Subgroups — #13128713

Order 1664.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719430	C₂	1	4	yes	maximal, complemented
3	#11719403	C₂	1	4	yes	maximal, complemented
4	#11719335	C₂	1	8	yes	maximal, complemented
5	#11719290	C₂	1	4	yes	maximal, complemented
6	#11719282	C₂	1	4	yes	maximal, complemented
7	#11719230	C₂	1	2	yes	maximal, complemented
8	#11719226	C₂	1	2	yes	maximal, complemented
9	C₂×#88128	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#6127	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#6131	C₂²	1	8	yes	complemented
12	C₂×#6127	C₂²	1	1	yes	characteristic
13	C₂²×D₅₂	C₂²	1	2	yes	—
14	C₂²×D₅₂	C₂²	1	4	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
16	C₁₃×#125	C₂²	1	2	yes	complemented
17	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₄×Q₈×C₁₃	C₂²	1	4	yes	complemented
19	C₁₃×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
21	#88201	C₂²	1	8	yes	complemented
22	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
23	#88180	C₂²	1	1	yes	characteristic
24	#88149	C₂²	1	8	yes	complemented
25	#88149	C₂²	1	8	yes	complemented
26	#88149	C₂²	1	16	yes	complemented
27	#88148	C₂²	1	16	yes	complemented
28	#88147	C₂²	1	8	yes	complemented
29	#88147	C₂²	1	8	yes	complemented
30	#88136	C₂²	1	8	yes	—
31	C₂²⋊C₄×D₁₃	C₂²	1	8	yes	—
32	#88129	C₂²	1	4	yes	complemented
33	#88128	C₂²	1	2	yes	—
34	#88128	C₂²	1	4	yes	—
35	#88128	C₂²	1	2	yes	—
36	#88128	C₂²	1	4	yes	complemented
37	C₄×D₅₂	C₂²	1	2	yes	complemented
38	C₄×D₅₂	C₂²	1	8	yes	complemented
39	#88126	C₂²	1	2	yes	complemented
40	C₄²×D₁₃	C₂²	1	2	yes	complemented
41	C₄×Dic₂₆	C₂²	1	2	yes	complemented
42	C₂³×D₁₃	C₂³	1	4	yes	—
43	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
47	#6131	—	4	16	no	—
48	#6127	—	2	4	no	—
49	C₂×D₅₂	—	2	8	no	—
50	C₂×D₅₂	—	4	8	no	—
51	C₂×D₅₂	C₂³	1	4	yes	—
52	C₂×D₅₂	C₂³	1	1	yes	characteristic
53	C₂×D₅₂	C₂³	1	4	yes	—
54	C₂×D₅₂	C₂³	1	8	yes	—
55	C₂×D₅₂	C₂³	1	8	yes	—
56	C₂×D₅₂	C₂³	1	16	yes	—
57	C₂×C₄×D₁₃	C₂³	1	2	yes	—
58	C₂×C₄×D₁₃	C₂³	1	4	yes	—
59	C₂×C₄×D₁₃	C₂³	1	16	yes	—
60	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
64	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
65	C₄²×C₁₃	C₂³	1	2	yes	—
66	C₄²×C₁₃	C₂³	1	1	yes	characteristic
67	C₄²×C₁₃	C₂³	1	1	yes	characteristic
68	C₄²×C₁₃	C₂³	1	4	yes	—
69	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
70	#6103	C₂³	1	4	yes	—
71	#6103	C₂³	1	16	yes	—
72	#6103	C₂³	1	16	yes	—
73	C₄⋊Dic₁₃	C₂³	1	2	yes	—
74	#6101	C₂³	1	4	yes	—
75	C₄×Dic₁₃	C₂³	1	4	yes	—
76	C₄×Dic₁₃	C₂³	1	2	yes	—
77	#3508	—	13	1	no	maximal, Sylow
78	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×D₁₃	—	2	8	no	—
80	C₂²×D₁₃	—	4	8	no	—
81	C₂²×D₁₃	C₂⁴	1	8	yes	—
82	C₂²×D₁₃	C₂⁴	1	2	yes	—
