Subgroups — C₁₃×#10538 #15696877

Order 3328.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1988	C₂	1	2	yes	maximal
3	C₁₃×#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#332	C₂	1	2	yes	maximal
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	—	2	1	no	—
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₁₃×#494	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₁₃×#340	—	2	2	no	—
16	C₁₃×#340	—	2	2	no	—
17	C₁₃×#336	—	2	1	no	—
18	C₁₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂×C₁₃×Q₁₆	—	2	4	no	—
21	C₂×C₁₃×Q₁₆	—	2	4	no	—
22	C₂×C₁₃×Q₁₆	—	4	2	no	—
23	C₂×C₁₃×Q₁₆	—	4	2	no	—
24	C₂×C₁₃×Q₁₆	—	4	2	no	—
25	C₂×C₁₃×Q₁₆	—	4	2	no	—
26	C₂²×C₈×C₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₁₃×C₄⋊Q₈	—	2	2	no	—
31	C₁₃×C₄⋊Q₈	—	2	2	no	—
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	4	2	no	—
44	C₄×C₈×C₁₃	—	2	2	no	—
45	C₁₃×#95	—	2	1	no	—
46	#10538	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂×Q₈×C₁₃	—	2	4	no	—
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₁₃×Q₁₆	—	4	8	no	—
60	C₁₃×Q₁₆	—	4	8	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	4	1	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	—	4	1	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₂×C₈×C₁₃	—	4	1	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	D₈	1	2	yes	—
72	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
73	C₁₃×C₄⋊C₄	—	2	1	no	—
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	Q₁₆	1	2	yes	complemented
80	C₄²×C₁₃	—	4	1	no	—
81	C₄²×C₁₃	—	4	1	no	—
82	C₄²×C₁₃	C₂×D₄	1	2	yes	—
83	C₄²×C₁₃	#41	1	2	yes	—
84	C₂×#494	C₂×C₁₃	1	1	yes	characteristic
85	#1988	C₂×C₁₃	1	2	yes	—
86	#1939	C₂×C₁₃	1	1	yes	characteristic
87	C₂×#332	C₂×C₁₃	1	2	yes	—
88	#1277	C₂×C₁₃	1	1	yes	characteristic
89	C₂³×C₁₃	#121	1	1	yes	characteristic
90	Q₈×C₁₃	—	4	4	no	—
91	Q₈×C₁₃	—	4	4	no	—
92	Q₈×C₁₃	—	8	2	no	—
93	Q₈×C₁₃	—	8	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	4	1	no	—
98	C₂×C₄×C₁₃	—	4	1	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	4	1	no	—
101	C₂×C₄×C₁₃	—	4	1	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#121	1	2	yes	—
107	C₂×C₄×C₁₃	#123	1	2	yes	—
108	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
113	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#137	1	2	yes	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	—	8	1	no	—
119	C₈×C₁₃	—	8	2	no	—
120	C₂²×Q₁₆	—	2	1	no	—
121	C₂²×Q₁₆	—	2	1	no	—
122	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
123	#494	—	2	4	no	—
124	C₂×C₄⋊C₈	—	2	1	no	—
125	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
128	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
129	#340	—	2	2	no	—
130	#340	—	2	2	no	—
131	#336	—	2	1	no	—
132	#332	—	2	4	no	—
133	C₂²×C₁₃	#394	1	1	yes	characteristic
134	C₂²×C₁₃	#459	1	1	yes	characteristic
135	C₂²×C₁₃	#460	1	1	yes	characteristic
136	C₂²×C₁₃	#467	1	1	yes	characteristic
137	C₂²×C₁₃	#470	1	1	yes	characteristic
138	C₂²×C₁₃	#473	1	1	yes	characteristic
139	C₂²×C₁₃	#473	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	—	8	1	no	—
143	C₄×C₁₃	—	8	4	no	—
