Subgroups — C₁₃×#9859 #15696198

Order 3328.

390 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1493	C₂	1	2	yes	maximal
4	C₁₃×#1493	C₂	1	2	yes	maximal
5	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	2	yes	—
7	C₁₃×#500	C₄	1	1	yes	characteristic
8	C₁₃×#500	C₄	1	1	yes	characteristic
9	C₁₃×#499	—	2	2	no	—
10	C₁₃×#498	C₄	1	2	yes	—
11	C₁₃×#427	—	2	2	no	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#341	—	2	2	no	—
17	C₁₃×Q₈⋊C₈	—	2	4	no	—
18	C₁₃×Q₈⋊C₈	—	2	4	no	—
19	C₁₃×#322	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	2	1	no	—
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	C₁₃×#125	—	2	2	no	—
27	C₁₃×C₂²⋊Q₈	—	2	2	no	—
28	C₁₃×C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈×C₁₃	—	2	2	no	—
30	C₄×Q₈×C₁₃	—	2	2	no	—
31	C₁₃×#117	—	2	2	no	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#107	—	2	2	no	—
34	C₁₃×#107	—	2	2	no	—
35	C₁₃×#106	—	2	4	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	2	no	—
38	C₄×C₈×C₁₃	—	2	1	no	—
39	C₄×C₈×C₁₃	—	2	1	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	D₄	1	2	yes	—
43	C₄×C₈×C₁₃	D₄	1	2	yes	—
44	C₄×C₈×C₁₃	D₄	1	2	yes	—
45	#9859	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×Q₈×C₁₃	—	2	2	no	—
47	C₂×Q₈×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
51	C₁₃×M₁₆	—	4	4	no	—
52	C₂×C₈×C₁₃	—	2	4	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	4	2	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₂²⋊C₄	—	4	2	no	—
67	C₄²×C₁₃	—	4	2	no	—
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	#3160	C₂×C₁₃	1	1	yes	characteristic
73	#1837	C₂×C₁₃	1	1	yes	characteristic
74	#1493	C₂×C₁₃	1	2	yes	—
75	#1493	C₂×C₁₃	1	2	yes	—
76	C₂×#322	C₂×C₁₃	1	1	yes	characteristic
77	C₂³×C₁₃	#124	1	1	yes	characteristic
78	Q₈×C₁₃	—	4	4	no	—
79	Q₈×C₁₃	—	4	4	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	—	4	2	no	—
84	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#137	1	4	yes	—
91	C₈×C₁₃	—	2	4	no	—
92	C₈×C₁₃	—	2	2	no	—
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	—	4	2	no	—
95	C₈×C₁₃	—	8	2	no	—
96	#533	C₂²×C₁₃	1	2	yes	—
97	#500	C₄×C₁₃	1	1	yes	characteristic
98	#500	C₄×C₁₃	1	1	yes	characteristic
99	#499	—	2	2	no	—
100	#498	C₄×C₁₃	1	2	yes	—
101	#427	—	2	2	no	—
102	#423	C₂²×C₁₃	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
106	#341	—	2	2	no	—
107	Q₈⋊C₈	—	2	4	no	—
108	Q₈⋊C₈	—	2	4	no	—
109	#322	—	2	2	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	#390	1	1	yes	characteristic
113	C₂²×C₁₃	#491	1	2	yes	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	8	2	no	—
117	C₄×C₁₃	—	8	2	no	—
118	C₄×C₁₃	#419	1	2	yes	—
119	C₄×C₁₃	#443	1	2	yes	—
120	C₄×C₁₃	#444	1	2	yes	—
121	C₄×C₁₃	#497	1	2	yes	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	D₄×C₁₃	1	2	yes	—
126	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
127	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
128	#125	—	2	2	no	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×Q₈	—	2	2	no	—
133	#117	—	2	2	no	—
134	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#106	—	2	4	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
143	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
144	C₄×C₈	D₄×C₁₃	1	2	yes	—
145	C₄×C₈	D₄×C₁₃	1	2	yes	—
146	C₄×C₈	D₄×C₁₃	1	2	yes	—
147	C₂×C₁₃	—	2	2	no	—
148	C₂×C₁₃	#1960	1	1	yes	characteristic
149	C₂×C₁₃	#1974	1	1	yes	characteristic
150	C₂×C₁₃	#1975	1	1	yes	characteristic
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
155	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
156	M₁₆	—	4	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
164	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
174	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
175	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
177	C₁₃	#9859	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³	C₁₃×#124	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
187	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
188	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
189	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
190	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
191	C₂×C₄	C₁₃×#137	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	8	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₁₃×#390	1	1	yes	characteristic
200	C₂²	C₁₃×#491	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	2	no	—
204	C₄	—	8	2	no	—
205	C₄	C₁₃×#419	1	2	yes	—
206	C₄	C₁₃×#443	1	2	yes	—
207	C₄	C₁₃×#444	1	2	yes	—
208	C₄	C₁₃×#497	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	C₁₃×#1960	1	1	yes	characteristic
211	C₂	C₁₃×#1974	1	1	yes	characteristic
212	C₂	C₁₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₂³×C₁₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
45. #9859	C₁₃	direct (N × H)
177. C₁₃	#9859	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×#3160	2
C₁₃×#1837	2
C₁₃×#1493	2
C₂×C₁₃×#322	2
#9859	13

Subgroups — C13×#9859 #15696198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9859 #15696198