Subgroups — #13131615

Order 1664.

452 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	#11719629	C₂	1	2	yes	maximal, complemented
2	D₁₃×#135	C₂	1	1	yes	maximal, characteristic
3	#11719166	C₂	1	2	yes	maximal, complemented
4	#11719112	C₂	1	2	yes	maximal, complemented
5	#11719107	C₂	1	4	yes	maximal, complemented
6	#11719104	C₂	1	2	yes	maximal, complemented
7	C₂×#88103	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719089	C₂	1	1	yes	maximal, characteristic, complemented
9	#41×D₁₃	C₂²	1	2	yes	—
10	#88240	C₂²	1	4	yes	complemented
11	#88238	—	2	4	no	—
12	#88238	C₂²	1	4	yes	complemented
13	#88237	C₂²	1	1	yes	characteristic
14	D₁₃⋊M₁₆	—	2	2	no	—
15	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
16	C₂×#6118	—	2	2	no	—
17	C₂×#6118	C₂²	1	2	yes	—
18	C₁₃×#135	C₄	1	1	yes	characteristic, complemented
19	#88204	C₄	1	2	yes	complemented
20	#88173	C₂²	1	1	yes	characteristic
21	Q₁₆×D₁₃	C₄	1	1	yes	characteristic, complemented
22	#88170	C₂²	1	2	yes	—
23	SD₁₆×D₁₃	C₄	1	2	yes	complemented
24	#88166	C₂²	1	1	yes	characteristic
25	D₈×D₁₃	C₄	1	1	yes	characteristic, complemented
26	#88158	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	#88118	C₂²	1	4	yes	complemented
29	#88116	C₂²	1	4	yes	complemented
30	#88103	C₂²	1	2	yes	complemented
31	#88103	C₂²	1	2	yes	complemented
32	#88100	C₂²	1	2	yes	complemented
33	C₈×#261	C₂²	1	2	yes	complemented
34	C₁₃×#41	C₂×C₄	1	2	yes	—
35	#6131	C₂³	1	2	yes	—
36	Q₈×D₁₃	C₂×C₄	1	2	yes	—
37	#6129	—	2	2	no	—
38	#6129	C₂³	1	2	yes	—
39	D₄×D₁₃	—	2	2	no	—
40	D₄×D₁₃	C₂×C₄	1	2	yes	—
41	#6127	—	2	2	no	—
42	#6127	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₃	—	2	2	no	—
44	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
45	C₂²⋊#261	—	2	1	no	—
46	#6122	—	2	4	no	—
47	#6122	D₄	1	2	yes	—
48	C₂×#1057	—	2	2	no	—
49	C₂×#1057	—	2	4	no	—
50	C₄⋊#261	—	2	1	no	—
51	C₄×#261	C₂³	1	2	yes	—
52	#6118	—	2	2	no	—
53	#6118	—	2	4	no	—
54	#6118	D₄	1	2	yes	—
55	#6118	C₂³	1	4	yes	—
56	D₁₃⋊C₈	—	2	4	no	—
57	D₁₃⋊C₈	C₂³	1	2	yes	—
58	C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
59	C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
60	C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	#6107	C₂×C₄	1	2	yes	—
63	#6106	C₂×C₄	1	2	yes	—
64	#6105	C₂×C₄	1	2	yes	—
65	#6104	C₂×C₄	1	2	yes	—
66	C₂×#1055	C₂×C₄	1	1	yes	characteristic
67	Dic₅₂	C₂×C₄	1	1	yes	characteristic
68	D₁₀₄	C₂×C₄	1	1	yes	characteristic
69	#6095	C₂×C₄	1	2	yes	—
70	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
71	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
72	C₈×D₁₃	C₂³	1	1	yes	characteristic
73	C₈×D₁₃	C₂³	1	1	yes	characteristic
74	#2957	—	13	1	no	maximal, Sylow
