Subgroups — #13131716

Order 1664.

410 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	#11719301	C₂	1	1	yes	maximal, characteristic
2	#11719147	C₂	1	2	yes	maximal, complemented
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	#11719125	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719119	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	C₂²×#1057	C₂²	1	2	yes	—
15	C₂×#6118	C₂²	1	2	yes	—
16	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
17	C₁₃×#121	C₄	1	1	yes	characteristic
18	#88194	C₄	1	1	yes	characteristic
19	#88193	C₄	1	2	yes	—
20	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	#88184	C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
23	#88147	C₂²	1	1	yes	characteristic
24	#88142	C₄	1	1	yes	characteristic
25	#88139	C₄	1	2	yes	—
26	#88135	C₂²	1	2	yes	—
27	C₂²⋊#1057	C₂²	1	2	yes	—
28	C₂²⋊#1057	C₂²	1	2	yes	—
29	C₂²⋊#1057	C₂²	1	2	yes	—
30	C₂²⋊#1057	C₂²	1	1	yes	characteristic
31	C₂²⋊#1057	C₂²	1	1	yes	characteristic
32	#88112	C₂²	1	1	yes	characteristic
33	#88112	C₂²	1	1	yes	characteristic
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	2	yes	—
39	C₄×#1057	C₂²	1	1	yes	characteristic
40	C₄×#1057	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
47	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	#6129	—	4	2	no	—
51	#6129	—	4	2	no	—
52	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₃	C₂³	1	2	yes	—
54	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
55	#6122	—	4	2	no	—
56	C₂×#1057	—	4	2	no	—
57	C₂×#1057	C₂³	1	2	yes	—
58	C₂×#1057	C₂³	1	2	yes	—
59	C₂×#1057	C₂³	1	1	yes	characteristic
60	C₂×#1057	C₂³	1	1	yes	characteristic
61	C₂×#1057	C₂³	1	1	yes	characteristic
62	C₂×#1057	C₂³	1	1	yes	characteristic
63	#6118	—	4	2	no	—
64	D₁₃⋊C₈	—	4	2	no	—
65	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
68	#6103	C₂×C₄	1	2	yes	—
69	#6103	C₂×C₄	1	2	yes	—
70	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	#6101	C₂×C₄	1	2	yes	—
72	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
75	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
76	#2979	—	13	1	no	maximal, Sylow
77	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
80	D₄×C₁₃	—	2	2	no	—
81	D₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊D₁₃	—	2	4	no	—
87	C₂²⋊D₁₃	—	4	2	no	—
88	C₂²⋊D₁₃	—	4	2	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
99	D₅₂	—	2	2	no	—
100	C₄×D₁₃	—	2	2	no	—
101	C₄×D₁₃	—	2	2	no	—
102	Dic₂₆	—	4	2	no	—
103	#1057	—	2	1	no	—
104	#1057	—	2	2	no	—
105	#1057	—	2	1	no	—
106	#1057	—	2	2	no	—
107	#1057	—	2	1	no	—
108	#1057	—	2	1	no	—
109	#532	—	13	1	no	—
110	#436	—	13	2	no	—
111	#436	—	13	1	no	—
112	#436	—	13	1	no	—
113	C₈×D₄	—	13	2	no	—
114	C₈×D₄	—	13	1	no	—
115	C₈×D₄	—	13	1	no	—
116	#431	—	13	1	no	—
117	#431	—	13	1	no	—
118	#408	—	13	2	no	—
119	#408	—	13	2	no	—
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×D₁₃	—	2	2	no	—
128	C₂×D₁₃	—	2	2	no	—
129	C₂×D₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	#131	1	2	yes	—
133	Dic₁₃	—	2	1	no	—
134	Dic₁₃	—	2	2	no	—
135	Dic₁₃	—	2	1	no	—
136	Dic₁₃	#131	1	2	yes	—
137	Dic₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	Dic₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×#41	—	13	2	no	—
140	C₂×M₁₆	—	13	2	no	—
141	C₂×M₁₆	—	13	2	no	—
142	C₂²×C₈	—	13	2	no	—
143	C₂²×C₈	—	13	2	no	—
144	C₄⋊Q₈	—	13	1	no	—
145	#127	—	13	1	no	—
146	#124	—	13	2	no	—
147	#121	—	13	2	no	—
148	#121	—	13	1	no	—
149	#121	#261	1	1	yes	characteristic
