Subgroups — C₁₃×#9696 #15696035

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2858	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#332	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#331	C₂	1	2	yes	maximal
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#124	—	2	1	no	—
9	C₂×C₁₃×#124	—	2	1	no	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	—	2	1	no	—
13	C₁₃×#390	—	2	2	no	—
14	C₁₃×#390	C₄	1	4	yes	—
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#336	—	2	1	no	—
17	C₁₃×#336	—	2	2	no	—
18	C₁₃×#332	—	2	4	no	—
19	C₁₃×#331	—	2	4	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×D₄×C₁₃	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₁₃×#127	—	2	2	no	—
29	C₁₃×#127	—	2	2	no	—
30	C₁₃×#124	—	4	4	no	—
31	C₁₃×#124	—	4	2	no	—
32	C₁₃×#124	—	4	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	C₁₃×C₄⋊C₈	—	2	2	no	—
45	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
46	C₄×C₈×C₁₃	—	4	2	no	—
47	#9696	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	—	2	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	1	no	—
51	C₂×Q₈×C₁₃	—	4	1	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×D₄×C₁₃	—	2	4	no	—
54	C₂×D₄×C₁₃	—	4	4	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	4	4	no	—
61	C₂×D₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	4	no	—
73	C₁₃×C₄⋊C₄	—	2	4	no	—
74	C₁₃×C₄⋊C₄	—	4	1	no	—
75	C₁₃×C₄⋊C₄	—	4	1	no	—
76	C₁₃×C₂²⋊C₄	—	4	8	no	—
77	C₁₃×C₂²⋊C₄	—	4	8	no	—
78	C₄²×C₁₃	—	2	4	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	4	2	no	—
81	C₄²×C₁₃	D₈	1	2	yes	—
82	C₄²×C₁₃	SD₁₆	1	2	yes	—
83	C₄²×C₁₃	C₂×D₄	1	2	yes	—
84	C₄²×C₁₃	#41	1	2	yes	—
85	#2858	C₂×C₁₃	1	1	yes	characteristic
86	#1831	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂×#332	C₂×C₁₃	1	2	yes	complemented
89	C₂×#331	C₂×C₁₃	1	2	yes	—
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	—	4	4	no	—
92	C₂³×C₁₃	—	4	2	no	—
93	C₂³×C₁₃	—	4	4	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₁₃	—	8	2	no	—
97	Q₈×C₁₃	—	8	2	no	—
98	D₄×C₁₃	—	8	4	no	—
99	D₄×C₁₃	—	8	2	no	—
100	D₄×C₁₃	—	8	2	no	—
101	D₄×C₁₃	—	8	2	no	—
102	D₄×C₁₃	—	8	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	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	#102	1	4	yes	—
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	#121	1	2	yes	—
115	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#135	1	4	yes	—
120	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₈×C₁₃	—	4	4	no	—
122	C₈×C₁₃	—	8	2	no	—
123	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×#124	—	2	1	no	—
126	C₂×#124	—	2	1	no	—
127	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄×C₈	—	2	1	no	—
130	#390	—	2	2	no	—
131	#390	C₄×C₁₃	1	4	yes	—
132	C₄³	C₂²×C₁₃	1	1	yes	characteristic
133	#336	—	2	2	no	—
134	#336	—	2	1	no	—
135	#332	—	2	4	no	—
136	#331	—	2	4	no	—
137	C₂²×C₁₃	—	8	4	no	—
138	C₂²×C₁₃	—	8	4	no	—
139	C₂²×C₁₃	—	8	2	no	—
140	C₂²×C₁₃	—	8	2	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	#390	1	1	yes	characteristic
143	C₂²×C₁₃	#413	1	2	yes	—
144	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
145	C₂²×C₁₃	#416	1	2	yes	—
146	C₂²×C₁₃	#418	1	1	yes	characteristic
147	C₄×C₁₃	—	2	4	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	—	8	2	no	—
151	C₄×C₁₃	#459	1	2	yes	—
152	C₄×C₁₃	#460	1	2	yes	—
153	C₄×C₁₃	#488	1	2	yes	—
154	C₄×C₁₃	#495	1	2	yes	—
155	C₂²×Q₈	—	2	1	no	—
156	C₂²×D₄	—	2	1	no	—
157	C₂²×D₄	—	2	1	no	—
158	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	#127	—	2	2	no	—
164	#127	—	2	2	no	—
165	#124	—	4	4	no	—
166	#124	—	4	2	no	—
167	#124	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
173	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₁₃	1	2	yes	—
175	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
181	C₄×C₈	—	4	2	no	—
182	C₂×C₁₃	—	16	2	no	—
183	C₂×C₁₃	#1540	1	2	yes	—
184	C₂×C₁₃	#1553	1	2	yes	—
185	C₂×C₁₃	#1958	1	1	yes	characteristic
186	C₂×C₁₃	#1967	1	1	yes	characteristic
187	C₂×C₁₃	#1982	1	1	yes	characteristic
188	C₂⁴	—	2	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
204	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₂²⋊C₄	—	4	8	no	—
217	C₂²⋊C₄	—	4	8	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	C₁₃×D₈	1	2	yes	—
222	C₄²	C₁₃×SD₁₆	1	2	yes	—
223	C₄²	C₂×D₄×C₁₃	1	2	yes	—
224	C₄²	C₁₃×#41	1	2	yes	—
225	C₁₃	#9696	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
232	Q₈	—	8	2	no	—
233	Q₈	—	8	2	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₁₃×#102	1	4	yes	—
249	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
250	C₂×C₄	C₁₃×#121	1	2	yes	—
251	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
252	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
253	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
254	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
255	C₂×C₄	C₁₃×#135	1	4	yes	—
256	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
257	C₈	—	4	4	no	—
258	C₈	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	4	no	—
261	C₂²	—	8	4	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	C₁₃×#390	1	1	yes	characteristic
265	C₂²	C₁₃×#413	1	2	yes	—
266	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
267	C₂²	C₁₃×#416	1	2	yes	—
268	C₂²	C₁₃×#418	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	8	2	no	—
273	C₄	C₁₃×#459	1	2	yes	—
274	C₄	C₁₃×#460	1	2	yes	—
275	C₄	C₁₃×#488	1	2	yes	—
276	C₄	C₁₃×#495	1	2	yes	—
277	C₂	—	16	2	no	—
278	C₂	C₁₃×#1540	1	2	yes	—
279	C₂	C₁₃×#1553	1	2	yes	—
280	C₂	C₁₃×#1958	1	1	yes	characteristic
281	C₂	C₁₃×#1967	1	1	yes	characteristic
282	C₂	C₁₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1831	C₂	semidirect
3. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
4. C₂×C₁₃×#332	C₂	semidirect
47. #9696	C₁₃	direct (N × H)
86. #1831	C₂×C₁₃	semidirect
87. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
88. C₂×#332	C₂×C₁₃	semidirect
225. C₁₃	#9696	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×#2858	2
C₁₃×#1831	2
C₄×C₁₃×C₄⋊C₈	2
C₂×C₁₃×#332	2
C₂×C₁₃×#331	2
#9696	13

Subgroups — C13×#9696 #15696035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9696 #15696035