Subgroups — C₁₃×#29143 #15715482

Order 3328.

890 conjugacy classes of subgroups.

The subgroup lattice has 282 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, complemented
2	C₄×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2025	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1957	C₂	1	2	yes	maximal
5	C₁₃×#1834	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1804	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#125	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₃	—	2	2	no	—
11	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#102	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#390	—	2	2	no	—
16	C₁₃×#390	C₂²	1	2	yes	—
17	C₁₃×#389	—	2	2	no	—
18	C₁₃×#389	—	2	2	no	—
19	C₁₃×#384	C₂²	1	1	yes	characteristic
20	C₁₃×#384	C₂²	1	2	yes	—
21	C₁₃×#382	C₂²	1	2	yes	—
22	C₄³×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#340	C₂²	1	8	yes	—
24	C₁₃×#336	C₂²	1	2	yes	—
25	C₂²×D₄×C₁₃	—	2	2	no	—
26	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
29	C₁₃×#127	—	2	2	no	—
30	C₁₃×#127	Q₈	1	4	yes	—
31	C₁₃×#125	—	4	4	no	—
32	C₄×D₄×C₁₃	—	2	8	no	—
33	C₄×D₄×C₁₃	—	4	8	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
42	C₂×C₄²×C₁₃	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	—	2	4	no	—
48	C₁₃×#102	—	4	8	no	—
49	C₄×C₈×C₁₃	—	2	4	no	—
50	C₁₃×#95	—	2	2	no	—
51	C₁₃×#95	—	2	2	no	—
52	#29143	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	—	2	8	no	—
55	C₂×D₄×C₁₃	—	2	4	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	2	4	no	—
58	C₂×D₄×C₁₃	—	4	4	no	—
59	C₂×D₄×C₁₃	—	4	4	no	—
60	C₂×D₄×C₁₃	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	4	8	no	—
64	C₂²×C₄×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
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	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	2	4	no	—
74	C₁₃×C₄⋊C₄	—	2	8	no	—
75	C₁₃×C₄⋊C₄	—	4	8	no	—
76	C₁₃×C₄⋊C₄	—	4	4	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₁₃×C₄⋊C₄	—	4	4	no	—
79	C₁₃×C₂²⋊C₄	—	4	8	no	—
80	C₁₃×C₂²⋊C₄	—	4	4	no	—
81	C₄²×C₁₃	—	2	4	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
85	C₄²×C₁₃	#41	1	2	yes	—
86	C₄²×C₁₃	#41	1	4	yes	—
87	#2859	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₄×#127	C₂×C₁₃	1	1	yes	characteristic
89	#2025	C₂×C₁₃	1	4	yes	complemented
90	#1957	C₂×C₁₃	1	2	yes	—
91	#1834	C₂×C₁₃	1	2	yes	complemented
92	#1804	C₂×C₁₃	1	4	yes	complemented
93	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
94	C₂³×C₁₃	—	2	4	no	—
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	—	4	2	no	—
97	C₂³×C₁₃	—	4	4	no	—
98	C₂³×C₁₃	—	4	4	no	—
99	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	D₄×C₁₃	—	2	8	no	—
101	D₄×C₁₃	—	4	4	no	—
102	D₄×C₁₃	—	4	16	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	8	8	no	—
114	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₁₃	#124	1	2	yes	—
116	C₂×C₄×C₁₃	#124	1	2	yes	—
117	C₂×C₄×C₁₃	#135	1	4	yes	—
118	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
123	C₈×C₁₃	—	4	4	no	—
124	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×#125	C₂²×C₁₃	1	2	yes	—
126	C₂×C₄×D₄	—	2	2	no	—
127	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
129	C₂×#102	C₂²×C₁₃	1	8	yes	—
130	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
131	#390	—	2	2	no	—
132	#390	C₂²×C₁₃	1	2	yes	—
133	#389	—	2	2	no	—
134	#389	—	2	2	no	—
135	#384	C₂²×C₁₃	1	1	yes	characteristic
136	#384	C₂²×C₁₃	1	2	yes	—
137	#382	C₂²×C₁₃	1	2	yes	—
138	C₄³	C₂²×C₁₃	1	1	yes	characteristic
139	#340	C₂²×C₁₃	1	8	yes	—
140	#336	C₂²×C₁₃	1	2	yes	—
141	C₂²×C₁₃	—	4	8	no	—
