Subgroups — C₂³×#88137 #16769566

Order 3328.

15150 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88137	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#6101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂²⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#6103	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₂²⋊D₁₃	—	2	16	no	—
14	C₂³×C₂²⋊D₁₃	C₂²	1	28	yes	—
15	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂³×C₄×D₁₃	—	2	16	no	—
19	C₂³×C₄×D₁₃	C₂²	1	28	yes	complemented
20	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	28	yes	complemented
21	C₂²×C₂²⋊Dic₁₃	C₂²	1	28	yes	complemented
22	C₂²×#6103	C₂²	1	28	yes	complemented
23	C₂²×C₄⋊Dic₁₃	C₂²	1	28	yes	complemented
24	C₂²×#6101	C₂²	1	28	yes	complemented
25	C₂×#88137	C₂²	1	448	yes	direct product factor
26	C₂⁵×C₁₃	—	2	8	no	—
27	C₂⁵×C₁₃	—	2	8	no	—
28	C₂⁵×C₁₃	—	2	8	no	—
29	C₂⁵×C₁₃	C₂³	1	14	yes	—
30	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
31	C₂⁴×D₁₃	—	2	8	no	—
32	C₂⁴×D₁₃	—	2	8	no	—
33	C₂⁴×D₁₃	D₄	1	16	yes	—
34	C₂⁴×D₁₃	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₃	—	2	8	no	—
36	C₂³×C₄×C₁₃	—	2	8	no	—
37	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
39	C₂²×C₂²⋊D₁₃	—	2	224	no	—
40	C₂²×C₂²⋊D₁₃	C₂³	1	112	yes	—
41	C₂³×Dic₁₃	—	2	8	no	—
42	C₂³×Dic₁₃	—	2	8	no	—
43	C₂³×Dic₁₃	—	2	8	no	—
44	C₂³×Dic₁₃	C₂³	1	14	yes	—
45	C₂³×Dic₁₃	C₂³	1	14	yes	—
46	C₂³×Dic₁₃	C₂³	1	14	yes	—
47	C₂²×C₄×D₁₃	—	2	224	no	—
48	C₂²×C₄×D₁₃	C₂³	1	112	yes	complemented
49	C₂×C₁₃×C₂²⋊C₄	C₂³	1	112	yes	complemented
50	C₂×C₂²⋊Dic₁₃	C₂³	1	112	yes	complemented
51	C₂×#6103	C₂³	1	112	yes	complemented
52	C₂×C₄⋊Dic₁₃	C₂³	1	112	yes	complemented
53	C₂×#6101	C₂³	1	112	yes	complemented
54	#88137	C₂³	1	512	yes	direct product factor
55	C₂³×#123	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	—	2	56	no	—
57	C₂⁴×C₁₃	—	2	56	no	—
58	C₂⁴×C₁₃	—	2	56	no	—
59	C₂⁴×C₁₃	—	4	64	no	—
60	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
61	C₂⁴×C₁₃	#41	1	8	yes	—
62	C₂⁴×C₁₃	#41	1	8	yes	—
63	C₂⁴×C₁₃	C₂⁴	1	28	yes	—
64	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
65	C₂³×D₁₃	—	2	128	no	—
66	C₂³×D₁₃	—	2	56	no	—
67	C₂³×D₁₃	—	2	56	no	—
68	C₂³×D₁₃	C₂×D₄	1	112	yes	—
69	C₂³×D₁₃	C₂⁴	1	28	yes	—
70	C₂²×C₄×C₁₃	—	2	56	no	—
71	C₂²×C₄×C₁₃	—	2	56	no	—
72	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
74	C₂×C₂²⋊D₁₃	—	2	448	no	—
75	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	—
76	C₂²×Dic₁₃	—	2	56	no	—
77	C₂²×Dic₁₃	—	2	56	no	—
78	C₂²×Dic₁₃	—	2	56	no	—
79	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
