Subgroups — C₁₃×#22739 #15709078

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2355	C₂	1	2	yes	maximal
3	C₁₃×#1949	C₂	1	4	yes	maximal
4	C₁₃×#1947	C₂	1	2	yes	maximal
5	C₁₃×#1835	C₂	1	2	yes	maximal
6	C₁₃×#1834	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₃×#107	C₂	1	2	yes	maximal
8	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
11	C₂×C₁₃×#107	C₂²	1	4	yes	—
12	C₂×C₁₃×#106	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₁₃×#391	—	2	2	no	—
18	C₁₃×#387	—	2	4	no	—
19	C₁₃×#384	C₂²	1	2	yes	—
20	C₁₃×#382	C₂²	1	4	yes	—
21	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
24	C₁₃×#340	C₂²	1	1	yes	characteristic
25	C₁₃×#340	C₂²	1	1	yes	characteristic
26	C₁₃×#340	C₂²	1	2	yes	—
27	C₁₃×#340	C₂²	1	2	yes	—
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₃	—	2	4	no	—
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#107	—	2	8	no	—
47	C₁₃×#106	—	2	4	no	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
52	C₁₃×#95	—	2	4	no	—
53	C₁₃×#95	—	2	2	no	—
54	C₁₃×#95	—	2	4	no	—
55	C₁₃×#95	—	2	4	no	—
56	#22739	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	4	4	no	—
79	C₁₃×C₄⋊C₄	—	8	2	no	—
80	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
81	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
82	C₁₃×C₄⋊C₄	#41	1	4	yes	—
83	C₄²×C₁₃	—	4	4	no	—
84	C₄²×C₁₃	—	8	4	no	—
85	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
88	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic
90	#2355	C₂×C₁₃	1	2	yes	—
91	#1949	C₂×C₁₃	1	4	yes	—
92	#1947	C₂×C₁₃	1	2	yes	—
93	#1835	C₂×C₁₃	1	2	yes	—
94	#1834	C₂×C₁₃	1	1	yes	characteristic
95	C₄×#107	C₂×C₁₃	1	2	yes	—
96	C₄×#106	C₂×C₁₃	1	1	yes	characteristic
97	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	#135	1	4	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
121	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₁₃	—	2	2	no	—
123	C₈×C₁₃	—	2	4	no	—
124	C₈×C₁₃	—	2	2	no	—
125	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
126	C₈×Q₈	C₄×C₁₃	1	8	yes	complemented
127	C₂×#107	C₂²×C₁₃	1	4	yes	—
128	C₂×#106	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
133	#391	—	2	2	no	—
134	#387	—	2	4	no	—
135	#384	C₂²×C₁₃	1	2	yes	—
136	#382	C₂²×C₁₃	1	4	yes	—
137	C₄×C₄⋊C₄	—	2	2	no	—
138	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
139	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
140	#340	C₂²×C₁₃	1	2	yes	—
141	#340	C₂²×C₁₃	1	2	yes	—
142	#340	C₂²×C₁₃	1	1	yes	characteristic
143	#340	C₂²×C₁₃	1	4	yes	—
144	#340	C₂²×C₁₃	1	1	yes	characteristic
145	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₁₃	#548	1	2	yes	—
147	C₂²×C₁₃	C₂×#135	1	2	yes	—
148	C₂²×C₁₃	#576	1	1	yes	characteristic
149	C₂²×C₁₃	#578	1	1	yes	characteristic
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	2	4	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	—	8	4	no	—
155	C₄×C₁₃	—	8	2	no	—
156	C₄×C₁₃	#427	1	4	yes	—
157	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
158	C₄×C₁₃	#519	1	1	yes	characteristic
159	C₄×C₁₃	#557	1	2	yes	—
160	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
164	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
165	C₄×Q₈	Q₈×C₁₃	1	4	yes	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
171	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	#107	—	2	8	no	—
178	#106	—	2	4	no	—
179	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
180	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
181	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
182	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
183	#95	—	2	4	no	—
184	#95	—	2	2	no	—
185	#95	—	2	4	no	—
186	#95	—	2	4	no	—
187	C₂×C₁₃	#2351	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
189	C₂×C₁₃	#2913	1	1	yes	characteristic
190	C₂×C₁₃	#3361	1	2	yes	—
191	C₂×C₁₃	#3364	1	2	yes	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	C₁₃×C₄⋊C₄	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
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	C₂⁴×C₁₃	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
207	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	8	2	no	—
215	C₄⋊C₄	C₁₃×C₄⋊C₄	1	8	yes	—
216	C₄⋊C₄	C₁₃×C₄⋊C₄	1	4	yes	—
217	C₄⋊C₄	C₁₃×#41	1	4	yes	—
218	C₄²	—	4	4	no	—
219	C₄²	—	8	4	no	—
220	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
221	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
222	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
223	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
224	C₁₃	#22739	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
242	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
243	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
244	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
245	C₂×C₄	C₁₃×#135	1	4	yes	—
246	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
249	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	4	no	—
252	C₈	—	2	2	no	—
253	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
254	C₂²	C₁₃×#548	1	2	yes	—
255	C₂²	C₂×C₁₃×#135	1	2	yes	—
256	C₂²	C₁₃×#576	1	1	yes	characteristic
257	C₂²	C₁₃×#578	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	2	no	—
264	C₄	C₁₃×#427	1	4	yes	—
265	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
266	C₄	C₁₃×#519	1	1	yes	characteristic
267	C₄	C₁₃×#557	1	2	yes	—
268	C₂	C₁₃×#2351	1	1	yes	characteristic
269	C₂	C₂×C₁₃×#427	1	1	yes	characteristic
270	C₂	C₁₃×#2913	1	1	yes	characteristic
271	C₂	C₁₃×#3361	1	2	yes	—
272	C₂	C₁₃×#3364	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₁₃	C₄	semidirect
56. #22739	C₁₃	direct (N × H)
126. C₈×Q₈	C₄×C₁₃	semidirect
224. C₁₃	#22739	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₈×Q₈×C₁₃	2
C₁₃×#2355	2
C₁₃×#1949	2
C₁₃×#1947	2
C₁₃×#1835	2
C₁₃×#1834	2
C₄×C₁₃×#107	2
C₄×C₁₃×#106	2
#22739	13

Subgroups — C13×#22739 #15709078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22739 #15709078