Subgroups — #13123576

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719537	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88155	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718790	C₂	1	1	yes	maximal, characteristic, complemented
6	M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88054	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718391	C₂	1	2	yes	maximal, complemented
9	#11718390	C₂	1	2	yes	maximal, complemented
10	#11718389	C₂	1	2	yes	maximal, complemented
11	#11718386	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×#6127	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#88202	—	2	4	no	—
17	#88201	C₄	1	1	yes	characteristic
18	#88198	C₄	1	1	yes	characteristic
19	#88194	C₄	1	1	yes	characteristic
20	#88193	C₄	1	2	yes	—
21	#88193	C₄	1	2	yes	—
22	#88190	C₄	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×#6099	C₂²	1	2	yes	—
27	C₂×#6099	C₂²	1	1	yes	characteristic
28	C₂²×#1055	C₂²	1	1	yes	characteristic
29	C₂²×#1055	C₂²	1	2	yes	—
30	#88155	—	2	4	no	—
31	C₂×#6094	C₂²	1	2	yes	—
32	C₂×C₈×D₁₃	C₂²	1	2	yes	—
33	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	#88060	C₂²	1	2	yes	—
36	#88060	C₂²	1	2	yes	—
37	#88055	C₂²	1	2	yes	—
38	#88054	C₂²	1	1	yes	characteristic
39	#88054	C₂²	1	1	yes	characteristic
40	#88054	C₂²	1	1	yes	characteristic
41	#88054	C₂²	1	1	yes	characteristic
42	C₈×Dic₁₃	C₂²	1	2	yes	—
43	C₁₃×#41	—	2	2	no	—
44	C₁₃×#41	—	2	2	no	—
45	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
53	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
55	#6127	—	2	2	no	—
56	#6127	—	2	2	no	—
57	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₃	C₂³	1	2	yes	—
59	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
60	C₁₃×M₁₆	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂³	1	2	yes	—
65	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
67	#6103	C₂×C₄	1	2	yes	—
68	#6103	C₂×C₄	1	2	yes	—
69	#6101	C₂×C₄	1	2	yes	—
70	#6101	C₂×C₄	1	2	yes	—
71	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
74	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
75	#6099	—	2	4	no	—
76	#6099	D₄	1	4	yes	complemented
77	C₂×#1055	—	2	2	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	C₂³	1	1	yes	characteristic
80	C₂×#1055	C₂³	1	1	yes	characteristic
81	C₂×#1055	C₂³	1	2	yes	—
82	#6094	—	2	4	no	—
83	C₈×D₁₃	—	2	4	no	—
84	#2935	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
88	Q₈×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	2	4	no	—
90	D₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	#41	1	2	yes	—
97	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₁₃	—	2	4	no	—
99	C₂²⋊D₁₃	—	2	4	no	—
100	C₂²⋊D₁₃	—	4	2	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	4	1	no	—
103	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
107	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	D₅₂	—	2	2	no	—
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	Dic₂₆	—	2	2	no	—
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	2	no	—
115	#1055	—	2	1	no	—
116	#1055	—	2	1	no	—
117	#1055	C₂×D₄	1	4	yes	—
118	C₂×#131	—	13	1	no	—
119	C₂×#131	—	13	1	no	—
120	#532	—	13	1	no	—
121	#436	—	13	2	no	—
122	#435	—	13	2	no	—
123	#435	—	13	2	no	—
124	C₈×D₄	—	13	2	no	—
125	C₂×C₄⋊C₈	—	13	1	no	—
126	#408	—	13	1	no	—
127	#408	D₁₃	1	1	yes	characteristic, complemented
128	C₄×M₁₆	—	13	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	C₄×D₄	1	2	yes	—
135	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×D₁₃	—	2	2	no	—
137	C₂×D₁₃	—	2	2	no	—
138	C₂×D₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	C₄×D₄	1	2	yes	—
141	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	Dic₁₃	—	2	2	no	—
144	Dic₁₃	—	2	1	no	—
145	Dic₁₃	—	2	1	no	—
146	Dic₁₃	—	2	1	no	—
147	Dic₁₃	#131	1	2	yes	complemented
148	C₂×#41	—	13	1	no	—
149	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
150	#131	—	26	4	no	—
