Subgroups — C₁₃×#17972 #15704311

Order 3328.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2985	C₂	1	2	yes	maximal
3	C₁₃×#1593	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1591	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1579	C₂	1	2	yes	maximal, complemented
6	C₈×C₁₃×Q₁₆	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1497	C₂	1	2	yes	maximal
9	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#533	C₂²	1	2	yes	—
11	C₁₃×#500	C₄	1	1	yes	characteristic
12	C₁₃×#494	C₄	1	1	yes	characteristic
13	C₁₃×#487	C₄	1	2	yes	—
14	C₁₃×#467	C₄	1	2	yes	—
15	C₁₃×#467	C₄	1	2	yes	—
16	C₁₃×#446	—	2	4	no	—
17	C₁₃×#446	C₂²	1	4	yes	—
18	C₈×Q₈×C₁₃	C₂²	1	4	yes	—
19	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
21	C₁₃×#433	—	2	4	no	—
22	C₁₃×#432	C₂²	1	2	yes	—
23	C₁₃×#431	C₂²	1	1	yes	characteristic
24	C₁₃×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#335	C₂²	1	2	yes	—
27	C₁₃×#334	C₂²	1	2	yes	—
28	C₁₃×Q₈⋊C₈	C₂²	1	4	yes	—
29	C₁₃×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₁₃×#322	C₂²	1	1	yes	characteristic
31	C₁₃×#322	C₂²	1	1	yes	characteristic
32	C₈²×C₁₃	C₂²	1	1	yes	characteristic
33	C₈²×C₁₃	C₂²	1	1	yes	characteristic
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₁₃×#125	—	2	2	no	—
41	C₁₃×C₂²⋊Q₈	—	2	2	no	—
42	C₁₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₁₃	—	2	2	no	—
44	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
45	C₁₃×#117	—	2	2	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#107	C₂×C₄	1	2	yes	—
48	C₁₃×#107	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₈	—	2	4	no	—
50	C₁₃×C₄⋊C₈	—	2	4	no	—
51	C₁₃×C₄⋊C₈	—	2	1	no	—
52	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
53	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
54	C₁₃×#103	C₂×C₄	1	4	yes	—
55	C₁₃×#103	C₂×C₄	1	4	yes	—
56	C₁₃×C₂²⋊C₈	—	2	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	1	no	—
58	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
59	C₁₃×#97	—	2	4	no	—
60	C₁₃×#97	—	2	2	no	—
61	C₄×C₈×C₁₃	—	2	4	no	—
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#17972	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
76	C₁₃×Q₁₆	—	2	4	no	—
77	C₁₃×Q₁₆	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
95	C₁₃×C₂²⋊C₄	—	2	4	no	—
96	C₄²×C₁₃	—	2	4	no	—
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	#41	1	1	yes	characteristic
100	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
101	#3170	C₂×C₁₃	1	1	yes	characteristic
102	#2985	C₂×C₁₃	1	2	yes	—
103	#1593	C₂×C₁₃	1	2	yes	complemented
104	#1591	C₂×C₁₃	1	2	yes	complemented
105	#1579	C₂×C₁₃	1	2	yes	complemented
106	C₈×Q₁₆	C₂×C₁₃	1	2	yes	complemented
107	#1568	C₂×C₁₃	1	1	yes	characteristic
108	#1497	C₂×C₁₃	1	2	yes	—
109	#1450	C₂×C₁₃	1	1	yes	characteristic
110	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	4	no	—
112	Q₈×C₁₃	—	2	4	no	—
113	Q₈×C₁₃	—	8	2	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₈×C₁₃	—	2	1	no	—
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	—	4	1	no	—
131	C₈×C₁₃	—	4	1	no	—
132	C₈×C₁₃	—	4	4	no	—
133	C₈×C₁₃	#131	1	2	yes	—
134	C₈×C₁₃	#131	1	2	yes	—
135	#533	C₂²×C₁₃	1	2	yes	—
136	#500	C₄×C₁₃	1	1	yes	characteristic
137	#494	C₄×C₁₃	1	1	yes	characteristic
138	#487	C₄×C₁₃	1	2	yes	—
139	#467	C₄×C₁₃	1	2	yes	—
140	#467	C₄×C₁₃	1	2	yes	—
141	#446	—	2	4	no	—
142	#446	C₂²×C₁₃	1	4	yes	—
143	C₈×Q₈	C₂²×C₁₃	1	4	yes	—
144	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
145	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
146	#433	—	2	4	no	—
147	#432	C₂²×C₁₃	1	2	yes	—
148	#431	C₂²×C₁₃	1	1	yes	characteristic
149	#431	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
151	#335	C₂²×C₁₃	1	2	yes	—
152	#334	C₂²×C₁₃	1	2	yes	—
153	Q₈⋊C₈	C₂²×C₁₃	1	4	yes	—
154	Q₈⋊C₈	C₂²×C₁₃	1	4	yes	—
155	#322	C₂²×C₁₃	1	1	yes	characteristic
156	#322	C₂²×C₁₃	1	1	yes	characteristic
157	C₈²	C₂²×C₁₃	1	1	yes	characteristic
158	C₈²	C₂²×C₁₃	1	1	yes	characteristic
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	4	4	no	—
168	C₄×C₁₃	—	4	4	no	—
169	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₁₃	#578	1	2	yes	—
172	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
173	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
174	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
176	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
177	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
178	#125	—	2	2	no	—
179	C₂²⋊Q₈	—	2	2	no	—
180	C₂²⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
183	#117	—	2	2	no	—
184	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	#107	C₂×C₄×C₁₃	1	2	yes	—
186	#107	C₂×C₄×C₁₃	1	2	yes	—
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
191	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
192	#103	C₂×C₄×C₁₃	1	4	yes	—
193	#103	C₂×C₄×C₁₃	1	4	yes	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
197	#97	—	2	4	no	—
198	#97	—	2	2	no	—
199	C₄×C₈	—	2	4	no	—
200	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
201	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₁₃	—	4	1	no	—
209	C₂×C₁₃	#2931	1	1	yes	characteristic
210	C₂×C₁₃	#2950	1	1	yes	characteristic
211	C₂×C₁₃	#2957	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
229	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₄²	C₁₃×#41	1	1	yes	characteristic
241	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
242	C₁₃	#17972	1	1	yes	characteristic, direct product factor, Sylow
243	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	8	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	4	no	—
266	C₈	C₁₃×#131	1	2	yes	—
267	C₈	C₁₃×#131	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
279	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
280	C₄	C₁₃×#578	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	C₁₃×#2931	1	1	yes	characteristic
283	C₂	C₁₃×#2950	1	1	yes	characteristic
284	C₂	C₁₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1593	C₂	semidirect
4. C₁₃×#1591	C₂	semidirect
5. C₁₃×#1579	C₂	semidirect
6. C₈×C₁₃×Q₁₆	C₂	semidirect
70. #17972	C₁₃	direct (N × H)
103. #1593	C₂×C₁₃	semidirect
104. #1591	C₂×C₁₃	semidirect
105. #1579	C₂×C₁₃	semidirect
106. C₈×Q₁₆	C₂×C₁₃	semidirect
242. C₁₃	#17972	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×#3170	2
C₁₃×#2985	2
C₁₃×#1593	2
C₁₃×#1591	2
C₁₃×#1579	2
C₈×C₁₃×Q₁₆	2
C₁₃×#1568	2
C₁₃×#1497	2
C₁₃×#1450	2
#17972	13

Subgroups — C13×#17972 #15704311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17972 #15704311