Subgroups — #13131598

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719631	C₂	1	2	yes	maximal, complemented
2	D₁₃×#135	C₂	1	1	yes	maximal, characteristic
3	#11719164	C₂	1	2	yes	maximal, complemented
4	Q₁₆×#261	C₂	1	2	yes	maximal, complemented
5	#11719106	C₂	1	4	yes	maximal, complemented
6	D₈×#261	C₂	1	2	yes	maximal, complemented
7	C₂×#88102	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719089	C₂	1	1	yes	maximal, characteristic, complemented
9	#41×D₁₃	C₂²	1	2	yes	—
10	Q₈×#261	C₂²	1	4	yes	—
11	D₄×#261	—	2	4	no	—
12	D₄×#261	C₂²	1	4	yes	—
13	#88237	—	2	2	no	—
14	#88237	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#261	—	2	2	no	—
16	C₂×C₄⋊#261	C₂²	1	2	yes	—
17	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#135	C₄	1	1	yes	characteristic, complemented
19	#88204	C₄	1	2	yes	complemented
20	#88173	C₄	1	1	yes	characteristic, complemented
21	Q₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	#88170	C₄	1	2	yes	complemented
23	SD₁₆×D₁₃	C₂²	1	2	yes	—
24	#88166	C₄	1	1	yes	characteristic, complemented
25	D₈×D₁₃	C₂²	1	1	yes	characteristic
26	#88158	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	#88117	C₂²	1	4	yes	—
29	#88115	C₂²	1	4	yes	—
30	#88102	C₂²	1	2	yes	—
31	#88102	C₂²	1	2	yes	—
32	#88100	C₂²	1	2	yes	—
33	C₈×#261	C₂²	1	2	yes	—
34	C₂²×#261	—	2	2	no	—
35	C₂²×#261	—	2	4	no	—
36	C₁₃×#41	C₂×C₄	1	2	yes	complemented
37	#6131	C₂×C₄	1	2	yes	complemented
38	Q₈×D₁₃	C₂³	1	2	yes	—
39	#6129	—	2	2	no	—
40	#6129	C₂×C₄	1	2	yes	complemented
41	D₄×D₁₃	—	2	2	no	—
42	D₄×D₁₃	C₂³	1	2	yes	—
43	#6127	—	2	2	no	—
44	#6127	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₁₃	—	2	2	no	—
46	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
47	C₂²⋊#261	—	2	4	no	—
48	C₂²⋊#261	D₄	1	2	yes	—
49	#6122	—	2	1	no	—
50	C₄⋊#261	—	2	4	no	—
51	C₄⋊#261	—	2	2	no	—
52	C₄⋊#261	D₄	1	1	yes	characteristic
53	C₄⋊#261	D₄	1	1	yes	characteristic
54	C₄⋊#261	C₂³	1	4	yes	—
55	C₄×#261	—	2	4	no	—
56	C₄×#261	C₂³	1	2	yes	—
57	#6118	—	2	1	no	—
58	D₁₃⋊C₈	C₂³	1	2	yes	—
59	C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
60	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
61	C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
62	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
63	#6107	C₂×C₄	1	2	yes	complemented
64	#6106	C₂×C₄	1	2	yes	complemented
65	#6105	C₂×C₄	1	2	yes	complemented
66	#6104	C₂×C₄	1	2	yes	complemented
67	C₂×#1055	C₂×C₄	1	1	yes	characteristic, complemented
68	Dic₅₂	C₂×C₄	1	1	yes	characteristic, complemented
69	D₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
70	#6095	C₂×C₄	1	2	yes	complemented
71	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
73	C₈×D₁₃	C₂³	1	1	yes	characteristic
74	C₈×D₁₃	C₂³	1	1	yes	characteristic
75	#2951	—	13	1	no	maximal, Sylow
76	C₂²×D₁₃	—	2	2	no	—
77	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×#261	—	2	4	no	—
79	C₂×#261	—	8	4	no	—
80	C₂×#261	—	8	2	no	—
81	C₂×#261	C₂×D₄	1	2	yes	—
82	C₂×#261	C₂×D₄	1	2	yes	—
