Subgroups — #13131599

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 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	#11719110	C₂	1	2	yes	maximal, complemented
5	SD₁₆×#261	C₂	1	4	yes	maximal, complemented
6	#11719102	C₂	1	2	yes	maximal, complemented
7	C₂×#88101	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	#88101	C₂²	1	2	yes	—
31	#88101	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	2	yes	—
53	C₄⋊#261	C₂³	1	4	yes	—
54	C₄×#261	—	2	4	no	—
55	C₄×#261	C₂³	1	2	yes	—
56	#6118	—	2	1	no	—
57	D₁₃⋊C₈	C₂³	1	2	yes	—
58	C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
59	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
60	C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
61	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
62	#6107	C₂×C₄	1	2	yes	complemented
63	#6106	C₂×C₄	1	2	yes	complemented
64	#6105	C₂×C₄	1	2	yes	complemented
65	#6104	C₂×C₄	1	2	yes	complemented
66	C₂×#1055	C₂×C₄	1	1	yes	characteristic, complemented
67	Dic₅₂	C₂×C₄	1	1	yes	characteristic, complemented
68	D₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
69	#6095	C₂×C₄	1	2	yes	complemented
70	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
71	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₁₃	C₂³	1	1	yes	characteristic
73	C₈×D₁₃	C₂³	1	1	yes	characteristic
74	#2952	—	13	1	no	maximal, Sylow
75	C₂²×D₁₃	—	2	2	no	—
76	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×#261	—	2	4	no	—
78	C₂×#261	—	8	4	no	—
79	C₂×#261	—	8	2	no	—
80	C₂×#261	C₂×D₄	1	4	yes	—
81	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
82	D₄×C₁₃	—	2	2	no	—
83	D₄×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊D₁₃	—	2	2	no	—
87	C₂²⋊D₁₃	—	2	2	no	—
88	C₂²⋊D₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	#41	1	1	yes	characteristic
91	D₅₂	—	2	2	no	—
92	D₅₂	C₂²×C₄	1	2	yes	—
93	C₄×D₁₃	—	2	2	no	—
94	C₄×D₁₃	—	2	2	no	—
95	C₄×D₁₃	—	2	2	no	—
96	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄×D₁₃	#41	1	1	yes	characteristic
99	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
100	Dic₂₆	—	2	2	no	—
101	Dic₂₆	C₂²×C₄	1	2	yes	—
102	#1057	—	2	2	no	—
103	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	#1055	C₂²×C₄	1	1	yes	characteristic
106	#1055	C₂²×C₄	1	1	yes	characteristic
107	C₂×#135	—	13	1	no	—
108	#520	—	13	2	no	—
109	#442	—	13	2	no	—
110	#441	—	13	2	no	—
111	C₄×SD₁₆	—	13	4	no	—
112	C₂×#106	—	13	1	no	—
113	#417	—	13	2	no	—
114	#405	—	13	1	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×D₁₃	—	2	2	no	—
118	C₂×D₁₃	—	2	2	no	—
119	C₂×D₁₃	—	4	2	no	—
120	C₂×D₁₃	—	4	2	no	—
121	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
123	#261	—	2	4	no	—
124	#261	—	4	4	no	—
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	Dic₁₃	—	2	2	no	—
129	Dic₁₃	C₄×D₄	1	1	yes	characteristic
130	Dic₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×#41	—	13	2	no	—
132	#135	—	13	2	no	—
133	#135	—	13	1	no	—
134	#135	—	13	2	no	—
135	#135	—	13	1	no	—
136	#135	—	13	1	no	—
137	#135	#261	1	1	yes	characteristic, complemented
138	C₂×Q₁₆	—	13	1	no	—
139	C₂×SD₁₆	—	13	2	no	—
140	C₂×D₈	—	13	1	no	—
141	C₂×M₁₆	—	13	1	no	—
142	C₂²×C₈	—	13	1	no	—
143	C₄×Q₈	—	13	4	no	—
144	C₄×D₄	—	13	4	no	—
145	C₄×D₄	—	26	4	no	—
146	#117	—	13	1	no	—
147	#117	—	26	2	no	—
148	C₂×C₄⋊C₄	—	13	2	no	—
149	C₂×C₄⋊C₄	—	26	2	no	—
150	#106	—	13	2	no	—
151	#106	—	13	2	no	—
152	#103	—	13	4	no	—
153	#102	—	13	4	no	—
