Subgroups — #13128239

Order 1664.

996 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#11719434	C₂	1	4	yes	maximal, complemented
3	#11719394	C₂	1	4	yes	maximal, complemented
4	#11719339	C₂	1	4	yes	maximal
5	#11719328	C₂	1	4	yes	maximal, complemented
6	#11719283	C₂	1	8	yes	maximal, complemented
7	D₁₃×#117	C₂	1	2	yes	maximal
8	#11719222	C₂	1	2	yes	maximal, complemented
9	C₂×#88126	C₂	1	1	yes	maximal, characteristic
10	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
13	C₁₃×#125	C₂²	1	2	yes	complemented
14	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
15	C₄×Q₈×C₁₃	C₂²	1	4	yes	complemented
16	C₁₃×#117	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	#88200	C₂²	1	8	yes	—
19	#88198	C₂²	1	8	yes	complemented
20	C₂×#6103	C₂²	1	2	yes	—
21	#88180	C₂²	1	2	yes	—
22	#88178	C₂²	1	4	yes	—
23	C₂×#6101	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	#88151	C₂²	1	8	yes	—
26	#88150	C₂²	1	8	yes	—
27	#88150	C₂²	1	8	yes	—
28	#88146	C₂²	1	4	yes	—
29	C₄⋊C₄×D₁₃	C₂²	1	4	yes	—
30	#88144	C₂²	1	4	yes	complemented
31	#88143	C₂²	1	8	yes	complemented
32	#88143	C₂²	1	8	yes	complemented
33	#88142	C₂²	1	4	yes	complemented
34	#88142	C₂²	1	8	yes	complemented
35	C₂²⋊C₄×D₁₃	C₂²	1	4	yes	—
36	#88132	C₂²	1	8	yes	—
37	#88131	C₂²	1	4	yes	—
38	#88126	C₂²	1	2	yes	—
39	#88126	C₂²	1	4	yes	—
40	#88126	C₂²	1	2	yes	—
41	#88126	C₂²	1	4	yes	—
42	C₄²×D₁₃	C₂²	1	4	yes	—
43	#88124	C₂²	1	2	yes	complemented
44	#88123	C₂²	1	2	yes	complemented
45	C₄×Dic₂₆	C₂²	1	4	yes	complemented
46	C₄×Dic₂₆	C₂²	1	8	yes	complemented
47	C₂³×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂³	1	2	yes	—
52	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₃	Q₈	1	8	yes	complemented
54	C₂×C₄×D₁₃	C₂³	1	8	yes	—
55	C₂×C₄×D₁₃	C₂³	1	2	yes	—
56	C₂×C₄×D₁₃	C₂³	1	2	yes	—
57	C₂×Dic₂₆	C₂³	1	4	yes	—
58	C₂×Dic₂₆	C₂³	1	8	yes	—
59	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
60	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
61	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
62	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
63	C₄²×C₁₃	C₂³	1	4	yes	—
64	C₄²×C₁₃	C₂³	1	2	yes	—
65	C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	C₄²×C₁₃	C₂³	1	1	yes	characteristic
67	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
68	#6103	C₂³	1	4	yes	—
69	#6103	C₂³	1	4	yes	—
70	#6103	C₂³	1	8	yes	—
71	C₄⋊Dic₁₃	C₂³	1	4	yes	—
72	C₄⋊Dic₁₃	C₂³	1	8	yes	—
73	C₄⋊Dic₁₃	C₂³	1	4	yes	—
74	#6101	C₂³	1	8	yes	—
75	#6101	C₂³	1	8	yes	—
76	#6101	C₂³	1	8	yes	—
77	#6101	C₂³	1	4	yes	—
78	#6101	C₂³	1	4	yes	—
79	#6101	C₂³	1	8	yes	—
80	C₄×Dic₁₃	C₂³	1	4	yes	—
81	C₄×Dic₁₃	C₂³	1	2	yes	—
82	C₄×Dic₁₃	C₂³	1	8	yes	—
83	C₄×Dic₁₃	C₂³	1	2	yes	—
84	#3458	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²×D₁₃	—	4	2	no	—