150	C₄×D₄	—	13	1	no	—
151	C₄×D₄	—	13	1	no	—
152	C₄×D₄	—	13	2	no	—
153	C₂×C₄²	—	13	1	no	—
154	C₄⋊C₈	—	13	1	no	—
155	C₄⋊C₈	—	13	1	no	—
156	C₄⋊C₈	—	13	1	no	—
157	C₄⋊C₈	—	13	1	no	—
158	C₂²⋊C₈	—	13	2	no	—
159	C₂²⋊C₈	—	13	2	no	—
160	C₂²⋊C₈	—	13	2	no	—
161	C₂²⋊C₈	—	13	1	no	—
162	C₂²⋊C₈	—	13	1	no	—
163	C₂²⋊C₈	—	13	2	no	—
164	C₂²⋊C₈	—	13	2	no	—
165	C₄×C₈	—	13	2	no	—
166	C₄×C₈	—	13	1	no	—
167	C₄×C₈	—	13	1	no	—
168	C₂×C₁₃	—	4	1	no	—
169	C₂×C₁₃	—	4	2	no	—
170	C₂×C₁₃	#518	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
172	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
173	D₁₃	—	4	2	no	—
174	#41	—	52	2	no	—
175	#41	—	52	2	no	—
176	C₂×Q₈	—	13	2	no	—
177	C₂×D₄	—	13	2	no	—
178	C₂×D₄	—	13	2	no	—
179	C₂×D₄	—	13	2	no	—
180	C₂×D₄	—	13	1	no	—
181	C₂×D₄	C₂×#261	1	1	yes	characteristic
182	C₂×D₄	C₂×#261	1	2	yes	—
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	2	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	C₂×#261	1	1	yes	characteristic
188	M₁₆	—	52	2	no	—
189	M₁₆	—	52	2	no	—
190	C₂×C₈	—	13	1	no	—
191	C₂×C₈	—	13	2	no	—
192	C₂×C₈	—	13	1	no	—
193	C₂×C₈	—	13	1	no	—
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	52	2	no	—
197	C₂×C₈	—	52	2	no	—
198	C₄⋊C₄	—	13	2	no	—
199	C₄⋊C₄	—	13	1	no	—
200	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
201	C₂²⋊C₄	—	13	2	no	—
202	C₂²⋊C₄	—	13	2	no	—
203	C₂²⋊C₄	—	13	2	no	—
204	C₂²⋊C₄	C₂×#261	1	2	yes	—
205	C₄²	—	13	1	no	—
206	C₄²	—	13	1	no	—
207	C₄²	—	13	1	no	—
208	C₄²	—	13	1	no	—
209	C₁₃	#2979	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	13	2	no	—
211	C₂³	C₂²×#261	1	2	yes	—
212	C₂³	C₂²×#261	1	1	yes	characteristic
213	Q₈	—	52	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	26	4	no	—
217	D₄	—	26	2	no	—
218	D₄	—	52	2	no	—
219	D₄	—	52	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	13	1	no	—
222	C₂×C₄	—	13	1	no	—
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	2	no	—
226	C₂×C₄	—	26	2	no	—
227	C₂×C₄	—	26	2	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	2	no	—
230	C₂×C₄	—	26	2	no	—
231	C₂×C₄	—	26	1	no	—
232	C₂×C₄	—	26	1	no	—
233	C₂×C₄	C₂²×#261	1	1	yes	characteristic
234	C₂×C₄	C₂²×#261	1	2	yes	—
235	C₂×C₄	C₂²×#261	1	1	yes	characteristic
236	C₈	—	26	1	no	—
237	C₈	—	26	2	no	—
238	C₈	—	26	1	no	—
239	C₈	—	26	1	no	—
240	C₈	—	26	2	no	—
241	C₈	—	26	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	26	2	no	—
249	C₂²	—	26	2	no	—
250	C₂²	—	26	2	no	—
251	C₂²	C₂³×#261	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	13	1	no	—
255	C₄	—	13	1	no	—
256	C₄	—	13	2	no	—
257	C₄	—	26	1	no	—
258	C₄	—	26	2	no	—
259	C₄	—	26	1	no	—
260	C₄	#88238	1	2	yes	—
261	C₂	—	4	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	52	2	no	—
264	C₂	C₂×#88238	1	1	yes	characteristic
265	C₂	#11719623	1	1	yes	characteristic
266	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	#2979
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
2. #11719147	C₂	semidirect
3. #11719146	C₂	semidirect
4. #11719145	C₂	semidirect
5. D₄×#1057	C₂	semidirect
6. #11719125	C₂	semidirect
7. #11719119	C₂	semidirect
8. #11719076	C₂	semidirect
9. #11719073	C₂	semidirect
10. #11719066	C₂	semidirect
11. #11719063	C₂	semidirect
209. C₁₃	#2979	semidirect

Maximal subgroups

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

subgroup	index
#11719301	2
#11719147	2
#11719146	2
#11719145	2
D₄×#1057	2
#11719125	2
#11719119	2
#11719076	2
#11719073	2
#11719066	2
#11719063	2
#2979	13