Subgroups — C₁₃×#29209 #15715548

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 272 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×#2058	C₂	1	2	yes	maximal
4	C₁₃×#2057	C₂	1	2	yes	maximal
5	C₁₃×#1991	C₂	1	2	yes	maximal
6	C₁₃×#1949	C₂	1	4	yes	maximal
7	C₁₃×#1846	C₂	1	2	yes	maximal
8	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	4	yes	—
12	C₂×C₁₃×#106	C₂²	1	4	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#391	—	2	2	no	—
16	C₁₃×#391	—	2	2	no	—
17	C₁₃×#389	C₂²	1	2	yes	—
18	C₁₃×#387	—	2	2	no	—
19	C₁₃×#382	C₂²	1	2	yes	—
20	C₁₃×#382	C₂²	1	1	yes	characteristic
21	C₁₃×#382	C₂²	1	2	yes	—
22	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄³×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#340	C₂²	1	4	yes	—
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#336	C₂²	1	1	yes	characteristic
28	C₁₃×#336	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₁₃×C₄⋊Q₈	—	4	4	no	—
33	C₁₃×#125	—	2	2	no	—
34	C₁₃×#125	Q₈	1	4	yes	complemented
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	C₂³	1	2	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	—	2	2	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×#107	—	4	4	no	—
48	C₁₃×#106	—	4	4	no	—
49	C₁₃×C₄⋊C₈	Q₈	1	8	yes	complemented
50	C₄×C₈×C₁₃	—	2	4	no	—
51	C₁₃×#95	—	2	2	no	—
52	C₁₃×#95	—	2	2	no	—
53	C₁₃×#95	—	2	2	no	—
54	C₁₃×#95	—	2	4	no	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	—	2	2	no	—
57	#29209	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×Q₈×C₁₃	—	4	4	no	—
59	C₂×Q₈×C₁₃	—	4	4	no	—
60	C₂×Q₈×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
75	C₁₃×C₄⋊C₄	—	2	8	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
83	C₄²×C₁₃	—	2	4	no	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₄²×C₁₃	—	4	4	no	—
87	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
88	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
89	C₄²×C₁₃	#41	1	2	yes	—
90	#2862	C₂×C₁₃	1	1	yes	characteristic
91	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
92	#2058	C₂×C₁₃	1	2	yes	—
93	#2057	C₂×C₁₃	1	2	yes	—
94	#1991	C₂×C₁₃	1	2	yes	—
95	#1949	C₂×C₁₃	1	4	yes	—
96	#1846	C₂×C₁₃	1	2	yes	—
97	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
98	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₁₃	—	8	8	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	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	4	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
116	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	#135	1	4	yes	—
119	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
123	C₈×C₁₃	—	2	4	no	—
124	C₈×C₁₃	—	2	4	no	—
125	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
126	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×#107	C₂²×C₁₃	1	4	yes	—
128	C₂×#106	C₂²×C₁₃	1	4	yes	—
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
131	#391	—	2	2	no	—
132	#391	—	2	2	no	—
133	#389	C₂²×C₁₃	1	2	yes	—
134	#387	—	2	2	no	—
135	#382	C₂²×C₁₃	1	1	yes	characteristic
136	#382	C₂²×C₁₃	1	2	yes	—
137	#382	C₂²×C₁₃	1	2	yes	—
138	C₄×C₄⋊C₄	—	2	2	no	—
139	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
140	C₄³	C₂²×C₁₃	1	1	yes	characteristic
141	#340	C₂²×C₁₃	1	4	yes	—
142	#340	C₂²×C₁₃	1	4	yes	—
143	#336	C₂²×C₁₃	1	1	yes	characteristic
144	#336	C₂²×C₁₃	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₁₃	#535	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#135	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	—	2	2	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	4	4	no	—
155	C₄×C₁₃	—	8	2	no	—
156	C₄×C₁₃	—	8	4	no	—
157	C₄×C₁₃	#554	1	2	yes	—
158	C₄×C₁₃	Q₈²	1	2	yes	—
159	C₂²×Q₈	—	2	2	no	—
160	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
161	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
162	C₄⋊Q₈	—	4	4	no	—
163	#125	—	2	2	no	—
164	#125	Q₈×C₁₃	1	4	yes	complemented
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	D₄×C₁₃	1	2	yes	—
173	C₂×C₄²	D₄×C₁₃	1	2	yes	—
174	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	#107	—	4	4	no	—
178	#106	—	4	4	no	—
179	C₄⋊C₈	Q₈×C₁₃	1	8	yes	complemented
180	C₄×C₈	—	2	4	no	—
181	#95	—	2	2	no	—
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	#95	—	2	4	no	—
185	#95	—	2	2	no	—
186	#95	—	2	2	no	—
187	C₂×C₁₃	#2567	1	1	yes	characteristic
188	C₂×C₁₃	#3076	1	2	yes	—
189	C₂×C₁₃	#3160	1	1	yes	characteristic
190	C₂×C₁₃	#3166	1	1	yes	characteristic
191	C₂×C₁₃	#3211	1	2	yes	—
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
208	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	2	no	—
220	C₄²	—	4	4	no	—
221	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
222	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
223	C₄²	C₁₃×#41	1	2	yes	—
224	C₁₃	#29209	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
226	Q₈	—	8	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
243	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
244	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
245	C₂×C₄	C₁₃×#135	1	4	yes	—
246	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
247	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
249	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	2	4	no	—
252	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
254	C₂²	C₁₃×#535	1	1	yes	characteristic
255	C₂²	C₂×C₁₃×#135	1	2	yes	—
256	C₂²	C₁₃×#575	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	4	no	—
264	C₄	C₁₃×#554	1	2	yes	—
265	C₄	Q₈²×C₁₃	1	2	yes	—
266	C₂	C₁₃×#2567	1	1	yes	characteristic
267	C₂	C₁₃×#3076	1	2	yes	—
268	C₂	C₁₃×#3160	1	1	yes	characteristic
269	C₂	C₁₃×#3166	1	1	yes	characteristic
270	C₂	C₁₃×#3211	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
34. C₁₃×#125	Q₈	semidirect
49. C₁₃×C₄⋊C₈	Q₈	semidirect
57. #29209	C₁₃	direct (N × H)
164. #125	Q₈×C₁₃	semidirect
179. C₄⋊C₈	Q₈×C₁₃	semidirect
224. C₁₃	#29209	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×#2058	2
C₁₃×#2057	2
C₁₃×#1991	2
C₁₃×#1949	2
C₁₃×#1846	2
C₄×C₁₃×C₄⋊C₈	2
#29209	13

Subgroups — C13×#29209 #15715548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29209 #15715548