151	#131	—	26	4	no	—
152	C₂×M₁₆	—	13	1	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂×M₁₆	—	13	2	no	—
155	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
156	C₂²×C₈	—	13	1	no	—
157	C₂²×C₈	—	13	2	no	—
158	C₂²×C₈	—	13	2	no	—
159	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	13	1	no	—
161	#127	—	13	1	no	—
162	#124	—	13	1	no	—
163	#124	—	13	1	no	—
164	#121	—	13	2	no	—
165	#121	—	13	2	no	—
166	C₄×D₄	—	13	2	no	—
167	C₄×D₄	—	13	2	no	—
168	C₂×C₄²	—	13	1	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	C₂²⋊C₈	—	13	2	no	—
174	C₂²⋊C₈	—	13	2	no	—
175	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
176	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
177	#97	—	13	2	no	—
178	C₄×C₈	—	13	2	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	—	4	2	no	—
181	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
183	C₂×C₁₃	#568	1	1	yes	characteristic
184	D₁₃	—	4	2	no	—
185	#41	—	2	2	no	—
186	#41	—	2	2	no	—
187	#41	—	26	2	no	—
188	#41	—	26	2	no	—
189	C₂×Q₈	—	13	1	no	—
190	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂×D₄	—	13	2	no	—
192	C₂×D₄	—	13	2	no	—
193	C₂×D₄	—	13	2	no	—
194	C₂×D₄	—	13	1	no	—
195	C₂×D₄	C₄×D₁₃	1	2	yes	—
196	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	13	4	no	—
204	M₁₆	—	26	4	no	—
205	M₁₆	—	26	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	1	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	26	4	no	—
211	C₂×C₈	—	26	4	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	C₂²×D₁₃	1	2	yes	—
214	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
216	C₄⋊C₄	—	13	2	no	—
217	C₄⋊C₄	—	13	2	no	—
218	C₂²⋊C₄	—	13	2	no	—
219	C₂²⋊C₄	—	13	2	no	—
220	C₂²⋊C₄	—	13	2	no	—
221	C₂²⋊C₄	—	13	2	no	—
222	C₄²	—	13	1	no	—
223	C₄²	—	13	1	no	—
224	C₄²	—	13	1	no	—
225	C₄²	—	13	1	no	—
226	C₁₃	#2935	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	13	2	no	—
228	C₂³	C₂×C₄×D₁₃	1	2	yes	—
229	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	26	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	26	4	no	—
235	D₄	—	26	4	no	—
236	D₄	—	26	2	no	—
237	D₄	—	52	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	13	1	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	2	no	—
245	C₂×C₄	—	26	2	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	26	2	no	—
248	C₂×C₄	—	52	1	no	—
249	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂×C₄	D₄×D₁₃	1	2	yes	—
252	C₂×C₄	#6129	1	2	yes	—
253	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	2	no	—
257	C₈	—	13	4	no	—
258	C₈	—	26	1	no	—
259	C₈	—	26	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	26	2	no	—
266	C₂²	—	26	2	no	—
267	C₂²	—	26	2	no	—
268	C₂²	#88135	1	2	yes	—
269	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	13	2	no	—
272	C₄	—	26	2	no	—
273	C₄	—	26	1	no	—
274	C₄	—	26	1	no	—
275	C₄	—	26	1	no	—
276	C₄	#88135	1	2	yes	—
277	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
278	C₄	C₂×#6129	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	52	2	no	—
282	C₂	C₂×#88135	1	1	yes	characteristic
283	C₂	D₁₃×#131	1	1	yes	characteristic
284	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#408	C₂
prime radical	#11719537	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#408	D₁₃
hypocenter	C₁₃	#2935
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂×#88155	C₂	semidirect
4. C₁₃×#408	C₂	semidirect
5. #11718790	C₂	semidirect
6. M₁₆×Dic₁₃	C₂	semidirect
7. C₂×#88054	C₂	semidirect
8. #11718391	C₂	semidirect
9. #11718390	C₂	semidirect
10. #11718389	C₂	semidirect
11. #11718386	C₂	semidirect
76. #6099	D₄	semidirect
127. #408	D₁₃	semidirect
147. Dic₁₃	#131	semidirect
226. C₁₃	#2935	semidirect

Maximal subgroups

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

subgroup	index
#11719537	2
C₂×#88202	2
C₂×#88155	2
C₁₃×#408	2
#11718790	2
M₁₆×Dic₁₃	2
C₂×#88054	2
#11718391	2
#11718390	2
#11718389	2
#11718386	2
#2935	13