142	C₂²×C₁₃	—	4	4	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	2	no	—
145	C₂²×C₁₃	—	8	4	no	—
146	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
148	C₂²×C₁₃	#535	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#135	1	2	yes	—
150	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	4	2	no	—
154	C₄×C₁₃	—	8	2	no	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	#478	1	8	yes	—
157	C₄×C₁₃	#554	1	2	yes	—
158	C₄×C₁₃	#555	1	2	yes	—
159	C₂²×D₄	—	2	2	no	—
160	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
161	C₂³×C₄	—	2	2	no	—
162	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
163	#127	—	2	2	no	—
164	#127	Q₈×C₁₃	1	4	yes	—
165	#125	—	4	4	no	—
166	C₄×D₄	—	2	8	no	—
167	C₄×D₄	—	4	8	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	C₂×C₄²	D₄×C₁₃	1	2	yes	—
177	C₂×C₄²	D₄×C₁₃	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	#102	—	4	8	no	—
183	C₄×C₈	—	2	4	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	C₂×C₁₃	—	8	4	no	—
187	C₂×C₁₃	#2563	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#478	1	2	yes	—
189	C₂×C₁₃	#3131	1	1	yes	characteristic
190	C₂×C₁₃	#3138	1	1	yes	characteristic
191	C₂×C₁₃	#3207	1	2	yes	—
192	C₂⁴	—	2	2	no	—
193	C₂×D₄	—	2	8	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	C₂×Q₈×C₁₃	1	4	yes	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	4	8	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
208	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	4	8	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₂²⋊C₄	—	4	8	no	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
224	C₄²	C₁₃×#41	1	2	yes	—
225	C₄²	C₁₃×#41	1	4	yes	—
226	C₁₃	#29143	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³	—	2	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	2	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
233	D₄	—	2	8	no	—
234	D₄	—	4	16	no	—
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	8	8	no	—
247	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
248	C₂×C₄	C₁₃×#124	1	2	yes	—
249	C₂×C₄	C₁₃×#124	1	2	yes	—
250	C₂×C₄	C₁₃×#135	1	4	yes	—
251	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
252	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
254	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
255	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	4	no	—
262	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
264	C₂²	C₁₃×#535	1	1	yes	characteristic
265	C₂²	C₂×C₁₃×#135	1	2	yes	—
266	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	4	no	—
272	C₄	C₁₃×#478	1	8	yes	—
273	C₄	C₁₃×#554	1	2	yes	—
274	C₄	C₁₃×#555	1	2	yes	—
275	C₂	—	8	4	no	—
276	C₂	C₁₃×#2563	1	1	yes	characteristic
277	C₂	C₂×C₁₃×#478	1	2	yes	—
278	C₂	C₁₃×#3131	1	1	yes	characteristic
279	C₂	C₁₃×#3138	1	1	yes	characteristic
280	C₂	C₁₃×#3207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
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
1. C₁₃×#2859	C₂	semidirect
3. C₁₃×#2025	C₂	semidirect
5. C₁₃×#1834	C₂	semidirect
6. C₁₃×#1804	C₂	semidirect
7. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
52. #29143	C₁₃	direct (N × H)
87. #2859	C₂×C₁₃	semidirect
89. #2025	C₂×C₁₃	semidirect
91. #1834	C₂×C₁₃	semidirect
92. #1804	C₂×C₁₃	semidirect
93. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
226. C₁₃	#29143	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×#127	2
C₁₃×#2025	2
C₁₃×#1957	2
C₁₃×#1834	2
C₁₃×#1804	2
C₄×C₁₃×C₄⋊C₈	2
#29143	13

Subgroups — C13×#29143 #15715482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29143 #15715482