Subgroups — C₁₃×#23674 #15710013

Order 3328.

766 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	C₁₃×#2911	C₂	1	2	yes	maximal
2	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2090	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2089	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1742	C₂	1	2	yes	maximal
7	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#117	C₂²	1	2	yes	—
13	C₁₃×#428	—	2	4	no	—
14	C₁₃×#428	C₂²	1	8	yes	—
15	C₁₃×#427	—	2	2	no	—
16	C₁₃×#427	—	2	2	no	—
17	C₁₃×#427	—	2	4	no	—
18	C₂×C₁₃×#107	C₂²	1	2	yes	—
19	C₂×C₁₃×#106	C₂²	1	2	yes	—
20	C₁₃×#424	—	2	2	no	—
21	C₁₃×#424	C₂²	1	2	yes	—
22	C₁₃×#424	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₁₃×#343	C₂²	1	8	yes	—
25	C₁₃×#341	C₂²	1	2	yes	—
26	C₁₃×#341	C₂²	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	4	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	4	no	—
32	C₂×C₁₃×M₁₆	Q₈	1	8	yes	—
33	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
34	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	—	2	4	no	—
37	C₂²×C₈×C₁₃	—	2	1	no	—
38	C₂²×C₈×C₁₃	D₄	1	2	yes	—
39	C₂²×C₈×C₁₃	D₄	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#117	—	2	1	no	—
43	C₁₃×#117	—	2	1	no	—
44	C₁₃×#117	—	2	2	no	—
45	C₁₃×#117	—	2	2	no	—
46	C₁₃×#117	—	2	2	no	—
47	C₁₃×#117	—	2	2	no	—
48	C₁₃×#117	D₄	1	2	yes	—
49	C₁₃×#117	C₂³	1	2	yes	—
50	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
51	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₄²×C₁₃	—	2	1	no	—
55	C₂×C₄²×C₁₃	—	2	2	no	—
56	C₁₃×#107	—	2	4	no	—
57	C₁₃×#107	—	2	8	no	—
58	C₁₃×#106	—	2	4	no	—
59	C₁₃×#106	—	2	8	no	—
60	C₁₃×C₄⋊C₈	—	2	4	no	—
61	C₁₃×C₄⋊C₈	—	2	4	no	—
62	#23674	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×M₁₆	—	2	8	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₁₃×M₁₆	C₂×Q₈	1	8	yes	—
79	C₂×C₈×C₁₃	—	2	8	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	4	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
93	C₁₃×C₂²⋊C₄	—	2	2	no	—
94	C₁₃×C₂²⋊C₄	—	2	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	—	4	2	no	—
101	#2911	C₂×C₁₃	1	2	yes	—
102	C₂×#427	C₂×C₁₃	1	1	yes	characteristic
103	C₂×#424	C₂×C₁₃	1	1	yes	characteristic
104	#2090	C₂×C₁₃	1	4	yes	complemented
105	#2089	C₂×C₁₃	1	4	yes	complemented
106	#1742	C₂×C₁₃	1	2	yes	—
107	C₂×#341	C₂×C₁₃	1	1	yes	characteristic
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	#121	1	1	yes	characteristic
110	C₂³×C₁₃	#121	1	1	yes	characteristic
111	C₂³×C₁₃	#125	1	4	yes	—
112	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	8	4	no	—
117	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₁₃	#123	1	2	yes	—
122	C₂×C₄×C₁₃	#123	1	2	yes	—
123	C₂×C₄×C₁₃	#125	1	4	yes	—
124	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
130	C₈×C₁₃	—	2	8	no	—
131	C₈×C₁₃	—	2	2	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	—	4	2	no	—
134	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
135	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
136	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×#117	C₂²×C₁₃	1	2	yes	—
139	#428	—	2	4	no	—
140	#428	C₂²×C₁₃	1	8	yes	—
141	#427	—	2	2	no	—
142	#427	—	2	2	no	—
143	#427	—	2	4	no	—
144	C₂×#107	C₂²×C₁₃	1	2	yes	—
145	C₂×#106	C₂²×C₁₃	1	2	yes	—
146	#424	—	2	2	no	—
147	#424	C₂²×C₁₃	1	2	yes	—
148	#424	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄⋊C₈	—	2	2	no	—
150	#343	C₂²×C₁₃	1	8	yes	—
151	#341	C₂²×C₁₃	1	2	yes	—
152	#341	C₂²×C₁₃	1	2	yes	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	#398	1	2	yes	—
158	C₂²×C₁₃	#398	1	2	yes	—
159	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#125	1	2	yes	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	—	8	4	no	—
163	C₄×C₁₃	#398	1	2	yes	—
164	C₄×C₁₃	#398	1	2	yes	—
165	C₄×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
166	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
167	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	Q₈×C₁₃	1	8	yes	—
174	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
175	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	D₄×C₁₃	1	2	yes	—
180	C₂²×C₈	D₄×C₁₃	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	#117	—	2	1	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	D₄×C₁₃	1	2	yes	—
190	#117	C₂³×C₁₃	1	2	yes	—
191	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
193	C₂×C₂²⋊C₄	—	2	1	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	—	2	2	no	—
197	#107	—	2	4	no	—
198	#107	—	2	8	no	—
199	#106	—	2	4	no	—
200	#106	—	2	8	no	—
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	C₂×#398	1	1	yes	characteristic
207	C₂×C₁₃	#3067	1	2	yes	—
208	C₂⁴	C₁₃×#41	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
216	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
217	C₂²×C₄	C₁₃×#41	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
219	M₁₆	—	2	8	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	C₂×Q₈×C₁₃	1	8	yes	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₁₃	#23674	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	4	2	no	—
248	C₂³	C₁₃×#121	1	1	yes	characteristic
249	C₂³	C₁₃×#121	1	1	yes	characteristic
250	C₂³	C₁₃×#125	1	4	yes	—
251	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
257	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
258	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
259	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
260	C₂×C₄	C₁₃×#123	1	2	yes	—
261	C₂×C₄	C₁₃×#123	1	2	yes	—
262	C₂×C₄	C₁₃×#125	1	4	yes	—
263	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
264	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
268	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
269	C₈	—	2	8	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₁₃×#398	1	2	yes	—
278	C₂²	C₁₃×#398	1	2	yes	—
279	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
280	C₂²	C₂×C₁₃×#125	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	8	4	no	—
283	C₄	C₁₃×#398	1	2	yes	—
284	C₄	C₁₃×#398	1	2	yes	—
285	C₄	C₂×C₁₃×C₂²⋊Q₈	1	2	yes	—
286	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
287	C₄	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₂×C₁₃×#398	1	1	yes	characteristic
292	C₂	C₁₃×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂⁴×C₁₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#2090	C₂	semidirect
5. C₁₃×#2089	C₂	semidirect
62. #23674	C₁₃	direct (N × H)
104. #2090	C₂×C₁₃	semidirect
105. #2089	C₂×C₁₃	semidirect
246. C₁₃	#23674	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2911	2
C₂×C₁₃×#427	2
C₂×C₁₃×#424	2
C₁₃×#2090	2
C₁₃×#2089	2
C₁₃×#1742	2
C₂×C₁₃×#341	2
#23674	13

Subgroups — C13×#23674 #15710013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23674 #15710013