83	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
84	D₄×C₁₃	—	2	2	no	—
85	D₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²⋊D₁₃	—	2	2	no	—
89	C₂²⋊D₁₃	—	2	2	no	—
90	C₂²⋊D₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	#41	1	1	yes	characteristic
93	D₅₂	—	2	2	no	—
94	D₅₂	C₂²×C₄	1	2	yes	—
95	C₄×D₁₃	—	2	2	no	—
96	C₄×D₁₃	—	2	2	no	—
97	C₄×D₁₃	—	2	2	no	—
98	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄×D₁₃	#41	1	1	yes	characteristic
101	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
102	Dic₂₆	—	2	2	no	—
103	Dic₂₆	C₂²×C₄	1	2	yes	—
104	#1057	—	2	2	no	—
105	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	#1055	C₂²×C₄	1	1	yes	characteristic
108	#1055	C₂²×C₄	1	1	yes	characteristic
109	C₂×#135	—	13	1	no	—
110	#520	—	13	2	no	—
111	#440	—	13	4	no	—
112	C₄×Q₁₆	—	13	2	no	—
113	C₄×D₈	—	13	2	no	—
114	C₂×#107	—	13	1	no	—
115	#417	—	13	2	no	—
116	#405	—	13	1	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×D₁₃	—	2	1	no	—
120	C₂×D₁₃	—	2	1	no	—
121	C₂×D₁₃	—	2	2	no	—
122	C₂×D₁₃	—	4	2	no	—
123	C₂×D₁₃	—	4	2	no	—
124	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
126	#261	—	2	2	no	—
127	#261	—	2	2	no	—
128	#261	—	4	4	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	Dic₁₃	—	2	2	no	—
133	Dic₁₃	C₄×D₄	1	1	yes	characteristic
134	Dic₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×#41	—	13	2	no	—
136	#135	—	13	2	no	—
137	#135	—	13	1	no	—
138	#135	—	13	2	no	—
139	#135	—	13	1	no	—
140	#135	—	13	1	no	—
141	#135	#261	1	1	yes	characteristic, complemented
142	C₂×Q₁₆	—	13	1	no	—
143	C₂×SD₁₆	—	13	2	no	—
144	C₂×D₈	—	13	1	no	—
145	C₂×M₁₆	—	13	1	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₄×Q₈	—	13	4	no	—
148	C₄×D₄	—	13	4	no	—
149	C₄×D₄	—	26	4	no	—
150	#117	—	13	1	no	—
151	#117	—	26	2	no	—
152	C₂×C₄⋊C₄	—	13	2	no	—
153	C₂×C₄⋊C₄	—	26	2	no	—
154	#107	—	13	2	no	—
155	#107	—	13	2	no	—
156	#103	—	13	4	no	—
157	#102	—	13	4	no	—
158	#97	—	13	2	no	—
159	C₄×C₈	—	13	2	no	—
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	—	4	2	no	—
162	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
163	D₁₃	—	2	1	no	—
164	D₁₃	—	4	2	no	—
165	D₁₃	#576	1	1	yes	characteristic
166	D₁₃	#578	1	1	yes	characteristic
167	#41	—	13	2	no	—
168	#41	—	13	2	no	—
169	#41	—	13	2	no	—
170	#41	—	26	2	no	—
171	#41	—	26	2	no	—
172	#41	C₂×#261	1	2	yes	complemented
173	C₂×Q₈	—	13	2	no	—
174	C₂×D₄	—	13	2	no	—
175	C₂×D₄	—	26	2	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	26	2	no	—
178	C₂²×C₄	—	26	4	no	—
179	C₂²×C₄	—	26	2	no	—
180	Q₁₆	—	13	2	no	—
181	Q₁₆	—	13	1	no	—
182	Q₁₆	C₂×#261	1	1	yes	characteristic, complemented
183	SD₁₆	—	13	2	no	—
184	SD₁₆	—	13	2	no	—
185	SD₁₆	—	13	2	no	—
186	SD₁₆	C₂×#261	1	2	yes	complemented
187	D₈	—	13	2	no	—
188	D₈	—	13	1	no	—
189	D₈	C₂×#261	1	1	yes	characteristic, complemented
190	M₁₆	—	26	1	no	—
191	M₁₆	—	26	1	no	—
192	C₂×C₈	—	13	1	no	—
193	C₂×C₈	—	13	1	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	C₂×#261	1	1	yes	characteristic, complemented