144	C₄×C₁₃	#447	1	1	yes	characteristic
145	C₄×C₁₃	#450	1	1	yes	characteristic
146	C₄×C₁₃	#462	1	2	yes	—
147	C₄×C₁₃	#464	1	2	yes	—
148	C₄×C₁₃	#486	1	1	yes	characteristic
149	C₄×C₁₃	#490	1	1	yes	characteristic
150	C₂²×Q₈	—	2	2	no	—
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	—	2	4	no	—
153	C₂×Q₁₆	—	4	2	no	—
154	C₂×Q₁₆	—	4	2	no	—
155	C₂×Q₁₆	—	4	2	no	—
156	C₂×Q₁₆	—	4	2	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	D₄×C₁₃	1	4	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	4	2	no	—
175	C₄×C₈	—	2	2	no	—
176	#95	—	2	1	no	—
177	C₂×C₁₃	#1669	1	1	yes	characteristic
178	C₂×C₁₃	#1670	1	1	yes	characteristic
179	C₂×C₁₃	#1689	1	1	yes	characteristic
180	C₂×C₁₃	#1690	1	1	yes	characteristic
181	C₂×C₁₃	#2041	1	1	yes	characteristic
182	C₂×C₁₃	#2047	1	1	yes	characteristic
183	C₂×C₁₃	#2056	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
196	Q₁₆	—	4	8	no	—
197	Q₁₆	—	4	8	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₁₃×D₈	1	2	yes	—
209	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	C₁₃×Q₁₆	1	2	yes	complemented
217	C₄²	—	4	1	no	—
218	C₄²	—	4	1	no	—
219	C₄²	C₂×D₄×C₁₃	1	2	yes	—
220	C₄²	C₁₃×#41	1	2	yes	—
221	C₁₃	#10538	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	C₁₃×#121	1	1	yes	characteristic
223	Q₈	—	4	4	no	—
224	Q₈	—	4	4	no	—
225	Q₈	—	8	2	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
239	C₂×C₄	C₁₃×#121	1	2	yes	—
240	C₂×C₄	C₁₃×#123	1	2	yes	—
241	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
242	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
243	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
244	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
245	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
246	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂×C₄	C₁₃×#137	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	2	no	—
253	C₂²	C₁₃×#394	1	1	yes	characteristic
254	C₂²	C₁₃×#459	1	1	yes	characteristic
255	C₂²	C₁₃×#460	1	1	yes	characteristic
256	C₂²	C₁₃×#467	1	1	yes	characteristic
257	C₂²	C₁₃×#470	1	1	yes	characteristic
258	C₂²	C₁₃×#473	1	1	yes	characteristic
259	C₂²	C₁₃×#473	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	1	no	—
264	C₄	C₁₃×#447	1	1	yes	characteristic
265	C₄	C₁₃×#450	1	1	yes	characteristic
266	C₄	C₁₃×#462	1	2	yes	—
267	C₄	C₁₃×#464	1	2	yes	—
268	C₄	C₁₃×#486	1	1	yes	characteristic
269	C₄	C₁₃×#490	1	1	yes	characteristic
270	C₂	C₁₃×#1669	1	1	yes	characteristic
271	C₂	C₁₃×#1670	1	1	yes	characteristic
272	C₂	C₁₃×#1689	1	1	yes	characteristic
273	C₂	C₁₃×#1690	1	1	yes	characteristic
274	C₂	C₁₃×#2041	1	1	yes	characteristic
275	C₂	C₁₃×#2047	1	1	yes	characteristic
276	C₂	C₁₃×#2056	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
46. #10538	C₁₃	direct (N × H)
79. C₁₃×C₄⋊C₄	Q₁₆	semidirect
216. C₄⋊C₄	C₁₃×Q₁₆	semidirect
221. C₁₃	#10538	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×C₁₃×#494	2
C₁₃×#1988	2
C₁₃×#1939	2
C₂×C₁₃×#332	2
C₁₃×#1277	2
#10538	13

Subgroups — C13×#10538 #15696877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10538 #15696877