	2	8	no	—
139	C₂×D₁₃	—	4	112	no	—
140	C₂×D₁₃	C₂³×D₄	1	32	yes	complemented
141	C₄×C₁₃	—	2	16	no	—
142	Dic₁₃	—	2	8	no	—
143	C₂⁵	—	2	16	no	—
144	C₂⁵	—	13	32	no	—
145	C₂⁵	—	13	28	no	—
146	C₂⁵	—	13	112	no	—
147	C₂⁵	—	13	14	no	—
148	C₂⁵	—	26	448	no	—
149	C₂⁵	—	26	512	no	—
150	C₂⁵	—	26	16	no	—
151	C₂⁵	—	26	224	no	—
152	C₂⁵	—	26	16	no	—
153	C₂⁵	—	26	8	no	—
154	C₂⁵	—	26	16	no	—
155	C₂⁵	D₅₂	1	16	yes	complemented
156	C₂⁵	C₂²×D₁₃	1	14	yes	—
157	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
158	C₂²×D₄	—	13	112	no	—
159	C₂²×D₄	—	13	224	no	—
160	C₂²×D₄	—	26	448	no	—
161	C₂²×D₄	—	26	224	no	—
162	C₂³×C₄	—	2	16	no	—
163	C₂³×C₄	—	13	14	no	—
164	C₂³×C₄	—	26	8	no	—
165	C₂³×C₄	C₂²×D₁₃	1	28	yes	complemented
166	#120	—	13	512	no	—
167	C₂×C₂²⋊C₄	—	13	224	no	—
168	C₂×C₂²⋊C₄	C₂²×D₁₃	1	112	yes	complemented
169	C₂×C₁₃	—	2	16	no	—
170	C₂×C₁₃	C₂²×#120	1	28	yes	complemented
171	C₂×C₁₃	C₂⁴×D₄	1	2	yes	—
172	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
173	D₁₃	—	2	32	no	—
174	D₁₃	—	4	8	no	—
175	C₂⁴	—	2	128	no	—
176	C₂⁴	—	2	112	no	—
177	C₂⁴	—	13	224	no	—
178	C₂⁴	—	13	56	no	—
179	C₂⁴	—	13	64	no	—
180	C₂⁴	—	13	28	no	—
181	C₂⁴	—	26	896	no	—
182	C₂⁴	—	26	3584	no	—
183	C₂⁴	—	26	256	no	—
184	C₂⁴	—	26	112	no	—
185	C₂⁴	—	26	448	no	—
186	C₂⁴	—	26	112	no	—
187	C₂⁴	—	26	56	no	—
188	C₂⁴	—	26	112	no	—
189	C₂⁴	—	52	64	no	—
190	C₂⁴	C₂×D₅₂	1	8	yes	—
191	C₂⁴	C₂×D₅₂	1	112	yes	complemented
192	C₂⁴	D₄×D₁₃	1	16	yes	—
193	C₂⁴	C₂³×D₁₃	1	28	yes	—
194	C₂⁴	C₂³×D₁₃	1	7	yes	—
195	C₂×D₄	—	13	64	no	—
196	C₂×D₄	—	13	128	no	—
197	C₂×D₄	—	26	896	no	—
198	C₂×D₄	—	26	448	no	—
199	C₂²×C₄	—	2	112	no	—
200	C₂²×C₄	—	13	28	no	—
201	C₂²×C₄	—	26	56	no	—
202	C₂²×C₄	C₂³×D₁₃	1	56	yes	complemented
203	C₂²⋊C₄	—	13	128	no	—
204	C₂²⋊C₄	C₂³×D₁₃	1	64	yes	complemented
205	C₁₃	C₂³×#120	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	2	448	no	—
207	C₂³	—	2	112	no	—
208	C₂³	—	13	224	no	—
209	C₂³	—	13	16	no	—
210	C₂³	—	13	8	no	—
211	C₂³	—	26	256	no	—
212	C₂³	—	26	3584	no	—
213	C₂³	—	26	896	no	—
214	C₂³	—	26	112	no	—
215	C₂³	—	26	128	no	—
216	C₂³	—	26	112	no	—
217	C₂³	—	26	56	no	—
218	C₂³	—	26	112	no	—
219	C₂³	—	52	224	no	—
220	C₂³	#88136	1	64	yes	direct product factor
221	C₂³	C₂²×D₅₂	1	28	yes	—
222	C₂³	C₂²×D₅₂	1	112	yes	complemented
223	C₂³	C₂×D₄×D₁₃	1	56	yes	—
224	C₂³	C₂⁴×D₁₃	1	8	yes	—
225	C₂³	C₂⁴×D₁₃	1	7	yes	—
226	D₄	—	26	256	no	—
227	D₄	—	26	128	no	—
228	C₂×C₄	—	2	112	no	—
229	C₂×C₄	—	13	8	no	—
230	C₂×C₄	—	26	56	no	—
231	C₂×C₄	C₂⁴×D₁₃	1	16	yes	complemented
232	C₂²	—	2	224	no	—
233	C₂²	—	2	16	no	—
234	C₂²	—	13	32	no	—
235	C₂²	—	26	512	no	—
236	C₂²	—	26	448	no	—
237	C₂²	—	26	16	no	—
238	C₂²	—	26	16	no	—
239	C₂²	—	26	16	no	—
240	C₂²	—	26	8	no	—
241	C₂²	—	52	112	no	—
242	C₂²	C₂×#88136	1	112	yes	direct product factor
