Subgroups — C₁₃×#29361 #15715700

Order 3328.

634 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2642	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2330	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2072	C₂	1	2	yes	maximal
4	C₁₃×#2026	C₂	1	2	yes	maximal
5	C₁₃×#1989	C₂	1	2	yes	maximal
6	C₁₃×#1981	C₂	1	2	yes	maximal
7	C₁₃×#1935	C₂	1	2	yes	maximal
8	C₁₃×#1933	C₂	1	2	yes	maximal
9	C₁₃×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×#103	C₂²	1	2	yes	—
15	C₂×C₁₃×#103	C₂²	1	2	yes	—
16	C₂×C₁₃×#103	C₂²	1	2	yes	—
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#400	—	2	2	no	—
21	C₁₃×#400	—	2	2	no	—
22	C₁₃×#391	C₂²	1	2	yes	—
23	C₁₃×#391	C₂²	1	2	yes	—
24	C₁₃×#389	—	2	2	no	—
25	C₁₃×#389	C₂²	1	2	yes	—
26	C₁₃×#387	—	2	2	no	—
27	C₁₃×#384	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#376	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₁₃×C₄⋊Q₈	—	2	4	no	—
39	C₁₃×#125	—	2	4	no	—
40	C₁₃×#125	—	2	4	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₈	—	2	4	no	—
54	C₁₃×#103	—	2	4	no	—
55	C₁₃×#103	—	2	4	no	—
56	C₁₃×#103	—	4	2	no	—
57	C₁₃×#103	—	4	2	no	—
58	C₁₃×#97	—	4	1	no	—
59	C₄×C₈×C₁₃	—	4	1	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	4	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	D₄	1	2	yes	—
66	C₁₃×#95	D₄	1	2	yes	—
67	#29361	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	8	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	4	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₁₃×C₄⋊C₄	—	8	2	no	—
97	C₁₃×C₄⋊C₄	—	8	2	no	—
98	C₁₃×C₄⋊C₄	#41	1	4	yes	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	2	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	8	2	no	—
103	C₄²×C₁₃	#41	1	4	yes	—
104	#2642	C₂×C₁₃	1	1	yes	characteristic
105	#2330	C₂×C₁₃	1	1	yes	characteristic
106	#2072	C₂×C₁₃	1	2	yes	—
107	#2026	C₂×C₁₃	1	2	yes	—
108	#1989	C₂×C₁₃	1	2	yes	—
109	#1981	C₂×C₁₃	1	2	yes	—
110	#1935	C₂×C₁₃	1	2	yes	—
111	#1933	C₂×C₁₃	1	2	yes	—
112	#1768	C₂×C₁₃	1	1	yes	characteristic
113	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₁₃	—	4	2	no	—
115	Q₈×C₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	#124	1	4	yes	—
140	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	—	4	2	no	—
146	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
150	C₂×#103	C₂²×C₁₃	1	2	yes	—
151	C₂×#103	C₂²×C₁₃	1	2	yes	—
152	C₂×#103	C₂²×C₁₃	1	2	yes	—
153	C₂×#103	C₂²×C₁₃	1	2	yes	—
154	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
156	#400	—	2	2	no	—
157	#400	—	2	2	no	—
158	#391	C₂²×C₁₃	1	2	yes	—
159	#391	C₂²×C₁₃	1	2	yes	—
160	#389	—	2	2	no	—
161	#389	C₂²×C₁₃	1	2	yes	—
162	#387	—	2	2	no	—
163	#384	C₂²×C₁₃	1	1	yes	characteristic
164	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
165	#376	C₂²×C₁₃	1	1	yes	characteristic
166	#340	C₂²×C₁₃	1	2	yes	—
167	#340	C₂²×C₁₃	1	2	yes	—
168	#340	C₂²×C₁₃	1	2	yes	—
169	#340	C₂²×C₁₃	1	2	yes	—
170	#336	C₂²×C₁₃	1	2	yes	—
171	C₂²×C₁₃	#525	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
173	C₂²×C₁₃	#537	1	1	yes	characteristic
174	C₂²×C₁₃	#577	1	2	yes	—
175	C₂²×C₁₃	#578	1	2	yes	—
176	C₄×C₁₃	—	2	4	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	—	4	1	no	—
180	C₄×C₁₃	—	4	1	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	#556	1	2	yes	—
185	C₄×C₁₃	#559	1	2	yes	—
186	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
189	C₄⋊Q₈	—	2	4	no	—
190	#125	—	2	4	no	—
191	#125	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
204	C₄⋊C₈	—	2	4	no	—
205	#103	—	2	4	no	—
206	#103	—	2	4	no	—
207	#103	—	4	2	no	—
208	#103	—	4	2	no	—
209	#97	—	4	1	no	—
210	C₄×C₈	—	4	1	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	—	2	4	no	—
215	#95	—	2	2	no	—
216	#95	D₄×C₁₃	1	2	yes	—
217	#95	D₄×C₁₃	1	2	yes	—
218	C₂×C₁₃	#2592	1	1	yes	characteristic
219	C₂×C₁₃	#3120	1	2	yes	—
220	C₂×C₁₃	#3137	1	2	yes	—
221	C₂×C₁₃	#3260	1	1	yes	characteristic
222	C₂×C₁₃	#3261	1	1	yes	characteristic
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	8	2	no	—
252	C₄⋊C₄	—	8	2	no	—
253	C₄⋊C₄	C₁₃×#41	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	8	2	no	—
258	C₄²	C₁₃×#41	1	4	yes	—
259	C₁₃	#29361	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₁₃×#124	1	4	yes	—
287	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
288	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
289	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
290	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₂²	C₁₃×#525	1	1	yes	characteristic
294	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
295	C₂²	C₁₃×#537	1	1	yes	characteristic
296	C₂²	C₁₃×#577	1	2	yes	—
297	C₂²	C₁₃×#578	1	2	yes	—
298	C₄	—	2	4	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₁₃×#556	1	2	yes	—
307	C₄	C₁₃×#559	1	2	yes	—
308	C₂	C₁₃×#2592	1	1	yes	characteristic
309	C₂	C₁₃×#3120	1	2	yes	—
310	C₂	C₁₃×#3137	1	2	yes	—
311	C₂	C₁₃×#3260	1	1	yes	characteristic
312	C₂	C₁₃×#3261	1	1	yes	characteristic

	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 ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
67. #29361	C₁₃	direct (N × H)
259. C₁₃	#29361	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×#2642	2
C₁₃×#2330	2
C₁₃×#2072	2
C₁₃×#2026	2
C₁₃×#1989	2
C₁₃×#1981	2
C₁₃×#1935	2
C₁₃×#1933	2
C₁₃×#1768	2
#29361	13

Subgroups — C13×#29361 #15715700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29361 #15715700