Subgroups — #13131559

Order 1664.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88238	C₂	1	2	yes	maximal, complemented
2	#11719301	C₂	1	1	yes	maximal, characteristic
3	#11719146	C₂	1	2	yes	maximal, complemented
4	#11719145	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#1057	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719129	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#1057	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719076	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719073	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719066	C₂	1	2	yes	maximal, complemented
11	#11719063	C₂	1	2	yes	maximal, complemented
12	C₂×#6129	C₂²	1	2	yes	—
13	C₂×#6122	C₂²	1	2	yes	—
14	C₂×#6122	C₂²	1	2	yes	—
15	C₂²×#1057	C₂²	1	2	yes	—
16	C₂²×#1057	C₂²	1	1	yes	characteristic
17	C₂²×#1057	C₂²	1	1	yes	characteristic
18	#88238	—	2	8	no	—
19	C₂×#6118	C₂²	1	2	yes	—
20	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
21	C₁₃×#121	C₄	1	1	yes	characteristic
22	#88194	C₄	1	1	yes	characteristic
23	#88193	C₄	1	2	yes	—
24	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	#88184	C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
27	#88147	C₂²	1	1	yes	characteristic
28	#88142	C₄	1	1	yes	characteristic
29	#88139	C₄	1	2	yes	—
30	#88135	C₂²	1	2	yes	—
31	C₂²⋊#1057	C₂²	1	2	yes	—
32	C₂²⋊#1057	C₂²	1	2	yes	—
33	#88112	C₂²	1	2	yes	—
34	#88111	C₂²	1	2	yes	—
35	#88111	C₂²	1	2	yes	—
36	C₄⋊#1057	C₂²	1	1	yes	characteristic
37	C₄⋊#1057	C₂²	1	1	yes	characteristic
38	C₄×#1057	C₂²	1	1	yes	characteristic
39	C₄×#1057	C₂²	1	1	yes	characteristic
40	C₄×#1057	C₂²	1	1	yes	characteristic
41	C₄×#1057	C₂²	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
48	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
51	#6129	—	2	4	no	—
52	#6129	—	2	4	no	—
53	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂³	1	2	yes	—
55	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
56	#6122	—	2	4	no	—
57	#6122	—	2	4	no	—
58	C₂×#1057	—	2	2	no	—
59	C₂×#1057	—	2	4	no	—
60	C₂×#1057	D₄	1	4	yes	complemented
61	C₂×#1057	C₂³	1	1	yes	characteristic
62	C₂×#1057	C₂³	1	1	yes	characteristic
63	C₂×#1057	C₂³	1	2	yes	—
64	C₂×#1057	C₂³	1	2	yes	—
65	C₂×#1057	C₂³	1	1	yes	characteristic
66	C₂×#1057	C₂³	1	1	yes	characteristic
67	#6118	—	2	4	no	—
68	D₁₃⋊C₈	—	2	4	no	—
69	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
70	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
71	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
72	#6103	C₂×C₄	1	2	yes	—
73	#6103	C₂×C₄	1	2	yes	—
74	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
75	#6101	C₂×C₄	1	2	yes	—
76	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
79	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
80	#2931	—	13	1	no	maximal, Sylow
81	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
84	D₄×C₁₃	—	2	4	no	—
85	D₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²⋊D₁₃	—	2	4	no	—
91	C₂²⋊D₁₃	—	2	4	no	—
92	C₂²⋊D₁₃	—	2	4	no	—
93	C₂×Dic₁₃	—	2	2	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
101	C₂×Dic₁₃	#41	1	2	yes	—
102	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
103	D₅₂	—	2	2	no	—
104	C₄×D₁₃	—	2	2	no	—
105	C₄×D₁₃	—	2	2	no	—
106	Dic₂₆	—	2	4	no	—
107	#1057	—	2	2	no	—
108	#1057	—	2	2	no	—
109	#1057	—	2	1	no	—
110	#1057	—	2	1	no	—
111	#1057	C₂×D₄	1	2	yes	—
112	#1057	C₂×D₄	1	2	yes	—
113	C₂×#131	—	13	2	no	—
114	#532	—	13	1	no	—
115	#436	—	13	2	no	—
116	#436	—	13	1	no	—
117	#436	—	13	1	no	—
118	C₈×D₄	—	13	2	no	—
119	C₈×D₄	—	13	1	no	—
120	C₈×D₄	—	13	1	no	—
121	#423	—	13	1	no	—
122	#408	—	13	2	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	C₄×D₄	1	2	yes	—
130	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₁₃	—	2	2	no	—
132	C₂×D₁₃	—	2	2	no	—
133	C₂×D₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	#131	1	2	yes	complemented
137	Dic₁₃	—	2	2	no	—
138	Dic₁₃	—	2	1	no	—
139	Dic₁₃	C₄×D₄	1	2	yes	—
140	Dic₁₃	#131	1	2	yes	complemented
141	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
