Subgroups — C₁₃×#12814 #15699153

Order 3328.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#358	C₂	1	2	yes	maximal
4	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₄×C₁₃×Q₁₆	—	2	2	no	—
10	C₄×C₁₃×Q₁₆	C₄	1	4	yes	complemented
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	—	2	1	no	—
14	C₂×C₁₃×#103	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#382	—	2	1	no	—
17	C₁₃×C₄⋊C₁₆	—	2	2	no	—
18	C₁₃×#358	—	2	4	no	—
19	C₂²×Q₈×C₁₃	—	2	1	no	—
20	C₂×C₁₃×Q₁₆	—	2	1	no	—
21	C₂×C₁₃×Q₁₆	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	—	2	1	no	—
23	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×Q₁₆	D₄	1	2	yes	—
25	C₂×C₁₃×Q₁₆	Q₈	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₃	—	2	4	no	—
29	C₄×Q₈×C₁₃	—	4	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
33	C₂×C₄²×C₁₃	—	4	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	—	2	2	no	—
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₁₃×#107	—	2	1	no	—
39	C₁₃×#107	—	2	2	no	—
40	C₁₃×#107	—	2	2	no	—
41	C₁₃×#107	C₂×C₄	1	2	yes	—
42	C₁₃×#106	—	4	2	no	—
43	C₁₃×#103	—	2	4	no	—
44	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₁₃×#95	—	4	1	no	—
47	C₁₃×#95	—	4	1	no	—
48	#12814	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×Q₈×C₁₃	—	2	2	no	—
50	C₂×Q₈×C₁₃	—	2	2	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₁₃×Q₁₆	—	2	2	no	—
62	C₁₃×Q₁₆	—	2	2	no	—
63	C₁₃×Q₁₆	—	2	2	no	—
64	C₁₃×Q₁₆	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₁₃×C₄⋊C₄	—	4	1	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₄²×C₁₃	—	4	4	no	—
80	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₃	SD₁₆	1	2	yes	—
82	C₁₃×C₁₆	—	4	2	no	—
83	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
84	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
85	C₂×#358	C₂×C₁₃	1	2	yes	—
86	#1846	C₂×C₁₃	1	1	yes	characteristic
87	#1380	C₂×C₁₃	1	2	yes	—
88	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₁₃	—	2	4	no	—
90	Q₈×C₁₃	—	4	2	no	—
91	Q₈×C₁₃	—	4	2	no	—
92	Q₈×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	8	2	no	—
100	C₂×C₄×C₁₃	—	8	2	no	—
101	C₂×C₄×C₁₃	—	8	2	no	—
102	C₂×C₄×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	#102	1	2	yes	—
104	C₂×C₄×C₁₃	#102	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₁₃	#117	1	1	yes	characteristic
114	C₈×C₁₃	#121	1	1	yes	characteristic
115	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×C₄×Q₈	—	2	1	no	—
118	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
119	C₄×Q₁₆	—	2	2	no	—
120	C₄×Q₁₆	C₄×C₁₃	1	4	yes	complemented
121	C₂×#107	C₂²×C₁₃	1	2	yes	—
122	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×#106	—	2	1	no	—
124	C₂×#103	—	2	1	no	—
125	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
126	#382	—	2	1	no	—
127	C₄⋊C₁₆	—	2	2	no	—
128	#358	—	2	4	no	—
129	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₃	#459	1	1	yes	characteristic
131	C₂²×C₁₃	#474	1	1	yes	characteristic
132	C₂²×C₁₃	#508	1	2	yes	—
133	C₂²×C₁₃	#510	1	2	yes	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	4	4	no	—
137	C₄×C₁₃	—	8	2	no	—
138	C₄×C₁₃	—	16	2	no	—
139	C₄×C₁₃	#380	1	1	yes	characteristic
140	C₄×C₁₃	#417	1	1	yes	characteristic
141	C₄×C₁₃	#461	1	1	yes	characteristic
142	C₄×C₁₃	#476	1	1	yes	characteristic
143	C₂²×Q₈	—	2	1	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	2	1	no	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
148	C₂×Q₁₆	D₄×C₁₃	1	2	yes	—
149	C₂×Q₁₆	Q₈×C₁₃	1	2	yes	—
150	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
152	C₄×Q₈	—	2	4	no	—
153	C₄×Q₈	—	4	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	4	1	no	—
157	C₂×C₄²	—	4	1	no	—
158	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	#107	—	2	1	no	—
163	#107	—	2	2	no	—
164	#107	—	2	2	no	—
165	#107	C₂×C₄×C₁₃	1	2	yes	—
166	#106	—	4	2	no	—
167	#103	—	2	4	no	—
168	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
169	C₄×C₈	D₄×C₁₃	1	2	yes	—
170	#95	—	4	1	no	—
171	#95	—	4	1	no	—
172	C₂×C₁₃	#1802	1	1	yes	characteristic
173	C₂×C₁₃	#2138	1	1	yes	characteristic
174	C₂×C₁₃	#2141	1	1	yes	characteristic
175	C₂×C₁₃	#2211	1	2	yes	—
176	C₂×C₁₃	#2229	1	2	yes	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
187	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
189	Q₁₆	—	2	2	no	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	—	2	2	no	—
192	Q₁₆	C₁₃×C₄⋊C₄	1	4	yes	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
196	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
200	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
201	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄²	—	4	4	no	—
208	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
209	C₄²	C₁₃×SD₁₆	1	2	yes	—
210	C₁₆	—	4	2	no	—
211	C₁₃	#12814	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	C₁₃×#102	1	2	yes	—
228	C₂×C₄	C₁₃×#102	1	2	yes	—
229	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
231	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
233	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
237	C₈	C₁₃×#117	1	1	yes	characteristic
238	C₈	C₁₃×#121	1	1	yes	characteristic
239	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
241	C₂²	C₁₃×#459	1	1	yes	characteristic
242	C₂²	C₁₃×#474	1	1	yes	characteristic
243	C₂²	C₁₃×#508	1	2	yes	—
244	C₂²	C₁₃×#510	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	16	2	no	—
250	C₄	C₁₃×#380	1	1	yes	characteristic
251	C₄	C₁₃×#417	1	1	yes	characteristic
252	C₄	C₁₃×#461	1	1	yes	characteristic
253	C₄	C₁₃×#476	1	1	yes	characteristic
254	C₂	C₁₃×#1802	1	1	yes	characteristic
255	C₂	C₁₃×#2138	1	1	yes	characteristic
256	C₂	C₁₃×#2141	1	1	yes	characteristic
257	C₂	C₁₃×#2211	1	2	yes	—
258	C₂	C₁₃×#2229	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₃×Q₁₆	C₄	semidirect
48. #12814	C₁₃	direct (N × H)
120. C₄×Q₁₆	C₄×C₁₃	semidirect
211. C₁₃	#12814	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×C₄×C₁₃×Q₁₆	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#358	2
C₁₃×#1846	2
C₁₃×#1380	2
#12814	13

Subgroups — C13×#12814 #15699153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12814 #15699153