Subgroups — C₁₃×#9705 #15696044

Order 3328.

538 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×#2868	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#332	C₂	1	4	yes	maximal
5	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#97	—	2	1	no	—
11	C₁₃×#384	—	2	2	no	—
12	C₁₃×#384	C₄	1	4	yes	—
13	C₄³×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#336	—	2	1	no	—
15	C₁₃×#336	—	2	2	no	—
16	C₁₃×#332	—	2	8	no	—
17	C₂²×Q₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₁₃×C₄⋊Q₈	—	2	4	no	—
21	C₁₃×C₄⋊Q₈	—	2	4	no	—
22	C₁₃×C₄⋊Q₈	—	4	4	no	—
23	C₁₃×C₄⋊Q₈	—	4	2	no	—
24	C₁₃×C₄⋊Q₈	—	4	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊C₈	—	2	4	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
37	C₁₃×#97	—	4	2	no	—
38	#9705	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×Q₈×C₁₃	—	4	4	no	—
40	C₂×Q₈×C₁₃	—	4	2	no	—
41	C₂×Q₈×C₁₃	—	4	2	no	—
42	C₂×Q₈×C₁₃	—	4	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	—	4	2	no	—
51	C₂×C₈×C₁₃	—	4	2	no	—
52	C₂×C₈×C₁₃	—	4	2	no	—
53	C₂×C₈×C₁₃	—	4	4	no	—
54	C₁₃×C₄⋊C₄	—	2	8	no	—
55	C₁₃×C₄⋊C₄	—	4	8	no	—
56	C₁₃×C₄⋊C₄	—	4	8	no	—
57	C₁₃×C₄⋊C₄	—	4	2	no	—
58	C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₄²×C₁₃	—	2	4	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	—	2	4	no	—
62	C₄²×C₁₃	D₈	1	2	yes	—
63	C₄²×C₁₃	SD₁₆	1	2	yes	—
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	C₄²×C₁₃	#41	1	2	yes	—
66	#2868	C₂×C₁₃	1	1	yes	characteristic
67	#1831	C₂×C₁₃	1	1	yes	characteristic
68	#1767	C₂×C₁₃	1	1	yes	characteristic
69	C₂×#332	C₂×C₁₃	1	4	yes	—
70	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	Q₈×C₁₃	—	8	4	no	—
72	Q₈×C₁₃	—	8	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	—	4	4	no	—
80	C₂×C₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	#101	1	4	yes	—
82	C₂×C₄×C₁₃	#102	1	4	yes	—
83	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₁₃	#121	1	2	yes	—
85	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₁₃	#137	1	4	yes	—
91	C₈×C₁₃	—	4	4	no	—
92	C₈×C₁₃	—	8	2	no	—
93	C₂×C₄⋊Q₈	—	2	1	no	—
94	C₂×C₄⋊Q₈	—	2	1	no	—
95	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
96	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
97	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
98	C₂×#97	—	2	1	no	—
99	#384	—	2	2	no	—
100	#384	C₄×C₁₃	1	4	yes	—
101	C₄³	C₂²×C₁₃	1	1	yes	characteristic
102	#336	—	2	2	no	—
103	#336	—	2	1	no	—
104	#332	—	2	8	no	—
105	C₂²×C₁₃	#390	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×#101	1	2	yes	—
107	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
108	C₂²×C₁₃	#418	1	1	yes	characteristic
109	C₂²×C₁₃	#419	1	2	yes	—
110	C₄×C₁₃	—	2	4	no	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	8	4	no	—
114	C₄×C₁₃	#459	1	2	yes	—
115	C₄×C₁₃	#460	1	2	yes	—
116	C₄×C₁₃	#491	1	2	yes	—
117	C₄×C₁₃	#497	1	2	yes	—
118	C₂²×Q₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	4	4	no	—
124	C₄⋊Q₈	—	4	2	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
130	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
131	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
132	C₂×C₄²	D₄×C₁₃	1	2	yes	—
133	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
134	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
138	#97	—	4	2	no	—
139	C₂×C₁₃	#1546	1	2	yes	—
140	C₂×C₁₃	#1556	1	2	yes	—
141	C₂×C₁₃	#1960	1	1	yes	characteristic
142	C₂×C₁₃	#1966	1	1	yes	characteristic
143	C₂×C₁₃	#1982	1	1	yes	characteristic
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
150	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	4	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	4	8	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	4	no	—
167	C₄²	C₁₃×D₈	1	2	yes	—
168	C₄²	C₁₃×SD₁₆	1	2	yes	—
169	C₄²	C₂×D₄×C₁₃	1	2	yes	—
170	C₄²	C₁₃×#41	1	2	yes	—
171	C₁₃	#9705	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
173	Q₈	—	8	4	no	—
174	Q₈	—	8	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	C₁₃×#101	1	4	yes	—
184	C₂×C₄	C₁₃×#102	1	4	yes	—
185	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
186	C₂×C₄	C₁₃×#121	1	2	yes	—
187	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
188	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
189	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
190	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
191	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
192	C₂×C₄	C₁₃×#137	1	4	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	8	2	no	—
195	C₂²	C₁₃×#390	1	1	yes	characteristic
196	C₂²	C₂×C₁₃×#101	1	2	yes	—
197	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
198	C₂²	C₁₃×#418	1	1	yes	characteristic
199	C₂²	C₁₃×#419	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	8	4	no	—
204	C₄	C₁₃×#459	1	2	yes	—
205	C₄	C₁₃×#460	1	2	yes	—
206	C₄	C₁₃×#491	1	2	yes	—
207	C₄	C₁₃×#497	1	2	yes	—
208	C₂	C₁₃×#1546	1	2	yes	—
209	C₂	C₁₃×#1556	1	2	yes	—
210	C₂	C₁₃×#1960	1	1	yes	characteristic
211	C₂	C₁₃×#1966	1	1	yes	characteristic
212	C₂	C₁₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
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
38. #9705	C₁₃	direct (N × H)
171. C₁₃	#9705	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×#2868	2
C₁₃×#1831	2
C₁₃×#1767	2
C₂×C₁₃×#332	2
#9705	13

Subgroups — C13×#9705 #15696044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9705 #15696044