243	C₂²	C₂³×D₅₂	1	14	yes	—
244	C₂²	C₂³×D₅₂	1	16	yes	complemented
245	C₂²	C₂²×D₄×D₁₃	1	28	yes	—
246	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
247	C₄	—	2	16	no	—
248	C₄	—	26	8	no	—
249	C₂	—	2	16	no	—
250	C₂	—	26	32	no	—
251	C₂	—	52	8	no	—
252	C₂	C₂²×#88136	1	28	yes	direct product factor
253	C₂	C₂³×D₄×D₁₃	1	2	yes	—
254	C₂	C₂⁴×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₂²⋊C₄	C₂
socle	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×#120
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₁₃	C₂	semidirect
2. C₂⁴×D₅₂	C₂	semidirect
3. C₂⁴×C₂²⋊D₁₃	C₂	semidirect
4. C₂²×#88136	C₂	direct (N × H)
5. C₂³×#6103	C₂	semidirect
6. C₂³×C₁₃×C₂²⋊C₄	C₂	semidirect
11. C₂⁵×D₁₃	C₂²	semidirect
12. C₂⁵×D₁₃	C₂²	semidirect
13. C₂⁴×C₄×C₁₃	C₂²	semidirect
15. C₂³×C₂²⋊D₁₃	C₂²	semidirect
16. C₂⁴×Dic₁₃	C₂²	semidirect
18. C₂³×D₅₂	C₂²	semidirect
19. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
20. C₂²×#6103	C₂²	semidirect
21. C₂×#88136	C₂²	direct (N × H)
23. C₂⁵×C₁₃	D₄	semidirect
33. C₂⁴×D₁₃	D₄	semidirect
35. C₂⁴×D₁₃	C₂³	semidirect
36. C₂⁴×D₁₃	C₂³	semidirect
38. C₂³×C₄×C₁₃	C₂³	semidirect
40. C₂²×C₂²⋊D₁₃	C₂³	semidirect
42. C₂³×Dic₁₃	C₂³	semidirect
44. C₂²×D₅₂	C₂³	semidirect
45. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
46. C₂×#6103	C₂³	semidirect
47. #88136	C₂³	direct (N × H)
53. C₂⁴×C₁₃	C₂×D₄	semidirect
65. C₂³×D₁₃	C₂×D₄	semidirect
67. C₂³×D₁₃	C₂⁴	semidirect
68. C₂³×D₁₃	C₂⁴	semidirect
70. C₂²×C₄×C₁₃	C₂⁴	semidirect
72. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
74. C₂²×Dic₁₃	C₂⁴	semidirect
76. C₂×D₅₂	C₂⁴	semidirect
77. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
78. #6103	C₂⁴	semidirect
84. C₂³×C₂²⋊C₄	D₁₃	semidirect
87. C₂³×C₁₃	#120	semidirect
90. C₂³×C₁₃	C₂²×D₄	semidirect
102. C₂²×D₁₃	C₂²×D₄	semidirect
104. C₂²×D₁₃	C₂⁵	semidirect
106. C₂×C₄×C₁₃	C₂⁵	semidirect
109. C₂×Dic₁₃	C₂⁵	semidirect
122. C₂⁴×C₄	C₂×D₁₃	semidirect
125. C₂²×C₂²⋊C₄	C₂×D₁₃	semidirect
128. C₂²×C₁₃	C₂×#120	semidirect
131. C₂²×C₁₃	C₂³×D₄	semidirect
140. C₂×D₁₃	C₂³×D₄	semidirect
155. C₂⁵	D₅₂	semidirect
165. C₂³×C₄	C₂²×D₁₃	semidirect
168. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
170. C₂×C₁₃	C₂²×#120	semidirect
191. C₂⁴	C₂×D₅₂	semidirect
202. C₂²×C₄	C₂³×D₁₃	semidirect
204. C₂²⋊C₄	C₂³×D₁₃	semidirect
205. C₁₃	C₂³×#120	semidirect
220. C₂³	#88136	direct (N × H)
222. C₂³	C₂²×D₅₂	semidirect
231. C₂×C₄	C₂⁴×D₁₃	semidirect
242. C₂²	C₂×#88136	direct (N × H)
244. C₂²	C₂³×D₅₂	semidirect
252. C₂	C₂²×#88136	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×D₁₃	2
C₂⁴×D₅₂	2
C₂⁴×C₂²⋊D₁₃	2
C₂²×#88136	2
C₂³×#6103	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂³×#120	13

Subgroups — C23×#88136 #16769551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88136 #16769551