Subgroups — #13123563

Order 1664.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6099	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718895	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718673	C₂	1	2	yes	maximal, complemented
7	#11718672	C₂	1	2	yes	maximal, complemented
8	#11718672	C₂	1	2	yes	maximal, complemented
9	D₄×#1055	C₂	1	2	yes	maximal, complemented
10	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718658	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#123	C₄	1	2	yes	—
15	C₁₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
16	C₁₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	C₁₃×#121	C₄	1	1	yes	characteristic
18	C₁₃×#121	C₄	1	1	yes	characteristic
19	C₁₃×#120	C₄	1	2	yes	—
20	C₄×D₄×C₁₃	C₂²	1	2	yes	—
21	C₄×D₄×C₁₃	C₂²	1	2	yes	—
22	C₁₃×#117	C₂²	1	1	yes	characteristic
23	#88202	—	2	4	no	—
24	C₂×#6099	—	2	4	no	—
25	C₂×#6099	C₂²	1	1	yes	characteristic
26	C₂×#6099	C₂²	1	2	yes	—
27	C₂×#6099	C₂²	1	1	yes	characteristic
28	C₂×#6099	C₂²	1	2	yes	complemented
29	C₂²×#1055	C₂²	1	1	yes	characteristic
30	C₂²×#1055	C₂²	1	1	yes	characteristic
31	C₂²×#1055	C₂²	1	2	yes	—
32	C₂²×#1055	C₂²	1	2	yes	complemented
33	C₂²⋊#1055	C₂²	1	1	yes	characteristic
34	C₂²⋊#1055	C₂²	1	1	yes	characteristic
35	C₂²⋊#1055	C₂²	1	1	yes	characteristic
36	C₂²⋊#1055	C₂²	1	1	yes	characteristic
37	C₂²⋊#1055	C₂²	1	2	yes	complemented
38	C₂²⋊#1055	C₂²	1	2	yes	complemented
39	C₄⋊#1055	C₂²	1	2	yes	complemented
40	C₄⋊#1055	C₂²	1	2	yes	complemented
41	#88043	C₂²	1	2	yes	complemented
42	C₄×#1055	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₁₃×#41	—	2	2	no	—
45	C₁₃×#41	—	2	2	no	—
46	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
48	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
49	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
65	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
66	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
67	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	C₄²×C₁₃	C₂³	1	2	yes	—
70	#6099	—	2	2	no	—
71	#6099	—	2	4	no	—
72	#6099	—	2	4	no	—
73	#6099	D₄	1	4	yes	complemented
74	C₂×#1055	—	2	2	no	—
75	C₂×#1055	—	2	2	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	—	2	4	no	—
78	C₂×#1055	C₂³	1	1	yes	characteristic
79	C₂×#1055	C₂³	1	1	yes	characteristic
80	C₂×#1055	C₂³	1	1	yes	characteristic
81	C₂×#1055	C₂³	1	1	yes	characteristic
82	C₂×#1055	C₂³	1	2	yes	—
83	C₂×#1055	C₂³	1	2	yes	—
84	#2933	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	4	1	no	—
88	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	Q₈×C₁₃	—	2	2	no	—
93	D₄×C₁₃	—	2	4	no	—
94	D₄×C₁₃	—	2	4	no	—
95	D₄×C₁₃	—	2	2	no	—
96	D₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	#41	1	2	yes	—
114	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
115	#1055	—	2	1	no	—
116	#1055	—	2	1	no	—
117	#1055	—	2	2	no	—
118	#1055	—	2	1	no	—
119	#1055	—	2	1	no	—
120	#1055	C₂×D₄	1	4	yes	—
121	C₂×#131	—	13	1	no	—
122	C₂²×M₁₆	—	13	1	no	—
123	#525	D₁₃	1	1	yes	characteristic
124	#436	—	13	2	no	—
125	#435	—	13	2	no	—
126	#435	—	13	2	no	—
127	C₈×D₄	—	13	2	no	—
128	#424	—	13	1	no	—
129	#408	—	13	1	no	—
130	C₂×C₂²⋊C₈	—	13	1	no	—
131	#405	—	13	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	1	no	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	4	4	no	—
143	C₂²×C₁₃	C₄×D₄	1	2	yes	—
144	C₂²×C₁₃	#131	1	2	yes	complemented
145	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	C₄×D₄	1	2	yes	—
152	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
156	#131	—	26	4	no	—
157	C₂×M₁₆	—	13	1	no	—
158	C₂×M₁₆	—	13	1	no	—
159	C₂×M₁₆	—	13	2	no	—
160	C₂×M₁₆	—	13	2	no	—
161	C₂×M₁₆	—	26	4	no	—
162	C₂²×C₈	—	13	2	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	—	13	1	no	—
166	#123	Dic₁₃	1	2	yes	—