142	Dic₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×#41	—	13	2	no	—
144	#131	—	26	8	no	—
145	C₂×M₁₆	—	13	2	no	—
146	C₂×M₁₆	—	13	2	no	—
147	C₂×M₁₆	—	13	2	no	—
148	C₂²×C₈	—	13	1	no	—
149	C₂²×C₈	—	13	2	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	—	13	1	no	—
152	C₄⋊Q₈	—	13	1	no	—
153	#127	—	13	1	no	—
154	#124	—	13	2	no	—
155	#121	—	13	2	no	—
156	#121	—	13	1	no	—
157	#121	#261	1	1	yes	characteristic
158	C₄×D₄	—	13	1	no	—
159	C₄×D₄	—	13	1	no	—
160	C₄×D₄	—	13	2	no	—
161	C₂×C₄²	—	13	1	no	—
162	C₄⋊C₈	—	13	1	no	—
163	C₄⋊C₈	—	13	1	no	—
164	C₄⋊C₈	—	13	2	no	—
165	C₂²⋊C₈	—	13	2	no	—
166	C₂²⋊C₈	—	13	2	no	—
167	C₂²⋊C₈	—	13	2	no	—
168	C₂²⋊C₈	—	13	2	no	—
169	C₄×C₈	—	13	1	no	—
170	C₄×C₈	—	13	1	no	—
171	C₄×C₈	—	13	1	no	—
172	C₄×C₈	—	13	1	no	—
173	C₂×C₁₃	—	2	2	no	—
174	C₂×C₁₃	—	4	2	no	—
175	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
178	D₁₃	—	4	2	no	—
179	#41	—	26	4	no	—
180	#41	—	26	4	no	—
181	C₂×Q₈	—	13	2	no	—
182	C₂×D₄	—	13	2	no	—
183	C₂×D₄	—	13	2	no	—
184	C₂×D₄	—	13	2	no	—
185	C₂×D₄	—	13	1	no	—
186	C₂×D₄	C₂×#261	1	1	yes	characteristic
187	C₂×D₄	C₂×#261	1	2	yes	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	C₂×#261	1	1	yes	characteristic
193	M₁₆	—	26	4	no	—
194	M₁₆	—	26	4	no	—
195	M₁₆	—	26	4	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	1	no	—
198	C₂×C₈	—	13	4	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	—	13	1	no	—
201	C₂×C₈	—	13	2	no	—
202	C₂×C₈	—	13	2	no	—
203	C₂×C₈	—	26	4	no	—
204	C₂×C₈	—	26	4	no	—
205	C₂×C₈	—	26	2	no	—
206	C₄⋊C₄	—	13	2	no	—
207	C₄⋊C₄	—	13	1	no	—
208	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
209	C₂²⋊C₄	—	13	2	no	—
210	C₂²⋊C₄	—	13	2	no	—
211	C₂²⋊C₄	—	13	2	no	—
212	C₂²⋊C₄	C₂×#261	1	2	yes	—
213	C₄²	—	13	1	no	—
214	C₄²	—	13	1	no	—
215	C₄²	—	13	1	no	—
216	C₄²	—	13	1	no	—
217	C₁₃	#2931	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	13	2	no	—
219	C₂³	C₂²×#261	1	2	yes	—
220	C₂³	C₂²×#261	1	1	yes	characteristic
221	Q₈	—	26	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	26	4	no	—
225	D₄	—	26	4	no	—
226	D₄	—	26	4	no	—
227	D₄	—	26	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	13	2	no	—
230	C₂×C₄	—	13	2	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	13	1	no	—
233	C₂×C₄	—	13	1	no	—
234	C₂×C₄	—	13	1	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	26	2	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	26	2	no	—
241	C₂×C₄	C₂²×#261	1	1	yes	characteristic
242	C₂×C₄	C₂²×#261	1	2	yes	—
243	C₂×C₄	C₂²×#261	1	1	yes	characteristic
244	C₈	—	13	2	no	—
245	C₈	—	13	2	no	—
246	C₈	—	26	1	no	—
247	C₈	—	26	1	no	—
248	C₈	—	26	2	no	—
249	C₈	—	26	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	26	2	no	—
256	C₂²	—	26	2	no	—
257	C₂²	—	26	2	no	—
258	C₂²	D₄×#261	1	2	yes	—
259	C₂²	C₂³×#261	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	13	2	no	—
263	C₄	—	13	1	no	—
264	C₄	—	13	1	no	—
265	C₄	—	13	2	no	—
266	C₄	—	26	2	no	—
267	C₄	—	26	1	no	—
268	C₄	#88238	1	2	yes	complemented
269	C₂	—	2	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	52	2	no	—
272	C₂	C₂×#88238	1	1	yes	characteristic
273	C₂	C₂×D₄×#261	1	1	yes	characteristic
274	C₂	D₁₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#121	C₄
prime radical	#11719301	C₂
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	#121	#261
hypocenter	C₁₃	#2931
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#88238	C₂	semidirect
3. #11719146	C₂	semidirect
4. #11719145	C₂	semidirect
5. D₄×#1057	C₂	semidirect
6. #11719129	C₂	semidirect
7. C₂×C₄×#1057	C₂	semidirect
8. #11719076	C₂	semidirect
9. #11719073	C₂	semidirect
10. #11719066	C₂	semidirect
11. #11719063	C₂	semidirect
60. C₂×#1057	D₄	semidirect
136. C₄×C₁₃	#131	semidirect
140. Dic₁₃	#131	semidirect
217. C₁₃	#2931	semidirect
268. C₄	#88238	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88238	2
#11719301	2
#11719146	2
#11719145	2
D₄×#1057	2
#11719129	2
C₂×C₄×#1057	2
#11719076	2
#11719073	2
#11719066	2
#11719063	2
#2931	13