87	C₂²×D₁₃	C₂×Q₈	1	4	yes	—
88	C₂²×D₁₃	C₂⁴	1	2	yes	—
89	Q₈×C₁₃	—	4	4	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
92	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
93	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
94	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
95	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₂×Dic₁₃	—	4	2	no	—
98	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
99	C₂×Dic₁₃	C₂⁴	1	4	yes	—
100	C₂×Dic₁₃	C₂⁴	1	8	yes	—
101	C₂×Dic₁₃	C₂⁴	1	2	yes	—
102	C₄×D₁₃	—	4	8	no	—
103	C₄×D₁₃	C₂×Q₈	1	8	yes	—
104	C₄×D₁₃	C₂×Q₈	1	8	yes	—
105	Dic₂₆	—	4	4	no	—
106	Dic₂₆	—	4	8	no	—
107	#544	—	13	4	no	—
108	#544	—	13	4	no	—
109	#544	—	13	4	no	—
110	#543	—	13	4	no	—
111	#533	—	13	8	no	—
112	#533	—	13	2	no	—
113	#533	—	13	1	no	—
114	#533	D₁₃	1	1	yes	characteristic, complemented
115	C₂×#117	—	13	2	no	—
116	C₂×#117	—	13	1	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
119	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
120	C₂×D₁₃	—	2	4	no	—
121	C₂×D₁₃	—	4	4	no	—
122	C₂×D₁₃	C₂²×Q₈	1	4	yes	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	8	no	—
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
127	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
128	Dic₁₃	—	2	8	no	—
129	Dic₁₃	—	2	4	no	—
130	Dic₁₃	C₂²×Q₈	1	4	yes	—
131	C₂³×C₄	—	13	1	no	—
132	C₄⋊Q₈	—	13	8	no	—
133	C₄⋊Q₈	—	13	8	no	—
134	C₄⋊Q₈	—	13	4	no	—
135	C₄⋊Q₈	—	13	2	no	—
136	C₄⋊Q₈	C₂×D₁₃	1	2	yes	complemented
137	#125	—	13	8	no	—
138	#125	—	13	8	no	—
139	#125	—	13	4	no	—
140	#125	—	13	2	no	—
141	#125	C₂×D₁₃	1	2	yes	complemented
142	C₂²⋊Q₈	—	13	8	no	—
143	C₂²⋊Q₈	—	13	4	no	—
144	C₂²⋊Q₈	—	13	8	no	—
145	C₂²⋊Q₈	—	13	8	no	—
146	C₂²⋊Q₈	—	13	8	no	—
147	C₂²⋊Q₈	—	13	8	no	—
148	C₂²⋊Q₈	C₂×D₁₃	1	4	yes	—
149	C₄×Q₈	—	13	8	no	—
150	C₄×Q₈	—	13	4	no	—
151	C₄×Q₈	C₂×D₁₃	1	4	yes	complemented
152	#117	—	13	2	no	—
153	#117	—	13	4	no	—
154	#117	—	13	4	no	—
155	#117	—	13	4	no	—
156	#117	—	13	4	no	—
157	#117	—	13	2	no	—
158	#117	—	13	2	no	—
159	#117	C₂×D₁₃	1	2	yes	—
160	C₂×C₄⋊C₄	—	13	2	no	—
161	C₂×C₄⋊C₄	—	13	4	no	—
162	C₂×C₂²⋊C₄	—	13	2	no	—
163	C₂×C₂²⋊C₄	—	13	4	no	—
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	—	13	4	no	—
166	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
169	C₂×C₁₃	#585	1	2	yes	—
170	D₁₃	—	2	4	no	—
171	C₂⁴	—	13	1	no	—
172	C₂×Q₈	—	13	8	no	—
173	C₂×Q₈	—	13	4	no	—
174	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	8	no	—
177	C₂²×C₄	—	13	2	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	8	no	—
181	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
182	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
183	C₄⋊C₄	—	13	8	no	—
184	C₄⋊C₄	—	13	4	no	—
185	C₄⋊C₄	—	13	4	no	—
186	C₄⋊C₄	—	13	4	no	—
187	C₄⋊C₄	—	13	8	no	—
188	C₄⋊C₄	—	13	8	no	—