83	Q₈×C₁₃	—	2	8	no	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	#41	1	4	yes	complemented
86	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
88	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
91	C₂²⋊D₁₃	—	2	4	no	—
92	C₂×Dic₁₃	C₂⁴	1	2	yes	—
93	C₂×Dic₁₃	C₂⁴	1	4	yes	—
94	D₅₂	—	2	8	no	—
95	D₅₂	—	2	8	no	—
96	D₅₂	—	2	2	no	—
97	D₅₂	—	2	16	no	—
98	D₅₂	—	4	8	no	—
99	D₅₂	—	4	16	no	—
100	C₄×D₁₃	—	2	4	no	—
101	C₄×D₁₃	—	2	8	no	—
102	C₄×D₁₃	—	4	16	no	—
103	Dic₂₆	—	2	2	no	—
104	#559	—	13	4	no	—
105	#550	—	13	4	no	—
106	#546	—	13	8	no	—
107	#540	—	13	4	no	—
108	#535	—	13	2	no	—
109	#533	D₁₃	1	1	yes	characteristic, complemented
110	#532	—	13	4	no	—
111	C₂×#127	—	13	1	no	—
112	#518	—	13	2	no	—
113	C₄×#41	—	13	1	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	C₂×#41	1	2	yes	—
116	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₃	—	2	2	no	—
118	C₂×D₁₃	—	2	8	no	—
119	C₂×D₁₃	—	2	4	no	—
120	C₂×D₁₃	—	2	16	no	—
121	C₂×D₁₃	—	8	8	no	—
122	C₄×C₁₃	—	2	8	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	C₂×#41	1	2	yes	—
125	C₄×C₁₃	C₂×#41	1	2	yes	—
126	C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
127	Dic₁₃	—	2	2	no	—
128	Dic₁₃	—	2	4	no	—
129	C₂×#41	—	13	8	no	—
130	C₂×#41	—	13	1	no	—
131	C₂²×D₄	—	13	2	no	—
132	C₂²×D₄	—	13	4	no	—
133	C₄⋊Q₈	C₂×D₁₃	1	2	yes	complemented
134	#127	—	13	2	no	—
135	#127	—	13	2	no	—
136	#127	—	13	4	no	—
137	#127	—	13	4	no	—
138	#125	C₂×D₁₃	1	2	yes	complemented
139	#124	—	13	8	no	—
140	#124	—	13	4	no	—
141	#123	—	13	16	no	—
142	C₂²⋊Q₈	C₂×D₁₃	1	4	yes	—
143	#121	—	13	8	no	—
144	#121	—	13	8	no	—
145	#121	—	13	16	no	—
146	#120	—	13	8	no	—
147	C₄×Q₈	—	13	2	no	—
148	C₄×Q₈	C₂×D₁₃	1	4	yes	complemented
149	C₄×D₄	—	13	4	no	—
150	C₄×D₄	—	13	8	no	—
151	C₄×D₄	—	13	8	no	—
152	C₄×D₄	—	13	2	no	—
153	C₄×D₄	—	13	8	no	—
154	#117	—	13	1	no	—
155	#117	—	13	2	no	—
156	#117	C₂×D₁₃	1	2	yes	—
157	C₂×C₂²⋊C₄	—	13	8	no	—
158	C₂×C₄²	—	13	2	no	—
159	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
160	C₂×C₁₃	—	2	2	no	—
161	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
162	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
163	D₁₃	—	4	2	no	—
164	D₁₃	—	4	8	no	—
165	C₂⁴	—	13	4	no	—
166	#41	—	26	4	no	—
167	#41	—	52	16	no	—
168	C₂×Q₈	—	13	1	no	—
169	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
170	C₂×D₄	—	13	2	no	—
171	C₂×D₄	—	13	4	no	—
172	C₂×D₄	—	13	4	no	—
173	C₂×D₄	—	13	1	no	—
174	C₂×D₄	—	13	8	no	—
175	C₂×D₄	—	13	16	no	—
176	C₂×D₄	—	13	8	no	—
177	C₂×D₄	—	26	8	no	—
178	C₂×D₄	—	52	8	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	4	no	—
181	C₂²×C₄	—	13	16	no	—
182	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₄⋊C₄	—	13	2	no	—
185	C₄⋊C₄	—	13	4	no	—
186	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
187	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