80	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
81	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
82	C₂×C₄×D₁₃	—	2	448	no	—
83	C₂×C₄×D₁₃	C₂⁴	1	64	yes	complemented
84	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
85	C₂²⋊Dic₁₃	C₂⁴	1	64	yes	complemented
86	#6103	C₂⁴	1	64	yes	complemented
87	C₄⋊Dic₁₃	C₂⁴	1	64	yes	complemented
88	#6101	C₂⁴	1	64	yes	complemented
89	C₂⁴×D₄	—	13	1	no	—
90	C₂⁵×C₄	—	13	1	no	—
91	C₂²×#123	—	13	56	no	—
92	C₂³×C₄⋊C₄	—	13	1	no	—
93	C₂³×C₄⋊C₄	—	13	1	no	—
94	C₂³×C₂²⋊C₄	—	13	1	no	—
95	C₂³×C₂²⋊C₄	—	13	1	no	—
96	C₂³×C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
97	C₂³×C₁₃	—	2	56	no	—
98	C₂³×C₁₃	—	2	56	no	—
99	C₂³×C₁₃	—	2	56	no	—
100	C₂³×C₁₃	—	4	224	no	—
101	C₂³×C₁₃	#123	1	64	yes	complemented
102	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
103	C₂³×C₁₃	C₂×#41	1	28	yes	—
104	C₂³×C₁₃	C₂×#41	1	28	yes	—
105	C₂³×C₁₃	C₂⁵	1	8	yes	—
106	C₂³×C₁₃	C₂⁵	1	7	yes	—
107	C₂²×D₁₃	—	2	448	no	—
108	C₂²×D₁₃	—	2	56	no	—
109	C₂²×D₁₃	—	2	56	no	—
110	C₂²×D₁₃	C₂²×D₄	1	112	yes	—
111	C₂²×D₁₃	C₂⁵	1	8	yes	—
112	C₂×C₄×C₁₃	—	2	56	no	—
113	C₂×C₄×C₁₃	—	2	56	no	—
114	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
115	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
116	C₂²⋊D₁₃	—	2	128	no	—
117	C₂×Dic₁₃	—	2	56	no	—
118	C₂×Dic₁₃	—	2	56	no	—
119	C₂×Dic₁₃	—	2	56	no	—
120	C₂×Dic₁₃	C₂⁵	1	8	yes	—
121	C₂×Dic₁₃	C₂⁵	1	8	yes	—
122	C₂×Dic₁₃	C₂⁵	1	8	yes	—
123	C₄×D₁₃	—	2	128	no	—
124	C₂⁶	—	13	1	no	—
125	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
126	C₂³×D₄	—	13	28	no	—
127	C₂³×D₄	—	26	16	no	—
128	C₂⁴×C₄	—	13	1	no	—
129	C₂⁴×C₄	—	13	1	no	—
130	C₂⁴×C₄	—	13	1	no	—
131	C₂⁴×C₄	—	13	28	no	—
132	C₂⁴×C₄	—	26	16	no	—
133	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
135	C₂×#123	—	13	448	no	—
136	C₂²×C₄⋊C₄	—	13	28	no	—
137	C₂²×C₄⋊C₄	—	13	28	no	—
138	C₂²×C₂²⋊C₄	—	13	28	no	—
139	C₂²×C₂²⋊C₄	—	13	28	no	—
140	C₂²×C₂²⋊C₄	C₂×D₁₃	1	28	yes	complemented
141	C₂²×C₁₃	—	2	8	no	—
142	C₂²×C₁₃	—	2	8	no	—
143	C₂²×C₁₃	—	2	8	no	—
144	C₂²×C₁₃	—	4	112	no	—
145	C₂²×C₁₃	C₂×#123	1	112	yes	complemented
146	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
147	C₂²×C₁₃	C₂²×#41	1	14	yes	—
148	C₂²×C₁₃	C₂²×#41	1	14	yes	—
149	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
150	C₂×D₁₃	—	2	8	no	—
151	C₂×D₁₃	—	2	8	no	—
152	C₂×D₁₃	—	2	224	no	—
153	C₂×D₁₃	C₂³×D₄	1	16	yes	—
154	C₄×C₁₃	—	2	8	no	—
155	C₄×C₁₃	—	2	8	no	—
156	Dic₁₃	—	2	8	no	—
157	Dic₁₃	—	2	8	no	—
158	Dic₁₃	—	2	8	no	—
159	C₂⁵	—	2	8	no	—
160	C₂⁵	—	2	8	no	—
161	C₂⁵	—	2	8	no	—
162	C₂⁵	—	13	16	no	—