154	#97	—	13	2	no	—
155	C₄×C₈	—	13	2	no	—
156	C₂×C₁₃	—	2	1	no	—
157	C₂×C₁₃	—	4	2	no	—
158	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	D₁₃	—	2	1	no	—
160	D₁₃	—	4	2	no	—
161	D₁₃	#577	1	2	yes	—
162	#41	—	13	2	no	—
163	#41	—	13	2	no	—
164	#41	—	13	2	no	—
165	#41	—	26	2	no	—
166	#41	—	26	2	no	—
167	#41	C₂×#261	1	2	yes	complemented
168	C₂×Q₈	—	13	2	no	—
169	C₂×D₄	—	13	2	no	—
170	C₂×D₄	—	26	2	no	—
171	C₂²×C₄	—	13	1	no	—
172	C₂²×C₄	—	26	2	no	—
173	C₂²×C₄	—	26	4	no	—
174	C₂²×C₄	—	26	2	no	—
175	Q₁₆	—	13	2	no	—
176	Q₁₆	—	13	1	no	—
177	Q₁₆	C₂×#261	1	1	yes	characteristic, complemented
178	SD₁₆	—	13	2	no	—
179	SD₁₆	—	13	2	no	—
180	SD₁₆	—	13	2	no	—
181	SD₁₆	C₂×#261	1	2	yes	complemented
182	D₈	—	13	2	no	—
183	D₈	—	13	1	no	—
184	D₈	C₂×#261	1	1	yes	characteristic, complemented
185	M₁₆	—	26	1	no	—
186	M₁₆	—	26	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	—	13	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	C₂×#261	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	13	2	no	—
195	C₄⋊C₄	—	13	4	no	—
196	C₄⋊C₄	—	26	2	no	—
197	C₄⋊C₄	—	26	4	no	—
198	C₂²⋊C₄	—	13	2	no	—
199	C₂²⋊C₄	—	26	4	no	—
200	C₄²	—	13	2	no	—
201	C₄²	—	26	4	no	—
202	C₁₃	#2952	1	1	yes	characteristic, complemented, Sylow
203	C₂³	—	13	1	no	—
204	C₂³	—	26	2	no	—
205	Q₈	—	13	2	no	—
206	Q₈	—	26	2	no	—
207	Q₈	C₂²×#261	1	2	yes	—
208	D₄	—	2	2	no	—
209	D₄	—	13	2	no	—
210	D₄	—	26	2	no	—
211	D₄	—	26	2	no	—
212	D₄	—	26	2	no	—
213	D₄	—	26	2	no	—
214	D₄	C₂²×#261	1	2	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	13	1	no	—
217	C₂×C₄	—	13	1	no	—
218	C₂×C₄	—	13	1	no	—
219	C₂×C₄	—	13	1	no	—
220	C₂×C₄	—	13	1	no	—
221	C₂×C₄	—	13	4	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	—	26	2	no	—
226	C₂×C₄	—	26	4	no	—
227	C₂×C₄	—	104	4	no	—
228	C₂×C₄	—	104	2	no	—
229	C₂×C₄	C₂²×#261	1	1	yes	characteristic
230	C₈	—	13	1	no	—
231	C₈	—	13	1	no	—
232	C₈	—	26	2	no	—
233	C₈	C₂²×#261	1	1	yes	characteristic
234	C₈	C₂²×#261	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	13	1	no	—
237	C₂²	—	13	1	no	—
238	C₂²	—	26	2	no	—
239	C₂²	—	26	2	no	—
240	C₂²	—	52	2	no	—
241	C₂²	—	52	2	no	—
242	C₂²	D₄×#261	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	13	1	no	—
245	C₄	—	13	1	no	—
246	C₄	—	26	4	no	—
247	C₄	—	26	2	no	—
248	C₄	—	52	4	no	—
249	C₄	D₄×#261	1	1	yes	characteristic
250	C₄	C₂³×#261	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	13	2	no	—
254	C₂	—	26	1	no	—
255	C₂	—	52	2	no	—
256	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	#2952
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. #11719110	C₂	semidirect
5. SD₁₆×#261	C₂	semidirect
6. #11719102	C₂	semidirect
7. C₂×#88101	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
58. C₁₃×Q₁₆	C₂×C₄	semidirect
59. C₁₃×SD₁₆	C₂×C₄	semidirect
60. C₁₃×D₈	C₂×C₄	semidirect
61. C₂×C₈×C₁₃	C₂×C₄	semidirect
62. #6107	C₂×C₄	semidirect
63. #6106	C₂×C₄	semidirect
64. #6105	C₂×C₄	semidirect
65. #6104	C₂×C₄	semidirect
66. C₂×#1055	C₂×C₄	semidirect
67. Dic₅₂	C₂×C₄	semidirect
68. D₁₀₄	C₂×C₄	semidirect
69. #6095	C₂×C₄	semidirect
70. C₈×D₁₃	C₂×C₄	semidirect
71. C₈×D₁₃	C₂×C₄	semidirect
137. #135	#261	semidirect
167. #41	C₂×#261	semidirect
177. Q₁₆	C₂×#261	semidirect
181. SD₁₆	C₂×#261	semidirect
184. D₈	C₂×#261	semidirect
193. C₂×C₈	C₂×#261	semidirect
202. C₁₃	#2952	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
#11719110	2
SD₁₆×#261	2
#11719102	2
C₂×#88101	2
#11719089	2
#2952	13