189	C₄⋊C₄	—	13	8	no	—
190	C₄⋊C₄	—	13	8	no	—
191	C₄⋊C₄	—	13	4	no	—
192	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
193	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
194	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
195	C₂²⋊C₄	—	13	4	no	—
196	C₂²⋊C₄	—	13	8	no	—
197	C₂²⋊C₄	—	13	4	no	—
198	C₂²⋊C₄	—	13	4	no	—
199	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
200	C₄²	—	13	8	no	—
201	C₄²	—	13	2	no	—
202	C₄²	—	13	2	no	—
203	C₄²	—	13	4	no	—
204	C₄²	C₂²×D₁₃	1	1	yes	characteristic
205	C₄²	C₂²×D₁₃	1	1	yes	characteristic
206	C₄²	C₂²×D₁₃	1	4	yes	—
207	C₄²	C₂²×D₁₃	1	2	yes	—
208	C₁₃	#3458	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	13	2	no	—
210	C₂³	—	13	4	no	—
211	C₂³	—	52	2	no	—
212	C₂³	C₂³×D₁₃	1	1	yes	characteristic
213	Q₈	—	4	4	no	—
214	Q₈	—	52	4	no	—
215	Q₈	—	52	8	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	13	2	no	—
219	C₂×C₄	—	13	8	no	—
220	C₂×C₄	—	13	8	no	—
221	C₂×C₄	—	13	8	no	—
222	C₂×C₄	—	13	4	no	—
223	C₂×C₄	—	52	8	no	—
224	C₂×C₄	—	52	2	no	—
225	C₂×C₄	Q₈×D₁₃	1	4	yes	—
226	C₂×C₄	C₂³×D₁₃	1	8	yes	—
227	C₂×C₄	C₂³×D₁₃	1	2	yes	—
228	C₂×C₄	C₂³×D₁₃	1	2	yes	—
229	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
230	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	13	4	no	—
233	C₂²	—	26	4	no	—
234	C₂²	—	52	4	no	—
235	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
236	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	8	no	—
240	C₄	—	13	4	no	—
241	C₄	—	26	8	no	—
242	C₄	—	26	4	no	—
243	C₄	C₂×Q₈×D₁₃	1	2	yes	—
244	C₄	C₂×Q₈×D₁₃	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	—	26	4	no	—
247	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
248	C₂	#11719648	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₁₃×#533	C₂
prime radical	C₂³×D₁₃	C₂³
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3458
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
2. #11719434	C₂	semidirect
3. #11719394	C₂	semidirect
5. #11719328	C₂	semidirect
6. #11719283	C₂	semidirect
8. #11719222	C₂	semidirect
10. #11719175	C₂	semidirect
12. C₁₃×C₄⋊Q₈	C₂²	semidirect
13. C₁₃×#125	C₂²	semidirect
15. C₄×Q₈×C₁₃	C₂²	semidirect
19. #88198	C₂²	semidirect
30. #88144	C₂²	semidirect
31. #88143	C₂²	semidirect
32. #88143	C₂²	semidirect
33. #88142	C₂²	semidirect
34. #88142	C₂²	semidirect
43. #88124	C₂²	semidirect
44. #88123	C₂²	semidirect
45. C₄×Dic₂₆	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
53. C₂×C₄×D₁₃	Q₈	semidirect
114. #533	D₁₃	semidirect
136. C₄⋊Q₈	C₂×D₁₃	semidirect
141. #125	C₂×D₁₃	semidirect
151. C₄×Q₈	C₂×D₁₃	semidirect
208. C₁₃	#3458	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
#11719434	2
#11719394	2
#11719339	2
#11719328	2
#11719283	2
D₁₃×#117	2
#11719222	2
C₂×#88126	2
#11719175	2
#3458	13