Subgroups — #13117912

Order 1664.

632 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88201	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718649	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#6101	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718331	C₂	1	2	yes	maximal, complemented
9	#11718330	C₂	1	2	yes	maximal, complemented
10	#11718329	C₂	1	2	yes	maximal, complemented
11	#11718327	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₅₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	#88201	C₄	1	8	yes	complemented
18	Q₈×Dic₁₃	—	2	4	no	—
19	C₂×#6103	C₂²	1	2	yes	—
20	C₂×#6103	C₂²	1	2	yes	—
21	C₂×#6103	C₂²	1	2	yes	—
22	C₂×#6103	C₂²	1	2	yes	—
23	C₂×#6103	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×#6101	C₂²	1	1	yes	characteristic
27	C₂×#6101	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
31	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
32	C₄×D₅₂	—	2	4	no	—
33	C₁₃×#95	C₂²	1	2	yes	—
34	C₁₃×#95	C₂²	1	2	yes	—
35	#88071	C₂²	1	1	yes	characteristic
36	#88071	C₂²	1	2	yes	—
37	#88071	C₂²	1	1	yes	characteristic
38	C₂³×D₁₃	C₂³	1	2	yes	—
39	C₂×Q₈×C₁₃	—	2	2	no	—
40	C₂×Q₈×C₁₃	—	2	2	no	—
41	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂³	1	2	yes	—
51	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂×D₅₂	—	2	2	no	—
53	C₂×D₅₂	—	2	2	no	—
54	C₂×D₅₂	C₂×C₄	1	4	yes	—
55	C₂×C₄×D₁₃	—	2	4	no	—
56	C₂×C₄×D₁₃	—	2	4	no	—
57	C₂×C₄×D₁₃	—	4	2	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	#6103	—	2	4	no	—
61	#6103	—	2	4	no	—
62	#6103	—	4	2	no	—
63	#6103	C₂×C₄	1	8	yes	complemented
64	#6103	C₂×C₄	1	8	yes	complemented
65	C₄⋊Dic₁₃	—	2	4	no	—
66	C₄⋊Dic₁₃	D₄	1	4	yes	—
67	#6101	—	2	2	no	—
68	#6101	—	2	2	no	—
69	C₄×Dic₁₃	—	2	2	no	—
70	C₄×Dic₁₃	—	2	4	no	—
71	C₄×Dic₁₃	—	4	2	no	—
72	C₄×Dic₁₃	C₂×C₄	1	4	yes	complemented
73	#2367	—	13	1	no	maximal, Sylow
74	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×D₁₃	—	2	2	no	—
76	C₂²×D₁₃	—	2	2	no	—
77	C₂²×D₁₃	—	2	2	no	—
78	C₂²×D₁₃	—	2	2	no	—
79	C₂²×D₁₃	—	2	2	no	—
80	C₂²×D₁₃	—	2	2	no	—
81	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
82	Q₈×C₁₃	—	2	8	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	#41	1	2	yes	—
97	C₂×C₄×C₁₃	#41	1	2	yes	—
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
110	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
111	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
112	C₂×Dic₁₃	#41	1	2	yes	—
113	D₅₂	—	2	8	no	—
114	C₄×D₁₃	—	4	8	no	—
115	C₂×#124	—	13	1	no	—
116	C₂×C₄×Q₈	—	13	1	no	—
117	C₂×C₄×D₄	—	13	1	no	—
118	#391	D₁₃	1	1	yes	characteristic, complemented
119	#390	—	13	1	no	—
120	#388	—	13	2	no	—
121	#388	—	13	2	no	—
122	#385	—	13	2	no	—
123	C₄×C₄⋊C₄	—	13	1	no	—
124	C₄×C₄⋊C₄	—	13	1	no	—
125	C₄×C₂²⋊C₄	—	13	2	no	—
126	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×D₁₃	—	2	4	no	—
134	C₂×D₁₃	—	2	4	no	—
135	C₂×D₁₃	—	2	4	no	—
136	C₂×D₁₃	—	4	2	no	—
137	C₂×D₁₃	—	4	2	no	—
138	C₂×D₁₃	—	4	2	no	—
139	C₂×D₁₃	—	4	2	no	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	—	4	2	no	—
144	C₄×C₁₃	C₄×D₄	1	4	yes	—
145	Dic₁₃	—	2	2	no	—
146	Dic₁₃	—	2	2	no	—
147	Dic₁₃	—	2	4	no	—
148	Dic₁₃	—	2	2	no	—
149	Dic₁₃	—	4	1	no	—
150	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂²×D₄	—	13	1	no	—
152	C₂³×C₄	—	13	2	no	—
153	#124	—	13	8	no	—
154	C₄×Q₈	—	26	4	no	—
155	C₄×D₄	—	26	4	no	—
156	C₂×C₄⋊C₄	—	13	1	no	—
157	C₂×C₄⋊C₄	—	13	2	no	—
158	C₂×C₄⋊C₄	—	13	1	no	—
159	C₂×C₄⋊C₄	—	13	1	no	—
160	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	—	13	2	no	—
162	C₂×C₂²⋊C₄	—	13	2	no	—
163	C₂×C₂²⋊C₄	—	13	2	no	—
164	C₂×C₂²⋊C₄	—	13	2	no	—
165	C₂×C₂²⋊C₄	—	13	2	no	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	13	2	no	—
168	C₂×C₄²	—	13	1	no	—
169	C₂×C₄²	—	13	2	no	—