75	C₂²×D₁₃	—	2	2	no	—
76	C₂²×D₁₃	#41	1	1	yes	characteristic
77	C₂×#261	—	2	2	no	—
78	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
79	D₄×C₁₃	—	2	2	no	—
80	D₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊D₁₃	—	2	2	no	—
84	C₂²⋊D₁₃	—	2	2	no	—
85	C₂²⋊D₁₃	—	2	2	no	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
88	D₅₂	—	2	2	no	—
89	D₅₂	C₂²×C₄	1	2	yes	—
90	C₄×D₁₃	—	2	2	no	—
91	C₄×D₁₃	—	2	2	no	—
92	C₄×D₁₃	—	2	2	no	—
93	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄×D₁₃	#41	1	1	yes	characteristic
96	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
97	Dic₂₆	—	2	2	no	—
98	Dic₂₆	C₂²×C₄	1	2	yes	—
99	#1057	—	2	4	no	—
100	#1057	C₂×D₄	1	4	yes	—
101	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	#1055	C₂²×C₄	1	1	yes	characteristic
104	#1055	C₂²×C₄	1	1	yes	characteristic
105	C₂×#135	—	13	1	no	—
106	#568	—	13	2	no	—
107	#444	—	13	2	no	—
108	#444	—	13	2	no	—
109	#443	—	13	4	no	—
110	C₂×#108	—	13	1	no	—
111	#421	—	13	2	no	—
112	#405	—	13	1	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×D₁₃	—	2	2	no	—
116	C₂×D₁₃	—	4	1	no	—
117	C₂×D₁₃	—	8	2	no	—
118	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×D₁₃	C₂×#41	1	1	yes	characteristic
120	#261	—	4	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	C₄×D₄	1	1	yes	characteristic
126	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×#41	—	13	2	no	—
128	#135	—	13	2	no	—
129	#135	—	13	1	no	—
130	#135	—	13	2	no	—
131	#135	—	13	1	no	—
132	#135	—	13	1	no	—
133	#135	#261	1	1	yes	characteristic, complemented
134	C₂×Q₁₆	—	13	1	no	—
135	C₂×SD₁₆	—	13	2	no	—
136	C₂×D₈	—	13	1	no	—
137	#131	—	13	4	no	—
138	#131	—	13	4	no	—
139	#131	—	26	4	no	—
140	C₂×M₁₆	—	13	1	no	—
141	C₂×M₁₆	—	13	2	no	—
142	C₂×M₁₆	—	26	2	no	—
143	C₂×M₁₆	—	26	2	no	—
144	C₂²×C₈	—	13	1	no	—
145	#117	—	13	1	no	—
146	#108	—	13	2	no	—
147	#108	—	13	2	no	—
148	#104	—	13	4	no	—
149	#104	—	13	4	no	—
150	#97	—	13	2	no	—
151	C₄×C₈	—	13	2	no	—
152	C₂×C₁₃	—	2	1	no	—
153	C₂×C₁₃	—	4	2	no	—
154	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
155	D₁₃	—	2	1	no	—
156	D₁₃	—	2	1	no	—
157	D₁₃	—	4	2	no	—
158	#41	—	13	2	no	—
159	#41	—	13	2	no	—
160	#41	—	13	2	no	—
161	#41	—	26	2	no	—
162	#41	—	26	2	no	—
163	#41	C₂×#261	1	2	yes	—
164	C₂×Q₈	—	13	2	no	—
165	C₂×D₄	—	13	2	no	—
166	C₂×D₄	—	26	2	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	26	2	no	—
169	Q₁₆	—	13	2	no	—
170	Q₁₆	—	13	1	no	—
171	Q₁₆	C₂×#261	1	1	yes	characteristic
172	SD₁₆	—	13	2	no	—
173	SD₁₆	—	13	2	no	—
174	SD₁₆	—	13	2	no	—
175	SD₁₆	C₂×#261	1	2	yes	—
176	D₈	—	13	2	no	—
177	D₈	—	13	1	no	—
178	D₈	C₂×#261	1	1	yes	characteristic
179	M₁₆	—	13	2	no	—
180	M₁₆	—	13	2	no	—
