Subgroups — C₁₃×#29357 #15715696

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2862	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2074	C₂	1	2	yes	maximal
4	C₁₃×#2026	C₂	1	2	yes	maximal
5	C₁₃×#1987	C₂	1	2	yes	maximal
6	C₁₃×#1981	C₂	1	2	yes	maximal
7	C₁₃×#1935	C₂	1	4	yes	maximal
8	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#103	C₂²	1	4	yes	—
14	C₂×C₁₃×#103	C₂²	1	4	yes	—
15	C₂×C₁₃×#97	C₂²	1	2	yes	—
16	C₁₃×#391	—	2	2	no	—
17	C₁₃×#391	C₂²	1	2	yes	—
18	C₁₃×#391	C₂²	1	2	yes	—
19	C₁₃×#389	C₂²	1	2	yes	—
20	C₁₃×#387	—	2	2	no	—
21	C₁₃×#382	—	2	2	no	—
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×Q₈×C₁₃	—	2	1	no	—
30	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₁₃×C₄⋊Q₈	—	2	4	no	—
33	C₁₃×C₄⋊Q₈	—	2	4	no	—
34	C₁₃×#125	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	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×C₄⋊C₈	—	2	4	no	—
50	C₁₃×#103	—	2	8	no	—
51	C₁₃×#103	—	4	4	no	—
52	C₁₃×#97	—	4	2	no	—
53	C₁₃×#95	—	2	2	no	—
54	C₁₃×#95	—	2	2	no	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	—	2	4	no	—
57	C₁₃×#95	—	2	4	no	—
58	#29357	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×Q₈×C₁₃	—	2	4	no	—
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	8	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	4	4	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₄⋊C₄	#41	1	4	yes	—
87	C₄²×C₁₃	—	2	4	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	4	2	no	—
92	C₄²×C₁₃	—	4	4	no	—
93	C₄²×C₁₃	#41	1	4	yes	—
94	#2862	C₂×C₁₃	1	1	yes	characteristic
95	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
96	#2074	C₂×C₁₃	1	2	yes	—
97	#2026	C₂×C₁₃	1	2	yes	—
98	#1987	C₂×C₁₃	1	2	yes	—
99	#1981	C₂×C₁₃	1	2	yes	—
100	#1935	C₂×C₁₃	1	4	yes	—
101	#1767	C₂×C₁₃	1	1	yes	characteristic
102	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₁₃	—	4	4	no	—
104	Q₈×C₁₃	—	4	4	no	—
105	Q₈×C₁₃	—	8	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	#124	1	2	yes	—
125	C₂×C₄×C₁₃	#124	1	2	yes	—
126	C₂×C₄×C₁₃	#137	1	4	yes	—
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
131	C₈×C₁₃	—	4	4	no	—
132	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
136	C₂×#103	C₂²×C₁₃	1	4	yes	—
137	C₂×#103	C₂²×C₁₃	1	4	yes	—
138	C₂×#97	C₂²×C₁₃	1	2	yes	—
139	#391	—	2	2	no	—
140	#391	C₂²×C₁₃	1	2	yes	—
141	#391	C₂²×C₁₃	1	2	yes	—
142	#389	C₂²×C₁₃	1	2	yes	—
143	#387	—	2	2	no	—
144	#382	—	2	2	no	—
145	#382	C₂²×C₁₃	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
148	C₄³	C₂²×C₁₃	1	1	yes	characteristic
149	#340	C₂²×C₁₃	1	4	yes	—
150	#340	C₂²×C₁₃	1	4	yes	—
151	#336	C₂²×C₁₃	1	2	yes	—
152	C₂²×C₁₃	#525	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
154	C₂²×C₁₃	#537	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#137	1	2	yes	—
156	C₂²×C₁₃	#575	1	2	yes	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	4	2	no	—
159	C₄×C₁₃	—	4	4	no	—
160	C₄×C₁₃	—	4	2	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	—	8	2	no	—
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	#491	1	4	yes	—
165	C₄×C₁₃	#556	1	2	yes	—
166	C₄×C₁₃	#559	1	2	yes	—
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	—	2	4	no	—
172	#125	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	D₄×C₁₃	1	2	yes	—
183	C₂×C₄²	D₄×C₁₃	1	2	yes	—
184	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	#103	—	2	8	no	—
189	#103	—	4	4	no	—
190	#97	—	4	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	4	no	—
195	#95	—	2	4	no	—
196	C₂×C₁₃	#2592	1	1	yes	characteristic
197	C₂×C₁₃	#3116	1	2	yes	—
198	C₂×C₁₃	C₂×#491	1	1	yes	characteristic
199	C₂×C₁₃	#3140	1	1	yes	characteristic
200	C₂×C₁₃	#3252	1	2	yes	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	4	4	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	C₁₃×#41	1	4	yes	—
229	C₄²	—	2	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	4	no	—
235	C₄²	C₁₃×#41	1	4	yes	—
236	C₁₃	#29357	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
238	Q₈	—	4	4	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	C₁₃×#124	1	2	yes	—
260	C₂×C₄	C₁₃×#124	1	2	yes	—
261	C₂×C₄	C₁₃×#137	1	4	yes	—
262	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
263	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
264	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₂²	C₁₃×#525	1	1	yes	characteristic
268	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
269	C₂²	C₁₃×#537	1	1	yes	characteristic
270	C₂²	C₂×C₁₃×#137	1	2	yes	—
271	C₂²	C₁₃×#575	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	8	2	no	—
279	C₄	C₁₃×#491	1	4	yes	—
280	C₄	C₁₃×#556	1	2	yes	—
281	C₄	C₁₃×#559	1	2	yes	—
282	C₂	C₁₃×#2592	1	1	yes	characteristic
283	C₂	C₁₃×#3116	1	2	yes	—
284	C₂	C₂×C₁₃×#491	1	1	yes	characteristic
285	C₂	C₁₃×#3140	1	1	yes	characteristic
286	C₂	C₁₃×#3252	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. #29357	C₁₃	direct (N × H)
236. C₁₃	#29357	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×#2862	2
C₄×C₁₃×#125	2
C₁₃×#2074	2
C₁₃×#2026	2
C₁₃×#1987	2
C₁₃×#1981	2
C₁₃×#1935	2
C₁₃×#1767	2
#29357	13

Subgroups — C13×#29357 #15715696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29357 #15715696