Subgroups — C₁₃×#10823 #15697162

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#341	C₂	1	2	yes	maximal
3	C₂×C₁₃×#341	C₂	1	2	yes	maximal
4	C₁₃×#1728	C₂	1	2	yes	maximal
5	C₂²×C₁₃×M₁₆	C₂²	1	4	yes	—
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#97	—	2	4	no	—
10	C₂×C₁₃×#97	C₄	1	4	yes	—
11	C₁₃×#341	—	2	4	no	—
12	C₁₃×#341	C₄	1	8	yes	complemented
13	C₁₃×#336	—	2	4	no	—
14	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₁₃×M₁₆	—	2	4	no	—
17	C₂×C₁₃×M₁₆	—	2	4	no	—
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	—	2	4	no	—
20	C₂×C₁₃×M₁₆	—	2	4	no	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	4	yes	—
31	C₂×C₄²×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₁₃×#97	—	2	8	no	—
36	#10823	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	—	4	4	no	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	#41	1	2	yes	—
46	C₂²×C₄×C₁₃	#41	1	2	yes	—
47	C₂²×C₄×C₁₃	#41	1	2	yes	—
48	C₂²×C₄×C₁₃	#41	1	2	yes	—
49	C₁₃×M₁₆	—	2	8	no	—
50	C₁₃×M₁₆	—	2	8	no	—
51	C₁₃×M₁₆	—	4	4	no	—
52	C₁₃×M₁₆	—	4	4	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	2	8	no	—
58	C₂×C₈×C₁₃	—	2	8	no	—
59	C₂×C₈×C₁₃	—	2	8	no	—
60	C₂×C₈×C₁₃	—	4	4	no	—
61	C₂×C₈×C₁₃	—	4	4	no	—
62	C₂×C₈×C₁₃	C₄²	1	4	yes	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	4	no	—
67	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic
68	C₂×#341	C₂×C₁₃	1	2	yes	—
69	C₂×#341	C₂×C₁₃	1	2	yes	—
70	#1728	C₂×C₁₃	1	2	yes	—
71	C₂³×C₁₃	—	2	4	no	—
72	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
74	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
75	C₂³×C₁₃	#125	1	2	yes	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	8	no	—
78	C₂×C₄×C₁₃	—	4	8	no	—
79	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#117	1	4	yes	—
84	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₃	#121	1	2	yes	—
89	C₂×C₄×C₁₃	#123	1	4	yes	—
90	C₂×C₄×C₁₃	#124	1	2	yes	—
91	C₂×C₄×C₁₃	#126	1	4	yes	—
92	C₈×C₁₃	—	2	4	no	—
93	C₈×C₁₃	—	2	4	no	—
94	C₈×C₁₃	—	4	8	no	—
95	C₂²×M₁₆	C₂²×C₁₃	1	4	yes	—
96	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
98	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
99	C₂×#97	—	2	4	no	—
100	C₂×#97	C₄×C₁₃	1	4	yes	—
101	#341	—	2	4	no	—
102	#341	C₄×C₁₃	1	8	yes	complemented
103	#336	—	2	4	no	—
104	C₂²×C₁₃	—	2	4	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	4	no	—
107	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₃	#380	1	1	yes	characteristic
109	C₂²×C₁₃	#382	1	1	yes	characteristic
110	C₂²×C₁₃	#387	1	4	yes	—
111	C₂²×C₁₃	#430	1	4	yes	—
112	C₄×C₁₃	—	4	4	no	—
113	C₄×C₁₃	#376	1	1	yes	characteristic
114	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₁₃	#381	1	1	yes	characteristic
116	C₄×C₁₃	#383	1	1	yes	characteristic
117	C₄×C₁₃	#388	1	4	yes	—
118	C₂³×C₄	Q₈×C₁₃	1	2	yes	—
119	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
132	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
133	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
134	C₂²×C₈	D₄×C₁₃	1	4	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
138	C₂×C₄²	D₄×C₁₃	1	2	yes	—
139	#97	—	2	8	no	—
140	C₂×C₁₃	—	2	4	no	—
141	C₂×C₁₃	#1434	1	1	yes	characteristic
142	C₂×C₁₃	#1778	1	2	yes	—
143	C₂×C₁₃	#1811	1	2	yes	—
144	C₂×C₁₃	#1838	1	2	yes	—
145	C₂⁴	C₂×Q₈×C₁₃	1	1	yes	characteristic
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
149	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₁₃×#41	1	2	yes	—
154	C₂²×C₄	C₁₃×#41	1	2	yes	—
155	C₂²×C₄	C₁₃×#41	1	2	yes	—
156	C₂²×C₄	C₁₃×#41	1	2	yes	—
157	M₁₆	—	2	8	no	—
158	M₁₆	—	2	8	no	—
159	M₁₆	—	4	4	no	—
160	M₁₆	—	4	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	C₄²×C₁₃	1	4	yes	—
171	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	4	no	—
175	C₁₃	#10823	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³	—	2	4	no	—
177	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
178	C₂³	C₄×Q₈×C₁₃	1	2	yes	—
179	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
180	C₂³	C₁₃×#125	1	2	yes	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
187	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
188	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
189	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
190	C₂×C₄	C₁₃×#117	1	4	yes	—
191	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
192	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
193	C₂×C₄	C₁₃×#121	1	2	yes	—
194	C₂×C₄	C₁₃×#123	1	4	yes	—
195	C₂×C₄	C₁₃×#124	1	2	yes	—
196	C₂×C₄	C₁₃×#126	1	4	yes	—
197	C₈	—	2	4	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	8	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
204	C₂²	C₁₃×#380	1	1	yes	characteristic
205	C₂²	C₁₃×#382	1	1	yes	characteristic
206	C₂²	C₁₃×#387	1	4	yes	—
207	C₂²	C₁₃×#430	1	4	yes	—
208	C₄	—	4	4	no	—
209	C₄	C₁₃×#376	1	1	yes	characteristic
210	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
211	C₄	C₁₃×#381	1	1	yes	characteristic
212	C₄	C₁₃×#383	1	1	yes	characteristic
213	C₄	C₁₃×#388	1	4	yes	—
214	C₂	—	2	4	no	—
215	C₂	C₁₃×#1434	1	1	yes	characteristic
216	C₂	C₁₃×#1778	1	2	yes	—
217	C₂	C₁₃×#1811	1	2	yes	—
218	C₂	C₁₃×#1838	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#341	C₄	semidirect
36. #10823	C₁₃	direct (N × H)
102. #341	C₄×C₁₃	semidirect
175. C₁₃	#10823	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₃×#97	2
C₂×C₁₃×#341	2
C₁₃×#1728	2
#10823	13

Subgroups — C13×#10823 #15697162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10823 #15697162