199	C₄⋊C₄	—	13	1	no	—
200	C₄⋊C₄	—	13	1	no	—
201	C₄⋊C₄	—	13	4	no	—
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	—	26	4	no	—
204	C₂²⋊C₄	—	13	2	no	—
205	C₂²⋊C₄	—	26	4	no	—
206	C₄²	—	13	2	no	—
207	C₄²	—	26	4	no	—
208	C₁₃	#2951	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	13	1	no	—
210	C₂³	—	26	2	no	—
211	Q₈	—	13	2	no	—
212	Q₈	—	26	2	no	—
213	Q₈	C₂²×#261	1	2	yes	—
214	D₄	—	2	2	no	—
215	D₄	—	13	2	no	—
216	D₄	—	26	2	no	—
217	D₄	—	26	2	no	—
218	D₄	—	26	2	no	—
219	D₄	—	26	2	no	—
220	D₄	C₂²×#261	1	2	yes	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	13	1	no	—
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	1	no	—
226	C₂×C₄	—	13	1	no	—
227	C₂×C₄	—	13	2	no	—
228	C₂×C₄	—	13	2	no	—
229	C₂×C₄	—	26	2	no	—
230	C₂×C₄	—	26	2	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	26	4	no	—
234	C₂×C₄	—	104	4	no	—
235	C₂×C₄	—	104	2	no	—
236	C₂×C₄	C₂²×#261	1	1	yes	characteristic
237	C₈	—	13	1	no	—
238	C₈	—	13	1	no	—
239	C₈	—	26	2	no	—
240	C₈	C₂²×#261	1	1	yes	characteristic
241	C₈	C₂²×#261	1	1	yes	characteristic
242	C₂²	—	2	2	no	—
243	C₂²	—	13	1	no	—
244	C₂²	—	13	1	no	—
245	C₂²	—	26	1	no	—
246	C₂²	—	26	1	no	—
247	C₂²	—	26	2	no	—
248	C₂²	—	52	2	no	—
249	C₂²	—	52	2	no	—
250	C₂²	D₄×#261	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	13	1	no	—
253	C₄	—	13	1	no	—
254	C₄	—	26	2	no	—
255	C₄	—	26	2	no	—
256	C₄	—	26	2	no	—
257	C₄	—	52	4	no	—
258	C₄	D₄×#261	1	1	yes	characteristic
259	C₄	C₂³×#261	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	13	1	no	—
263	C₂	—	13	1	no	—
264	C₂	—	26	1	no	—
265	C₂	—	52	2	no	—
266	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	#2951
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. #11719631	C₂	semidirect
3. #11719164	C₂	semidirect
4. Q₁₆×#261	C₂	semidirect
5. #11719106	C₂	semidirect
6. D₈×#261	C₂	semidirect
7. C₂×#88102	C₂	semidirect
8. #11719089	C₂	semidirect
18. C₁₃×#135	C₄	semidirect
19. #88204	C₄	semidirect
20. #88173	C₄	semidirect
22. #88170	C₄	semidirect
24. #88166	C₄	semidirect
26. #88158	C₄	semidirect
36. C₁₃×#41	C₂×C₄	semidirect
37. #6131	C₂×C₄	semidirect
40. #6129	C₂×C₄	semidirect
44. #6127	C₂×C₄	semidirect
59. C₁₃×Q₁₆	C₂×C₄	semidirect
60. C₁₃×SD₁₆	C₂×C₄	semidirect
61. C₁₃×D₈	C₂×C₄	semidirect
62. C₂×C₈×C₁₃	C₂×C₄	semidirect
63. #6107	C₂×C₄	semidirect
64. #6106	C₂×C₄	semidirect
65. #6105	C₂×C₄	semidirect
66. #6104	C₂×C₄	semidirect
67. C₂×#1055	C₂×C₄	semidirect
68. Dic₅₂	C₂×C₄	semidirect
69. D₁₀₄	C₂×C₄	semidirect
70. #6095	C₂×C₄	semidirect
71. C₈×D₁₃	C₂×C₄	semidirect
72. C₈×D₁₃	C₂×C₄	semidirect
141. #135	#261	semidirect
172. #41	C₂×#261	semidirect
182. Q₁₆	C₂×#261	semidirect
186. SD₁₆	C₂×#261	semidirect
189. D₈	C₂×#261	semidirect
198. C₂×C₈	C₂×#261	semidirect
208. C₁₃	#2951	semidirect

Maximal subgroups

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

subgroup	index
#11719631	2
D₁₃×#135	2
#11719164	2
Q₁₆×#261	2
#11719106	2
D₈×#261	2
C₂×#88102	2
#11719089	2
#2951	13