163	C₂⁵	—	13	14	no	—
164	C₂⁵	—	26	8	no	—
165	C₂⁵	—	26	8	no	—
166	C₂⁵	C₂²×D₁₃	1	14	yes	—
167	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
168	C₂²×D₄	—	13	112	no	—
169	C₂²×D₄	—	26	224	no	—
170	C₂³×C₄	—	2	8	no	—
171	C₂³×C₄	—	2	8	no	—
172	C₂³×C₄	—	13	14	no	—
173	C₂³×C₄	—	13	14	no	—
174	C₂³×C₄	—	13	14	no	—
175	C₂³×C₄	—	13	112	no	—
176	C₂³×C₄	—	26	224	no	—
177	C₂³×C₄	—	26	8	no	—
178	C₂³×C₄	—	26	8	no	—
179	C₂³×C₄	—	26	8	no	—
180	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
181	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
182	#123	—	13	512	no	—
183	C₂×C₄⋊C₄	—	13	112	no	—
184	C₂×C₄⋊C₄	—	13	112	no	—
185	C₂×C₂²⋊C₄	—	13	112	no	—
186	C₂×C₂²⋊C₄	—	13	112	no	—
187	C₂×C₂²⋊C₄	C₂²×D₁₃	1	112	yes	complemented
188	C₂×C₁₃	—	4	8	no	—
189	C₂×C₁₃	C₂²×#123	1	28	yes	complemented
190	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
191	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
192	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
193	D₁₃	—	2	16	no	—
194	C₂⁴	—	2	56	no	—
195	C₂⁴	—	2	56	no	—
196	C₂⁴	—	2	56	no	—
197	C₂⁴	—	4	64	no	—
198	C₂⁴	—	13	112	no	—
199	C₂⁴	—	13	28	no	—
200	C₂⁴	—	26	128	no	—
201	C₂⁴	—	26	56	no	—
202	C₂⁴	—	26	56	no	—
203	C₂⁴	#6127	1	8	yes	—
204	C₂⁴	D₄×D₁₃	1	8	yes	—
205	C₂⁴	#6129	1	8	yes	—
206	C₂⁴	C₂³×D₁₃	1	28	yes	—
207	C₂⁴	C₂³×D₁₃	1	7	yes	—
208	C₂×D₄	—	13	64	no	—
209	C₂×D₄	—	26	448	no	—
210	C₂²×C₄	—	2	56	no	—
211	C₂²×C₄	—	2	56	no	—
212	C₂²×C₄	—	13	28	no	—
213	C₂²×C₄	—	13	28	no	—
214	C₂²×C₄	—	13	28	no	—
215	C₂²×C₄	—	13	64	no	—
216	C₂²×C₄	—	26	448	no	—
217	C₂²×C₄	—	26	56	no	—
218	C₂²×C₄	—	26	56	no	—
219	C₂²×C₄	—	26	56	no	—
220	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
221	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
222	C₄⋊C₄	—	13	64	no	—
223	C₄⋊C₄	—	13	64	no	—
224	C₂²⋊C₄	—	13	64	no	—
225	C₂²⋊C₄	—	13	64	no	—
226	C₂²⋊C₄	C₂³×D₁₃	1	64	yes	complemented
227	C₁₃	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	2	56	no	—
229	C₂³	—	2	56	no	—
230	C₂³	—	2	56	no	—
231	C₂³	—	4	224	no	—
232	C₂³	—	13	112	no	—
233	C₂³	—	13	8	no	—
234	C₂³	—	26	448	no	—
235	C₂³	—	26	56	no	—
236	C₂³	—	26	56	no	—
237	C₂³	#88137	1	64	yes	direct product factor
238	C₂³	C₂×#6127	1	28	yes	—
239	C₂³	C₂×D₄×D₁₃	1	28	yes	—
240	C₂³	C₂×#6129	1	28	yes	—
241	C₂³	C₂⁴×D₁₃	1	8	yes	—
242	C₂³	C₂⁴×D₁₃	1	7	yes	—
243	D₄	—	26	128	no	—
244	C₂×C₄	—	2	56	no	—
245	C₂×C₄	—	2	56	no	—
246	C₂×C₄	—	13	8	no	—
247	C₂×C₄	—	13	8	no	—
248	C₂×C₄	—	13	8	no	—
249	C₂×C₄	—	26	128	no	—
250	C₂×C₄	—	26	56	no	—
251	C₂×C₄	—	26	56	no	—
252	C₂×C₄	—	26	56	no	—
