Subgroups — C₁₃×#29149 #15715488

Order 3328.

878 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#1961	C₂	1	2	yes	maximal
5	C₁₃×#1836	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1802	C₂	1	4	yes	maximal, complemented
7	C₁₃×#1767	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×C₁₃×#97	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	C₁₃×#97	—	2	4	no	—
50	C₁₃×#95	—	2	4	no	—
51	#29149	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	2	8	no	—
54	C₂×D₄×C₁₃	—	2	4	no	—
55	C₂×D₄×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	2	4	no	—
57	C₂×D₄×C₁₃	—	4	4	no	—
58	C₂×D₄×C₁₃	—	4	4	no	—
59	C₂×D₄×C₁₃	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	4	4	no	—
63	C₂²×C₄×C₁₃	—	4	8	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
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	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	2	8	no	—
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₁₃×C₄⋊C₄	—	4	8	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₁₃×C₂²⋊C₄	—	4	8	no	—
79	C₁₃×C₂²⋊C₄	—	4	4	no	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	4	2	no	—
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₃	#41	1	2	yes	—
85	C₄²×C₁₃	#41	1	4	yes	—
86	#2859	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₄×#127	C₂×C₁₃	1	1	yes	characteristic
88	#2025	C₂×C₁₃	1	4	yes	complemented
89	#1961	C₂×C₁₃	1	2	yes	—
90	#1836	C₂×C₁₃	1	2	yes	complemented
91	#1802	C₂×C₁₃	1	4	yes	complemented
92	#1767	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂³×C₁₃	—	2	4	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	—	4	4	no	—
97	C₂³×C₁₃	—	4	4	no	—
98	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	D₄×C₁₃	—	2	8	no	—
100	D₄×C₁₃	—	4	16	no	—
101	D₄×C₁₃	—	4	4	no	—
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	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	AGL(1,ℤ/8)	1	4	yes	—
118	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₈×C₁₃	—	4	4	no	—
123	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×#125	C₂²×C₁₃	1	2	yes	—
125	C₂×C₄×D₄	—	2	2	no	—
126	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
128	C₂×#102	C₂²×C₁₃	1	8	yes	—
129	C₂×#97	C₂²×C₁₃	1	2	yes	—
130	#390	—	2	2	no	—
131	#390	C₂²×C₁₃	1	2	yes	—
132	#389	—	2	2	no	—
133	#389	—	2	2	no	—
134	#384	C₂²×C₁₃	1	1	yes	characteristic
135	#384	C₂²×C₁₃	1	2	yes	—
136	#382	C₂²×C₁₃	1	2	yes	—
137	C₄³	C₂²×C₁₃	1	1	yes	characteristic
138	#340	C₂²×C₁₃	1	8	yes	—
139	#336	C₂²×C₁₃	1	2	yes	—
140	C₂²×C₁₃	—	4	4	no	—
141	C₂²×C₁₃	—	4	8	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	4	no	—
145	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
147	C₂²×C₁₃	#535	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₂²×C₁₃	#575	1	2	yes	—
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	4	2	no	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	—	8	4	no	—
155	C₄×C₁₃	#490	1	4	yes	—
156	C₄×C₁₃	#554	1	2	yes	—
157	C₄×C₁₃	#555	1	2	yes	—
158	C₂²×D₄	—	2	2	no	—
159	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
160	C₂³×C₄	—	2	2	no	—
161	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
162	#127	—	2	2	no	—
163	#127	Q₈×C₁₃	1	4	yes	—
164	#125	—	4	4	no	—
165	C₄×D₄	—	2	8	no	—
166	C₄×D₄	—	4	8	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	D₄×C₁₃	1	2	yes	—
176	C₂×C₄²	D₄×C₁₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	#102	—	4	8	no	—
182	#97	—	2	4	no	—
183	#95	—	2	4	no	—
184	C₂×C₁₃	—	8	4	no	—
185	C₂×C₁₃	#2563	1	1	yes	characteristic
186	C₂×C₁₃	#3077	1	2	yes	—
187	C₂×C₁₃	C₂×#490	1	1	yes	characteristic
188	C₂×C₁₃	#3142	1	1	yes	characteristic
189	C₂×C₁₃	#3214	1	2	yes	—
190	C₂⁴	—	2	2	no	—
191	C₂×D₄	—	2	8	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	C₂×Q₈×C₁₃	1	4	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	8	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	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	8	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	4	8	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
222	C₄²	C₁₃×#41	1	2	yes	—
223	C₄²	C₁₃×#41	1	4	yes	—
224	C₁₃	#29149	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	4	no	—
230	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
231	D₄	—	2	8	no	—
232	D₄	—	4	16	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	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	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	8	no	—
246	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
247	C₂×C₄	C₁₃×#124	1	2	yes	—
248	C₂×C₄	C₁₃×#124	1	2	yes	—
249	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
250	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
252	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
253	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
254	C₈	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	8	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
261	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
262	C₂²	C₁₃×#535	1	1	yes	characteristic
263	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
264	C₂²	C₁₃×#575	1	2	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	4	no	—
270	C₄	C₁₃×#490	1	4	yes	—
271	C₄	C₁₃×#554	1	2	yes	—
272	C₄	C₁₃×#555	1	2	yes	—
273	C₂	—	8	4	no	—
274	C₂	C₁₃×#2563	1	1	yes	characteristic
275	C₂	C₁₃×#3077	1	2	yes	—
276	C₂	C₂×C₁₃×#490	1	1	yes	characteristic
277	C₂	C₁₃×#3142	1	1	yes	characteristic
278	C₂	C₁₃×#3214	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×#1836	C₂	semidirect
6. C₁₃×#1802	C₂	semidirect
7. C₁₃×#1767	C₂	semidirect
51. #29149	C₁₃	direct (N × H)
86. #2859	C₂×C₁₃	semidirect
88. #2025	C₂×C₁₃	semidirect
90. #1836	C₂×C₁₃	semidirect
91. #1802	C₂×C₁₃	semidirect
92. #1767	C₂×C₁₃	semidirect
224. C₁₃	#29149	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×#1961	2
C₁₃×#1836	2
C₁₃×#1802	2
C₁₃×#1767	2
#29149	13

Subgroups — C13×#29149 #15715488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29149 #15715488