Subgroups — #13131614

Order 1664.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719625	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719621	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88170	C₂	1	1	yes	maximal, characteristic
4	#11719155	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719140	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719108	C₂	1	4	yes	maximal, complemented
7	#11719107	C₂	1	4	yes	maximal, complemented
8	D₁₃⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719089	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#6131	C₂²	1	1	yes	characteristic
11	C₂×#6129	C₂²	1	1	yes	characteristic
12	C₂×#6122	—	2	2	no	—
13	#88240	—	2	2	no	—
14	#88240	C₂²	1	4	yes	complemented
15	#88238	—	2	2	no	—
16	#88238	C₂²	1	4	yes	complemented
17	#88237	C₂²	1	1	yes	characteristic
18	D₁₃⋊M₁₆	—	2	2	no	—
19	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
20	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
21	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×SD₁₆	C₄	1	1	yes	characteristic, complemented
23	C₂×#6107	C₄	1	1	yes	characteristic, complemented
24	C₂×#6104	C₄	1	1	yes	characteristic, complemented
25	#88170	C₄	1	2	yes	complemented
26	#88170	C₄	1	2	yes	complemented
27	#88170	C₂²	1	2	yes	—
28	#88170	C₂²	1	2	yes	—
29	C₂×#6095	C₄	1	1	yes	characteristic, complemented
30	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#88118	C₂²	1	4	yes	complemented
32	#88116	C₂²	1	4	yes	complemented
33	#88104	C₂²	1	2	yes	complemented
34	#88103	C₂²	1	2	yes	complemented
35	#88100	C₂²	1	2	yes	complemented
36	C₈×#261	C₂²	1	2	yes	complemented
37	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₃	—	2	1	no	—
40	C₂²×Dic₁₃	—	2	1	no	—
41	#6131	—	2	1	no	—
42	#6131	—	2	1	no	—
43	#6131	C₂×C₄	1	2	yes	—
44	#6131	C₂³	1	2	yes	—
45	#6129	—	2	1	no	—
46	#6129	—	2	1	no	—
47	#6129	C₂×C₄	1	2	yes	—
48	#6129	C₂³	1	2	yes	—
49	C₂×D₅₂	—	2	1	no	—
50	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃	—	2	1	no	—
52	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊#261	—	2	1	no	—
55	#6122	—	2	1	no	—
56	#6122	—	2	1	no	—
57	#6122	—	2	4	no	—
58	#6122	D₄	1	2	yes	—
59	C₂×#1057	—	2	4	no	—
60	C₄⋊#261	—	2	1	no	—
61	C₄×#261	C₂³	1	2	yes	—
62	#6118	—	2	4	no	—
63	#6118	D₄	1	2	yes	—
64	#6118	C₂³	1	2	yes	—
65	#6118	C₂³	1	2	yes	—
66	D₁₃⋊C₈	—	2	1	no	—
67	D₁₃⋊C₈	—	2	1	no	—
68	D₁₃⋊C₈	—	2	4	no	—
69	D₁₃⋊C₈	C₂³	1	2	yes	—
70	C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
71	C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
73	#6107	C₂×C₄	1	2	yes	—
74	#6107	C₂×C₄	1	2	yes	—
75	#6104	C₂×C₄	1	2	yes	—
76	#6104	C₂×C₄	1	2	yes	—
77	C₂×#1055	C₂×C₄	1	1	yes	characteristic
78	#6095	C₂×C₄	1	2	yes	—
79	#6095	C₂×C₄	1	2	yes	—
80	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
81	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
82	C₈×D₁₃	C₂³	1	1	yes	characteristic