253	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
254	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	4	112	no	—
259	C₂²	—	13	16	no	—
260	C₂²	—	26	224	no	—
261	C₂²	—	26	8	no	—
262	C₂²	—	26	8	no	—
263	C₂²	C₂×#88137	1	112	yes	direct product factor
264	C₂²	C₂²×#6127	1	14	yes	—
265	C₂²	C₂²×D₄×D₁₃	1	14	yes	—
266	C₂²	C₂²×#6129	1	14	yes	—
267	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	2	8	no	—
270	C₄	—	26	8	no	—
271	C₄	—	26	8	no	—
272	C₄	—	26	8	no	—
273	C₂	—	4	8	no	—
274	C₂	—	26	16	no	—
275	C₂	C₂²×#88137	1	28	yes	direct product factor
276	C₂	C₂³×D₄×D₁₃	1	1	yes	characteristic
277	C₂	C₂³×#6129	1	1	yes	characteristic
278	C₂	C₂³×#6127	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
prime radical	C₂⁴×C₂²⋊D₁₃	C₂
hypercenter	C₂³×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×#123
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
2. C₂⁴×C₄×D₁₃	C₂	semidirect
3. C₂²×#88137	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₃	C₂	semidirect
5. C₂³×#6101	C₂	semidirect
6. C₂³×C₂²⋊Dic₁₃	C₂	semidirect
7. C₂³×#6103	C₂	semidirect
8. C₂³×C₁₃×C₂²⋊C₄	C₂	semidirect
19. C₂³×C₄×D₁₃	C₂²	semidirect
20. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
21. C₂²×C₂²⋊Dic₁₃	C₂²	semidirect
22. C₂²×#6103	C₂²	semidirect
23. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
24. C₂²×#6101	C₂²	semidirect
25. C₂×#88137	C₂²	direct (N × H)
48. C₂²×C₄×D₁₃	C₂³	semidirect
49. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
50. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
51. C₂×#6103	C₂³	semidirect
52. C₂×C₄⋊Dic₁₃	C₂³	semidirect
53. C₂×#6101	C₂³	semidirect
54. #88137	C₂³	direct (N × H)
83. C₂×C₄×D₁₃	C₂⁴	semidirect
84. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
85. C₂²⋊Dic₁₃	C₂⁴	semidirect
86. #6103	C₂⁴	semidirect
87. C₄⋊Dic₁₃	C₂⁴	semidirect
88. #6101	C₂⁴	semidirect
96. C₂³×C₂²⋊C₄	D₁₃	semidirect
101. C₂³×C₁₃	#123	semidirect
140. C₂²×C₂²⋊C₄	C₂×D₁₃	semidirect
145. C₂²×C₁₃	C₂×#123	semidirect
187. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
189. C₂×C₁₃	C₂²×#123	semidirect
226. C₂²⋊C₄	C₂³×D₁₃	semidirect
227. C₁₃	C₂³×#123	semidirect
237. C₂³	#88137	direct (N × H)
263. C₂²	C₂×#88137	direct (N × H)
275. C₂	C₂²×#88137	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×C₂²⋊D₁₃	2
C₂⁴×C₄×D₁₃	2
C₂²×#88137	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6101	2
C₂³×C₂²⋊Dic₁₃	2
C₂³×#6103	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂³×#123	13

Subgroups — C23×#88137 #16769566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88137 #16769566