Subgroups — C₁₃×#29210 #15715549

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2059	C₂	1	4	yes	maximal
4	C₁₃×#1991	C₂	1	2	yes	maximal
5	C₁₃×#1948	C₂	1	2	yes	maximal
6	C₁₃×#1947	C₂	1	2	yes	maximal
7	C₁₃×#1846	C₂	1	2	yes	maximal
8	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₂²	1	4	yes	—
12	C₂×C₁₃×#106	C₂²	1	4	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#389	—	2	2	no	—
17	C₁₃×#389	—	2	2	no	—
18	C₁₃×#389	C₂²	1	2	yes	—
19	C₁₃×#387	—	2	2	no	—
20	C₁₃×#384	C₂²	1	2	yes	—
21	C₁₃×#382	C₂²	1	2	yes	—
22	C₁₃×#382	C₂²	1	1	yes	characteristic
23	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄³×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₁₃×#125	—	2	2	no	—
32	C₁₃×#125	—	4	2	no	—
33	C₁₃×#125	—	4	2	no	—
34	C₁₃×#125	Q₈	1	4	yes	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃	—	2	2	no	—
44	C₂×C₄²×C₁₃	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#107	—	4	4	no	—
50	C₁₃×#106	—	4	4	no	—
51	C₁₃×C₄⋊C₈	Q₈	1	4	yes	—
52	C₁₃×C₄⋊C₈	Q₈	1	4	yes	—
53	C₄×C₈×C₁₃	—	2	4	no	—
54	C₁₃×#95	—	2	2	no	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	—	2	2	no	—
57	C₁₃×#95	—	2	4	no	—
58	#29210	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
73	C₁₃×C₄⋊C₄	—	2	8	no	—
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	4	4	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₄⋊C₄	—	4	8	no	—
82	C₁₃×C₄⋊C₄	—	4	4	no	—
83	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	—	4	2	no	—
88	C₄²×C₁₃	—	4	4	no	—
89	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
92	C₄²×C₁₃	#41	1	2	yes	—
93	#2859	C₂×C₁₃	1	1	yes	characteristic
94	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
95	#2059	C₂×C₁₃	1	4	yes	—
96	#1991	C₂×C₁₃	1	2	yes	—
97	#1948	C₂×C₁₃	1	2	yes	—
98	#1947	C₂×C₁₃	1	2	yes	—
99	#1846	C₂×C₁₃	1	2	yes	—
100	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	#135	1	4	yes	—
121	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₈×C₁₃	—	2	4	no	—
128	C₈×C₁₃	—	2	4	no	—
129	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×#125	C₂²×C₁₃	1	2	yes	—
131	C₂×#107	C₂²×C₁₃	1	4	yes	—
132	C₂×#106	C₂²×C₁₃	1	4	yes	—
133	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
136	#389	—	2	2	no	—
137	#389	—	2	2	no	—
138	#389	C₂²×C₁₃	1	2	yes	—
139	#387	—	2	2	no	—
140	#384	C₂²×C₁₃	1	2	yes	—
141	#382	C₂²×C₁₃	1	1	yes	characteristic
142	#382	C₂²×C₁₃	1	2	yes	—
143	C₄×C₄⋊C₄	—	2	2	no	—
144	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
145	C₄³	C₂²×C₁₃	1	1	yes	characteristic
146	#340	C₂²×C₁₃	1	4	yes	—
147	#340	C₂²×C₁₃	1	4	yes	—
148	#336	C₂²×C₁₃	1	2	yes	—
149	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×C₁₃	#535	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×#135	1	2	yes	—
153	C₂²×C₁₃	#575	1	2	yes	—
154	C₄×C₁₃	—	2	4	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	4	4	no	—
157	C₄×C₁₃	—	4	2	no	—
158	C₄×C₁₃	—	8	4	no	—
159	C₄×C₁₃	—	8	2	no	—
160	C₄×C₁₃	#499	1	4	yes	—
161	C₄×C₁₃	#554	1	1	yes	characteristic
162	C₄×C₁₃	#554	1	1	yes	characteristic
163	C₄×C₁₃	Q₈²	1	1	yes	characteristic
164	C₄×C₁₃	Q₈²	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
167	#125	—	2	2	no	—
168	#125	—	4	2	no	—
169	#125	—	4	2	no	—
170	#125	Q₈×C₁₃	1	4	yes	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	D₄×C₁₃	1	2	yes	—
181	C₂×C₄²	D₄×C₁₃	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	#107	—	4	4	no	—
186	#106	—	4	4	no	—
187	C₄⋊C₈	Q₈×C₁₃	1	4	yes	—
188	C₄⋊C₈	Q₈×C₁₃	1	4	yes	—
189	C₄×C₈	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	4	no	—
194	C₂×C₁₃	#2567	1	1	yes	characteristic
195	C₂×C₁₃	#3076	1	2	yes	—
196	C₂×C₁₃	C₂×#499	1	1	yes	characteristic
197	C₂×C₁₃	#3166	1	1	yes	characteristic
198	C₂×C₁₃	#3212	1	2	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
212	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	8	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	4	2	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
230	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
231	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
232	C₄²	C₁₃×#41	1	2	yes	—
233	C₁₃	#29210	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
251	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
252	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
253	C₂×C₄	C₁₃×#135	1	4	yes	—
254	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
258	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	2	4	no	—
262	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
264	C₂²	C₁₃×#535	1	1	yes	characteristic
265	C₂²	C₂×C₁₃×#135	1	2	yes	—
266	C₂²	C₁₃×#575	1	2	yes	—
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	4	no	—
272	C₄	—	8	2	no	—
273	C₄	C₁₃×#499	1	4	yes	—
274	C₄	C₁₃×#554	1	1	yes	characteristic
275	C₄	C₁₃×#554	1	1	yes	characteristic
276	C₄	Q₈²×C₁₃	1	1	yes	characteristic
277	C₄	Q₈²×C₁₃	1	1	yes	characteristic
278	C₂	C₁₃×#2567	1	1	yes	characteristic
279	C₂	C₁₃×#3076	1	2	yes	—
280	C₂	C₂×C₁₃×#499	1	1	yes	characteristic
281	C₂	C₁₃×#3166	1	1	yes	characteristic
282	C₂	C₁₃×#3212	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
58. #29210	C₁₃	direct (N × H)
233. C₁₃	#29210	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×#2859	2
C₄×C₁₃×#125	2
C₁₃×#2059	2
C₁₃×#1991	2
C₁₃×#1948	2
C₁₃×#1947	2
C₁₃×#1846	2
C₄×C₁₃×C₄⋊C₈	2
#29210	13

Subgroups — C13×#29210 #15715549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29210 #15715549