83	C₈×D₁₃	C₂³	1	1	yes	characteristic
84	#2957	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	—	2	1	no	—
86	C₂²×D₁₃	—	2	1	no	—
87	C₂²×D₁₃	#41	1	1	yes	characteristic
88	C₂×#261	—	2	2	no	—
89	Q₈×C₁₃	—	2	1	no	—
90	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
91	D₄×C₁₃	—	2	1	no	—
92	D₄×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²⋊D₁₃	—	2	2	no	—
96	C₂²⋊D₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
101	D₅₂	—	2	2	no	—
102	D₅₂	—	2	2	no	—
103	D₅₂	—	2	1	no	—
104	D₅₂	C₂²×C₄	1	2	yes	—
105	C₄×D₁₃	—	2	2	no	—
106	C₄×D₁₃	—	2	2	no	—
107	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄×D₁₃	#41	1	1	yes	characteristic
110	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
111	Dic₂₆	—	2	1	no	—
112	Dic₂₆	C₂²×C₄	1	2	yes	—
113	#1057	—	2	2	no	—
114	#1057	—	2	2	no	—
115	#1057	C₂×D₄	1	4	yes	—
116	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	#1055	C₂²×C₄	1	1	yes	characteristic
119	#1055	C₂²×C₄	1	1	yes	characteristic
120	C₂×#135	—	13	1	no	—
121	#568	—	13	1	no	—
122	#568	—	13	1	no	—
123	#444	—	13	4	no	—
124	#443	—	13	4	no	—
125	C₂×#108	—	13	1	no	—
126	#421	—	13	1	no	—
127	#421	—	13	1	no	—
128	#405	—	13	1	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	8	1	no	—
131	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×D₁₃	—	2	2	no	—
133	C₂×D₁₃	—	4	1	no	—
134	C₂×D₁₃	—	8	1	no	—
135	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂×D₁₃	C₂×#41	1	1	yes	characteristic
137	#261	—	4	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	Dic₁₃	—	2	2	no	—
142	Dic₁₃	C₄×D₄	1	1	yes	characteristic
143	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×#41	—	13	1	no	—
145	C₂×#41	—	13	1	no	—
146	#135	—	13	2	no	—
147	#135	—	13	2	no	—
148	#135	—	13	2	no	—
149	#135	—	13	2	no	—
150	C₂×Q₁₆	—	13	1	no	—
151	C₂×SD₁₆	—	13	1	no	—
152	C₂×SD₁₆	#261	1	1	yes	characteristic, complemented
153	C₂×D₈	—	13	1	no	—
154	#131	—	13	4	no	—
155	#131	—	13	4	no	—
156	#131	—	26	2	no	—
157	#131	—	26	2	no	—
158	C₂×M₁₆	—	13	1	no	—
159	C₂×M₁₆	—	13	1	no	—
160	C₂×M₁₆	—	13	1	no	—
161	C₂×M₁₆	—	26	2	no	—
162	C₂×M₁₆	—	26	2	no	—
163	C₂²×C₈	—	13	1	no	—
164	#117	—	13	1	no	—
165	#108	—	13	2	no	—
166	#108	—	13	2	no	—
167	#104	—	13	4	no	—
168	#104	—	13	4	no	—
169	#97	—	13	2	no	—
170	C₄×C₈	—	13	2	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	4	2	no	—
173	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
174	D₁₃	—	2	1	no	—
175	D₁₃	—	2	1	no	—
176	D₁₃	—	4	2	no	—
177	#41	—	13	2	no	—
178	#41	—	13	2	no	—
179	#41	—	13	2	no	—
180	#41	—	13	2	no	—
181	#41	—	26	1	no	—
182	#41	—	26	1	no	—
183	#41	—	26	1	no	—
184	#41	—	26	1	no	—
185	C₂×Q₈	—	13	1	no	—
186	C₂×Q₈	C₂×#261	1	1	yes	characteristic
187	C₂×D₄	—	13	1	no	—
188	C₂×D₄	—	26	1	no	—
189	C₂×D₄	—	26	1	no	—
190	C₂×D₄	C₂×#261	1	1	yes	characteristic
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	26	1	no	—
193	C₂²×C₄	—	26	1	no	—
194	Q₁₆	—	13	2	no	—