170	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
171	#95	—	13	1	no	—
172	#95	—	13	2	no	—
173	#95	—	13	1	no	—
174	#95	C₂×D₁₃	1	2	yes	—
175	#95	C₂×D₁₃	1	2	yes	—
176	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
178	C₂×C₁₃	#520	1	1	yes	characteristic
179	C₂×C₁₃	#546	1	1	yes	characteristic
180	C₂×C₁₃	#548	1	1	yes	characteristic
181	C₂×C₁₃	#556	1	1	yes	characteristic
182	C₂×C₁₃	#559	1	1	yes	characteristic
183	D₁₃	—	4	4	no	—
184	C₂⁴	—	13	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	C₄×D₁₃	1	4	yes	complemented
188	C₂×D₄	—	13	4	no	—
189	C₂×D₄	—	26	2	no	—
190	C₂×D₄	—	26	2	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	1	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	26	4	no	—
197	C₂²×C₄	—	26	4	no	—
198	C₂²×C₄	—	52	2	no	—
199	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
200	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	13	4	no	—
206	C₄⋊C₄	—	26	4	no	—
207	C₄⋊C₄	—	26	2	no	—
208	C₄⋊C₄	—	26	2	no	—
209	C₂²⋊C₄	—	13	8	no	—
210	C₂²⋊C₄	—	13	8	no	—
211	C₂²⋊C₄	—	26	4	no	—
212	C₂²⋊C₄	—	26	4	no	—
213	C₂²⋊C₄	—	52	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	13	4	no	—
216	C₄²	—	26	4	no	—
217	C₄²	—	26	2	no	—
218	C₄²	—	52	2	no	—
219	C₁₃	#2367	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	13	4	no	—
221	C₂³	—	26	2	no	—
222	C₂³	—	26	2	no	—
223	C₂³	—	26	2	no	—
224	C₂³	—	26	2	no	—
225	C₂³	—	26	2	no	—
226	C₂³	—	26	2	no	—
227	C₂³	C₂³×D₁₃	1	1	yes	characteristic
228	Q₈	—	2	8	no	—
229	D₄	—	26	8	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	2	no	—
242	C₂×C₄	—	13	4	no	—
243	C₂×C₄	—	13	2	no	—
244	C₂×C₄	—	26	1	no	—
245	C₂×C₄	—	26	1	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	26	1	no	—
248	C₂×C₄	—	26	1	no	—
249	C₂×C₄	—	26	1	no	—
250	C₂×C₄	—	26	1	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	52	8	no	—
256	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
257	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
258	C₂×C₄	D₄×D₁₃	1	2	yes	—
259	C₂×C₄	#6129	1	2	yes	—
260	C₂×C₄	#6131	1	2	yes	—
261	C₂²	—	26	4	no	—
262	C₂²	—	26	4	no	—
263	C₂²	—	26	4	no	—
264	C₂²	—	52	2	no	—
265	C₂²	—	52	2	no	—
266	C₂²	—	52	2	no	—
267	C₂²	—	52	2	no	—
268	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
269	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
270	C₂²	C₂×#6129	1	1	yes	characteristic
271	C₂²	C₂×#6131	1	1	yes	characteristic
272	C₂²	#88254	1	1	yes	characteristic
273	C₂²	#41×D₁₃	1	1	yes	characteristic
274	C₂²	#88256	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	26	2	no	—
280	C₄	—	26	2	no	—
281	C₄	—	26	4	no	—
282	C₄	—	26	2	no	—
283	C₄	—	52	1	no	—
284	C₄	#88135	1	4	yes	—
285	C₂	—	52	4	no	—
286	C₂	C₂×#88135	1	1	yes	characteristic
287	C₂	C₄×#6131	1	1	yes	characteristic
288	C₂	#11719279	1	1	yes	characteristic
289	C₂	#11719335	1	1	yes	characteristic
290	C₂	#11719338	1	1	yes	characteristic
291	C₂	#11719386	1	1	yes	characteristic
292	C₂	#11719437	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#391	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₅₂	C₂²
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2367
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₁₃	C₂	semidirect
4. C₁₃×#391	C₂	semidirect
6. C₄⋊C₄×Dic₁₃	C₂	semidirect
7. C₄×#6101	C₂	semidirect
8. #11718331	C₂	semidirect
9. #11718330	C₂	semidirect
10. #11718329	C₂	semidirect
11. #11718327	C₂	semidirect
17. #88201	C₄	semidirect
41. C₂×Q₈×C₁₃	C₂×C₄	semidirect
63. #6103	C₂×C₄	semidirect
64. #6103	C₂×C₄	semidirect
72. C₄×Dic₁₃	C₂×C₄	semidirect
118. #391	D₁₃	semidirect
187. C₂×Q₈	C₄×D₁₃	semidirect
219. C₁₃	#2367	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88201	2
C₂×Q₈×Dic₁₃	2
C₂×C₄×D₅₂	2
C₁₃×#391	2
#11718649	2
C₄⋊C₄×Dic₁₃	2
C₄×#6101	2
#11718331	2
#11718330	2
#11718329	2
#11718327	2
#2367	13