181	M₁₆	—	13	4	no	—
182	M₁₆	—	26	4	no	—
183	M₁₆	—	26	4	no	—
184	M₁₆	—	26	2	no	—
185	C₂×C₈	—	13	2	no	—
186	C₂×C₈	—	13	1	no	—
187	C₂×C₈	—	13	1	no	—
188	C₂×C₈	—	13	1	no	—
189	C₂×C₈	—	13	1	no	—
190	C₂×C₈	—	13	1	no	—
191	C₂×C₈	—	26	4	no	—
192	C₂×C₈	—	26	4	no	—
193	C₂×C₈	—	26	2	no	—
194	C₂×C₈	C₂×#261	1	1	yes	characteristic
195	C₄⋊C₄	—	26	1	no	—
196	C₂²⋊C₄	—	26	1	no	—
197	C₄²	—	13	2	no	—
198	C₁₃	#2957	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	13	1	no	—
200	C₂³	—	26	2	no	—
201	Q₈	—	13	2	no	—
202	Q₈	—	26	2	no	—
203	Q₈	C₂²×#261	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	13	2	no	—
206	D₄	—	26	2	no	—
207	D₄	—	26	2	no	—
208	D₄	—	26	2	no	—
209	D₄	—	26	2	no	—
210	D₄	C₂²×#261	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	13	1	no	—
213	C₂×C₄	—	13	1	no	—
214	C₂×C₄	—	13	1	no	—
215	C₂×C₄	—	13	1	no	—
216	C₂×C₄	—	13	1	no	—
217	C₂×C₄	—	26	2	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	—	26	2	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	C₂²×#261	1	1	yes	characteristic
223	C₈	—	13	4	no	—
224	C₈	—	13	1	no	—
225	C₈	—	13	1	no	—
226	C₈	—	26	4	no	—
227	C₈	C₂²×#261	1	1	yes	characteristic
228	C₈	C₂²×#261	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	13	1	no	—
231	C₂²	—	13	1	no	—
232	C₂²	—	26	2	no	—
233	C₂²	—	52	1	no	—
234	C₂²	—	104	2	no	—
235	C₂²	D₄×#261	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	13	1	no	—
238	C₄	—	13	1	no	—
239	C₄	—	26	2	no	—
240	C₄	—	52	2	no	—
241	C₄	D₄×#261	1	1	yes	characteristic
242	C₄	C₂³×#261	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	26	1	no	—
246	C₂	—	26	1	no	—
247	C₂	—	52	2	no	—
248	C₂	C₂×D₄×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₄	C₂³×#261
Fitting	C₁₃×#135	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	D₁₃×#135	C₂
cosocle	C₄×D₁₃	C₂⁴
hypercenter	#135	#261
hypocenter	C₁₃	#2957
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #11719629	C₂	semidirect
3. #11719166	C₂	semidirect
4. #11719112	C₂	semidirect
5. #11719107	C₂	semidirect
6. #11719104	C₂	semidirect
7. C₂×#88103	C₂	semidirect
8. #11719089	C₂	semidirect
10. #88240	C₂²	semidirect
12. #88238	C₂²	semidirect
18. C₁₃×#135	C₄	semidirect
19. #88204	C₄	semidirect
21. Q₁₆×D₁₃	C₄	semidirect
23. SD₁₆×D₁₃	C₄	semidirect
25. D₈×D₁₃	C₄	semidirect
26. #88158	C₄	semidirect
28. #88118	C₂²	semidirect
29. #88116	C₂²	semidirect
30. #88103	C₂²	semidirect
31. #88103	C₂²	semidirect
32. #88100	C₂²	semidirect
33. C₈×#261	C₂²	semidirect
133. #135	#261	semidirect
198. C₁₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
#11719629	2
D₁₃×#135	2
#11719166	2
#11719112	2
#11719107	2
#11719104	2
C₂×#88103	2
#11719089	2
#2957	13