195	Q₁₆	—	13	2	no	—
196	SD₁₆	—	13	2	no	—
197	SD₁₆	—	13	2	no	—
198	SD₁₆	C₂×#261	1	2	yes	—
199	SD₁₆	C₂×#261	1	2	yes	—
200	D₈	—	13	2	no	—
201	D₈	—	13	2	no	—
202	M₁₆	—	13	2	no	—
203	M₁₆	—	13	2	no	—
204	M₁₆	—	13	2	no	—
205	M₁₆	—	13	2	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	—	26	4	no	—
208	M₁₆	—	26	1	no	—
209	M₁₆	—	26	1	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	26	4	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	—	26	1	no	—
219	C₂×C₈	—	26	1	no	—
220	C₂×C₈	C₂×#261	1	1	yes	characteristic
221	C₄⋊C₄	—	26	1	no	—
222	C₂²⋊C₄	—	26	1	no	—
223	C₄²	—	13	2	no	—
224	C₁₃	#2957	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	1	no	—
226	C₂³	—	13	1	no	—
227	C₂³	—	26	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	13	2	no	—
230	Q₈	—	26	1	no	—
231	Q₈	C₂²×#261	1	2	yes	—
232	D₄	—	2	1	no	—
233	D₄	—	13	2	no	—
234	D₄	—	26	2	no	—
235	D₄	—	26	2	no	—
236	D₄	—	26	2	no	—
237	D₄	—	26	2	no	—
238	D₄	—	26	1	no	—
239	D₄	C₂²×#261	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	26	2	no	—
248	C₂×C₄	—	26	2	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	1	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	C₂²×#261	1	1	yes	characteristic
253	C₈	—	13	4	no	—
254	C₈	—	13	1	no	—
255	C₈	—	13	1	no	—
256	C₈	—	26	2	no	—
257	C₈	—	26	2	no	—
258	C₈	C₂²×#261	1	1	yes	characteristic
259	C₈	C₂²×#261	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	13	1	no	—
263	C₂²	—	13	1	no	—
264	C₂²	—	26	2	no	—
265	C₂²	—	52	1	no	—
266	C₂²	—	104	1	no	—
267	C₂²	D₄×#261	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	13	1	no	—
270	C₄	—	13	1	no	—
271	C₄	—	26	2	no	—
272	C₄	—	52	2	no	—
273	C₄	D₄×#261	1	1	yes	characteristic
274	C₄	C₂³×#261	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	4	2	no	—
277	C₂	—	26	1	no	—
278	C₂	—	26	1	no	—
279	C₂	—	52	2	no	—
280	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	C₂×C₁₃×SD₁₆	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88170	C₂
cosocle	C₄×D₁₃	C₂⁴
hypercenter	C₂×SD₁₆	#261
hypocenter	C₁₃	#2957
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #11719625	C₂	semidirect
2. #11719621	C₂	semidirect
4. #11719155	C₂	semidirect
5. #11719140	C₂	semidirect
6. #11719108	C₂	semidirect
7. #11719107	C₂	semidirect
8. D₁₃⋊#108	C₂	semidirect
9. #11719089	C₂	semidirect
14. #88240	C₂²	semidirect
16. #88238	C₂²	semidirect
22. C₂×C₁₃×SD₁₆	C₄	semidirect
23. C₂×#6107	C₄	semidirect
24. C₂×#6104	C₄	semidirect
25. #88170	C₄	semidirect
26. #88170	C₄	semidirect
29. C₂×#6095	C₄	semidirect
31. #88118	C₂²	semidirect
32. #88116	C₂²	semidirect
33. #88104	C₂²	semidirect
34. #88103	C₂²	semidirect
35. #88100	C₂²	semidirect
36. C₈×#261	C₂²	semidirect
152. C₂×SD₁₆	#261	semidirect
224. C₁₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
#11719625	2
#11719621	2
C₂×#88170	2
#11719155	2
#11719140	2
#11719108	2
#11719107	2
D₁₃⋊#108	2
#11719089	2
#2957	13