167	C₂²⋊Q₈	Dic₁₃	1	1	yes	characteristic
168	C₂²⋊Q₈	Dic₁₃	1	1	yes	characteristic
169	#121	Dic₁₃	1	1	yes	characteristic
170	#121	Dic₁₃	1	1	yes	characteristic
171	#120	Dic₁₃	1	2	yes	—
172	C₄×D₄	C₂×D₁₃	1	2	yes	—
173	C₄×D₄	C₂×D₁₃	1	2	yes	—
174	#117	C₂×D₁₃	1	1	yes	characteristic
175	C₄⋊C₈	—	13	2	no	—
176	C₄⋊C₈	—	13	2	no	—
177	C₂²⋊C₈	—	13	1	no	—
178	C₂²⋊C₈	—	13	1	no	—
179	C₂²⋊C₈	—	13	1	no	—
180	C₂²⋊C₈	—	13	1	no	—
181	C₂²⋊C₈	—	13	2	no	—
182	C₂²⋊C₈	—	13	2	no	—
183	#97	—	13	2	no	—
184	C₄×C₈	—	13	2	no	—
185	C₂×C₁₃	—	2	2	no	—
186	C₂×C₁₃	—	2	2	no	—
187	C₂×C₁₃	—	4	2	no	—
188	C₂×C₁₃	—	4	1	no	—
189	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
191	C₂×C₁₃	#568	1	1	yes	characteristic
192	C₂⁴	C₂×Dic₁₃	1	1	yes	characteristic
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
196	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
197	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
198	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
199	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
209	M₁₆	—	13	4	no	—
210	M₁₆	—	26	4	no	—
211	M₁₆	—	26	4	no	—
212	M₁₆	—	26	2	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	—	26	4	no	—
220	C₂×C₈	—	26	4	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	26	2	no	—
223	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
224	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
225	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
226	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
227	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
228	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
229	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
230	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
231	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
232	C₄²	C₂²×D₁₃	1	2	yes	—
233	C₁₃	#2933	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
238	C₂³	C₂²×Dic₁₃	1	2	yes	—
239	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
240	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	D₄×D₁₃	1	2	yes	—
256	C₂×C₄	#6129	1	2	yes	—
257	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
258	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
259	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
262	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₈	—	13	4	no	—
265	C₈	—	26	1	no	—
266	C₈	—	26	2	no	—
267	C₈	—	26	1	no	—
268	C₈	—	26	1	no	—
269	C₈	—	26	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	D₄×Dic₁₃	1	2	yes	—
282	C₂²	#88202	1	2	yes	complemented
283	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	D₄×Dic₁₃	1	2	yes	—
290	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
291	C₄	C₂×#6129	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic
297	C₂	C₂×#88202	1	1	yes	characteristic
298	C₂	#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting, prime radical	C₁₃×#525	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#525	D₁₃
hypocenter	C₁₃	#2933
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂²×#6099	C₂	semidirect
4. #11718895	C₂	semidirect
5. C₂×C₂²⋊#1055	C₂	semidirect
6. #11718673	C₂	semidirect
7. #11718672	C₂	semidirect
8. #11718672	C₂	semidirect
9. D₄×#1055	C₂	semidirect
10. #11718660	C₂	semidirect
11. #11718658	C₂	semidirect
28. C₂×#6099	C₂²	semidirect
32. C₂²×#1055	C₂²	semidirect
37. C₂²⋊#1055	C₂²	semidirect
38. C₂²⋊#1055	C₂²	semidirect
39. C₄⋊#1055	C₂²	semidirect
40. C₄⋊#1055	C₂²	semidirect
41. #88043	C₂²	semidirect
42. C₄×#1055	C₂²	semidirect
73. #6099	D₄	semidirect
144. C₂²×C₁₃	#131	semidirect
233. C₁₃	#2933	semidirect
282. C₂²	#88202	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#525	2
C₂×#88202	2
C₂²×#6099	2
#11718895	2
C₂×C₂²⋊#1055	2
#11718673	2
#11718672	2
D₄×#1055	2
#11718660	2
#11718658	2
#2933	13