Subgroups — C₁₃×#13538 #15699877

Order 3328.

294 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#1942	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1380	C₂	1	2	yes	maximal
4	C₁₃×#1380	C₂	1	2	yes	maximal
5	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃×#107	—	2	1	no	—
8	C₂×C₁₃×#107	C₂²	1	2	yes	—
9	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#389	—	2	1	no	—
13	C₁₃×#384	—	2	1	no	—
14	C₁₃×#340	—	2	1	no	—
15	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
20	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
21	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
22	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₁₃×C₁₆	—	2	2	no	—
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₁₃×#107	—	2	1	no	—
28	C₁₃×#107	—	2	2	no	—
29	C₁₃×#107	—	2	4	no	—
30	C₁₃×#107	—	4	2	no	—
31	C₁₃×#107	Q₈	1	2	yes	—
32	C₄×C₈×C₁₃	—	2	2	no	—
33	C₁₃×#95	—	4	1	no	—
34	#13538	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₄×C₁₃	—	4	1	no	—
36	C₂²×C₄×C₁₃	—	4	1	no	—
37	C₂²×C₄×C₁₃	—	4	2	no	—
38	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
44	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃	#41	1	2	yes	—
46	C₂×C₈×C₁₃	#41	1	2	yes	—
47	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
48	C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₁₃×C₄⋊C₄	—	4	1	no	—
50	C₁₃×C₄⋊C₄	—	4	1	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	2	no	—
53	C₁₃×C₄⋊C₄	—	4	1	no	—
54	C₁₃×C₄⋊C₄	—	8	2	no	—
55	C₁₃×C₄⋊C₄	—	8	2	no	—
56	C₁₃×C₄⋊C₄	—	8	2	no	—
57	C₄²×C₁₃	—	2	1	no	—
58	C₄²×C₁₃	—	2	1	no	—
59	C₁₃×C₁₆	—	4	2	no	—
60	#1942	C₂×C₁₃	1	1	yes	characteristic
61	#1845	C₂×C₁₃	1	1	yes	characteristic
62	#1380	C₂×C₁₃	1	2	yes	—
63	#1380	C₂×C₁₃	1	2	yes	—
64	#1374	C₂×C₁₃	1	1	yes	characteristic
65	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	2	1	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	8	1	no	—
72	C₂×C₄×C₁₃	—	8	1	no	—
73	C₂×C₄×C₁₃	—	8	1	no	—
74	C₂×C₄×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	—	8	1	no	—
76	C₂×C₄×C₁₃	—	8	2	no	—
77	C₂×C₄×C₁₃	—	8	2	no	—
78	C₂×C₄×C₁₃	—	8	1	no	—
79	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
80	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	#125	1	2	yes	—
89	C₈×C₁₃	#126	1	2	yes	—
90	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
91	C₂×#107	—	2	1	no	—
92	C₂×#107	C₂²×C₁₃	1	2	yes	—
93	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
94	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
96	#389	—	2	1	no	—
97	#384	—	2	1	no	—
98	#340	—	2	1	no	—
99	C₂²×C₁₃	#474	1	1	yes	characteristic
100	C₂²×C₁₃	#480	1	1	yes	characteristic
101	C₂²×C₁₃	#486	1	1	yes	characteristic
102	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
103	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
104	C₂²×C₁₃	#509	1	1	yes	characteristic
105	C₂²×C₁₃	#509	1	1	yes	characteristic
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	8	2	no	—
109	C₄×C₁₃	—	16	1	no	—
110	C₄×C₁₃	—	16	2	no	—
111	C₄×C₁₃	#400	1	1	yes	characteristic
112	C₄×C₁₃	#475	1	1	yes	characteristic
113	C₄×C₁₃	#483	1	1	yes	characteristic
114	C₄×C₁₃	#491	1	1	yes	characteristic
115	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	1	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	4	1	no	—
122	C₂×C₄⋊C₄	—	4	1	no	—
123	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	#107	—	2	1	no	—
128	#107	—	2	2	no	—
129	#107	—	2	4	no	—
130	#107	—	4	2	no	—
131	#107	Q₈×C₁₃	1	2	yes	—
132	C₄×C₈	—	2	2	no	—
133	#95	—	4	1	no	—
134	C₂×C₁₃	#2099	1	1	yes	characteristic
135	C₂×C₁₃	#2226	1	1	yes	characteristic
136	C₂×C₁₃	#2227	1	1	yes	characteristic
137	C₂×C₁₃	#2233	1	1	yes	characteristic
138	C₂×C₁₃	#2236	1	1	yes	characteristic
139	C₂×C₁₃	#2241	1	1	yes	characteristic
140	C₂×C₁₃	#2244	1	1	yes	characteristic
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
145	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
150	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
151	C₂×C₈	C₁₃×#41	1	2	yes	—
152	C₂×C₈	C₁₃×#41	1	2	yes	—
153	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	4	1	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₄⋊C₄	—	8	2	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₄⋊C₄	—	8	2	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	4	2	no	—
166	C₁₃	#13538	1	1	yes	characteristic, direct product factor, Sylow
167	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	C₁₃×SD₃₂	1	2	yes	—
182	C₂×C₄	C₁₃×Q₃₂	1	2	yes	—
183	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
184	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
185	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
186	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
187	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
188	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	C₁₃×#125	1	2	yes	—
191	C₈	C₁₃×#126	1	2	yes	—
192	C₂²	C₁₃×#474	1	1	yes	characteristic
193	C₂²	C₁₃×#480	1	1	yes	characteristic
194	C₂²	C₁₃×#486	1	1	yes	characteristic
195	C₂²	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
196	C₂²	C₂×C₁₃×Q₃₂	1	1	yes	characteristic
197	C₂²	C₁₃×#509	1	1	yes	characteristic
198	C₂²	C₁₃×#509	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	8	2	no	—
202	C₄	—	16	2	no	—
203	C₄	—	16	1	no	—
204	C₄	C₁₃×#400	1	1	yes	characteristic
205	C₄	C₁₃×#475	1	1	yes	characteristic
206	C₄	C₁₃×#483	1	1	yes	characteristic
207	C₄	C₁₃×#491	1	1	yes	characteristic
208	C₂	C₁₃×#2099	1	1	yes	characteristic
209	C₂	C₁₃×#2226	1	1	yes	characteristic
210	C₂	C₁₃×#2227	1	1	yes	characteristic
211	C₂	C₁₃×#2233	1	1	yes	characteristic
212	C₂	C₁₃×#2236	1	1	yes	characteristic
213	C₂	C₁₃×#2241	1	1	yes	characteristic
214	C₂	C₁₃×#2244	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
LCS	G ▷ C₂²×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
34. #13538	C₁₃	direct (N × H)
166. C₁₃	#13538	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×#1942	2
C₁₃×#1845	2
C₁₃×#1380	2
C₁₃×#1374	2
#13538	13

Subgroups — C13×#13538 #15699877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13538 #15699877