188	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
189	C₂²⋊C₄	—	13	2	no	—
190	C₂²⋊C₄	—	13	16	no	—
191	C₂²⋊C₄	—	13	4	no	—
192	C₂²⋊C₄	—	13	16	no	—
193	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
194	C₄²	—	13	2	no	—
195	C₄²	—	13	4	no	—
196	C₄²	C₂²×D₁₃	1	1	yes	characteristic
197	C₄²	C₂²×D₁₃	1	1	yes	characteristic
198	C₄²	C₂²×D₁₃	1	2	yes	—
199	C₄²	C₂²×D₁₃	1	4	yes	—
200	C₁₃	#3508	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	13	8	no	—
202	C₂³	—	13	2	no	—
203	C₂³	—	26	8	no	—
204	C₂³	—	52	8	no	—
205	C₂³	C₂³×D₁₃	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	Q₈	—	26	2	no	—
208	D₄	—	26	4	no	—
209	D₄	—	26	8	no	—
210	D₄	—	26	8	no	—
211	D₄	—	26	2	no	—
212	D₄	—	26	16	no	—
213	D₄	—	52	8	no	—
214	D₄	—	52	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	26	4	no	—
219	C₂×C₄	—	26	8	no	—
220	C₂×C₄	—	52	16	no	—
221	C₂×C₄	#6131	1	4	yes	complemented
222	C₂×C₄	C₂³×D₁₃	1	2	yes	—
223	C₂×C₄	C₂³×D₁₃	1	8	yes	—
224	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₁₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	26	2	no	—
229	C₂²	—	26	8	no	—
230	C₂²	—	26	4	no	—
231	C₂²	—	26	16	no	—
232	C₂²	—	104	8	no	—
233	C₂²	C₂×#6131	1	2	yes	—
234	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	8	no	—
237	C₄	—	26	2	no	—
238	C₄	—	26	4	no	—
239	C₄	C₂×#6131	1	2	yes	—
240	C₄	C₂×#6131	1	2	yes	—
241	C₄	#88256	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	52	8	no	—
244	C₂	—	52	2	no	—
245	C₂	C₂²×#6131	1	1	yes	characteristic
246	C₂	C₂×#88256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3508
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
2. #11719430	C₂	semidirect
3. #11719403	C₂	semidirect
4. #11719335	C₂	semidirect
5. #11719290	C₂	semidirect
6. #11719282	C₂	semidirect
7. #11719230	C₂	semidirect
8. #11719226	C₂	semidirect
9. C₂×#88128	C₂	semidirect
10. C₄×#6127	C₂	semidirect
11. C₂×#6131	C₂²	semidirect
15. C₁₃×C₄⋊Q₈	C₂²	semidirect
16. C₁₃×#125	C₂²	semidirect
18. C₄×Q₈×C₁₃	C₂²	semidirect
21. #88201	C₂²	semidirect
24. #88149	C₂²	semidirect
25. #88149	C₂²	semidirect
26. #88149	C₂²	semidirect
27. #88148	C₂²	semidirect
28. #88147	C₂²	semidirect
29. #88147	C₂²	semidirect
32. #88129	C₂²	semidirect
36. #88128	C₂²	semidirect
37. C₄×D₅₂	C₂²	semidirect
38. C₄×D₅₂	C₂²	semidirect
39. #88126	C₂²	semidirect
40. C₄²×D₁₃	C₂²	semidirect
41. C₄×Dic₂₆	C₂²	semidirect
85. C₂×C₄×C₁₃	#41	semidirect
109. #533	D₁₃	semidirect
133. C₄⋊Q₈	C₂×D₁₃	semidirect
138. #125	C₂×D₁₃	semidirect
148. C₄×Q₈	C₂×D₁₃	semidirect
200. C₁₃	#3508	semidirect
221. C₂×C₄	#6131	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
#11719430	2
#11719403	2
#11719335	2
#11719290	2
#11719282	2
#11719230	2
#11719226	2
C₂×#88128	2
C₄×#6127	2
#3508	13