Subgroups — C₁₃×#17352 #15703691

Order 3328.

1162 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#519	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2982	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1461	C₂	1	6	yes	maximal, complemented
4	C₁₃×#1458	C₂	1	6	yes	maximal
5	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₁₃×#519	—	2	8	no	—
7	C₂×C₄×Q₈×C₁₃	C₄	1	3	yes	—
8	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₄	1	3	yes	—
12	C₂×C₁₃×#117	C₂²	1	3	yes	—
13	C₈×Q₈×C₁₃	—	2	8	no	—
14	C₁₃×#431	C₂²	1	12	yes	—
15	C₁₃×#324	C₂²	1	12	yes	—
16	C₁₃×#323	C₂²	1	4	yes	—
17	C₂²×Q₈×C₁₃	—	2	4	no	—
18	C₂²×Q₈×C₁₃	—	2	4	no	—
19	C₂²×Q₈×C₁₃	C₈	1	8	yes	complemented
20	C₂²×Q₈×C₁₃	C₂×C₄	1	3	yes	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₃	—	2	4	no	—
24	C₄×Q₈×C₁₃	—	2	12	no	—
25	C₄×Q₈×C₁₃	—	2	12	no	—
26	C₄×Q₈×C₁₃	D₄	1	8	yes	—
27	C₁₃×#117	—	2	12	no	—
28	C₁₃×#117	—	2	12	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
33	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	3	yes	—
34	C₂×C₁₃×C₂²⋊C₄	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
38	C₁₃×C₄⋊C₈	—	2	12	no	—
39	C₁₃×C₂²⋊C₈	C₂³	1	8	yes	—
40	C₄×C₈×C₁₃	—	2	12	no	—
41	#17352	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	—	2	32	no	—
44	C₂×Q₈×C₁₃	—	2	6	no	—
45	C₂×Q₈×C₁₃	—	2	6	no	—
46	C₂×Q₈×C₁₃	C₂²⋊C₄	1	4	yes	—
47	C₂×Q₈×C₁₃	C₂×C₈	1	12	yes	—
48	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
49	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
50	C₂²×C₄×C₁₃	—	2	6	no	—
51	C₂²×C₄×C₁₃	—	2	6	no	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	12	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₁₃	—	2	8	no	—
60	C₁₃×C₄⋊C₄	—	2	6	no	—
61	C₁₃×C₄⋊C₄	—	2	6	no	—
62	C₁₃×C₄⋊C₄	—	2	6	no	—
63	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
64	C₁₃×C₂²⋊C₄	—	2	6	no	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₄²×C₁₃	—	2	12	no	—
67	C₄²×C₁₃	—	2	6	no	—
68	C₄²×C₁₃	C₂×D₄	1	12	yes	—
69	C₂×#519	C₂×C₁₃	1	1	yes	characteristic
70	#2982	C₂×C₁₃	1	2	yes	complemented
71	#1461	C₂×C₁₃	1	6	yes	complemented
72	#1458	C₂×C₁₃	1	6	yes	—
73	C₂³×C₁₃	—	2	1	no	—
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	4	1	no	—
76	C₂³×C₁₃	—	4	1	no	—
77	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
78	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₁₃	—	2	24	no	—
80	Q₈×C₁₃	C₂²⋊C₈	1	16	yes	—
81	C₂×C₄×C₁₃	—	2	3	no	—
82	C₂×C₄×C₁₃	—	2	3	no	—
83	C₂×C₄×C₁₃	—	2	24	no	—
84	C₂×C₄×C₁₃	—	2	3	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	4	3	no	—
87	C₂×C₄×C₁₃	—	4	1	no	—
88	C₂×C₄×C₁₃	—	4	3	no	—
89	C₂×C₄×C₁₃	—	4	1	no	—
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
91	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
93	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
94	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
95	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
96	C₈×C₁₃	—	4	8	no	—
97	C₂³×Q₈	C₄×C₁₃	1	1	yes	characteristic
98	#519	—	2	8	no	—
99	C₂×C₄×Q₈	C₄×C₁₃	1	3	yes	—
100	C₂×C₄×Q₈	C₄×C₁₃	1	1	yes	characteristic
101	C₂×C₄×Q₈	C₂²×C₁₃	1	3	yes	—
102	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
103	C₂×#117	C₄×C₁₃	1	3	yes	—
104	C₂×#117	C₂²×C₁₃	1	3	yes	—
105	C₈×Q₈	—	2	8	no	—
106	#431	C₂²×C₁₃	1	12	yes	—
107	#324	C₂²×C₁₃	1	12	yes	—
108	#323	C₂²×C₁₃	1	4	yes	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	4	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	1	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
119	C₄×C₁₃	—	2	6	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	6	no	—
122	C₄×C₁₃	—	4	3	no	—
123	C₄×C₁₃	—	4	1	no	—
124	C₄×C₁₃	C₂×C₂²⋊C₈	1	12	yes	—
125	C₂²×Q₈	—	2	4	no	—
126	C₂²×Q₈	—	2	4	no	—
127	C₂²×Q₈	C₈×C₁₃	1	8	yes	complemented
128	C₂²×Q₈	C₂×C₄×C₁₃	1	3	yes	—
129	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
130	C₂³×C₄	C₂×C₄×C₁₃	1	3	yes	—
131	C₄×Q₈	—	2	4	no	—
132	C₄×Q₈	—	2	12	no	—
133	C₄×Q₈	—	2	12	no	—
134	C₄×Q₈	D₄×C₁₃	1	8	yes	—
135	#117	—	2	12	no	—
136	#117	—	2	12	no	—
137	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
138	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
139	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
140	C₂×C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
141	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
142	C₂×C₂²⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
143	C₂×C₄²	C₂×C₄×C₁₃	1	6	yes	—
144	C₂×C₄²	C₂×C₄×C₁₃	1	3	yes	—
145	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
146	C₄⋊C₈	—	2	12	no	—
147	C₂²⋊C₈	C₂³×C₁₃	1	8	yes	—
148	C₄×C₈	—	2	12	no	—
149	C₂×C₁₃	—	2	1	no	—
150	C₂×C₁₃	—	2	1	no	—
151	C₂×C₁₃	—	4	2	no	—
152	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
153	C₂×C₁₃	#2884	1	1	yes	characteristic
154	C₂×C₁₃	#2889	1	1	yes	characteristic
155	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
156	C₂×Q₈	—	2	32	no	—
157	C₂×Q₈	—	2	6	no	—
158	C₂×Q₈	—	2	6	no	—
159	C₂×Q₈	C₁₃×C₂²⋊C₄	1	4	yes	—
160	C₂×Q₈	C₂×C₈×C₁₃	1	12	yes	—
161	C₂×Q₈	C₂×C₈×C₁₃	1	4	yes	—
162	C₂×Q₈	C₁₃×M₁₆	1	4	yes	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	C₂×C₈×C₁₃	1	12	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
168	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
171	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
172	C₂×C₈	—	2	8	no	—
173	C₄⋊C₄	—	2	6	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	—	2	6	no	—
176	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	12	yes	—
177	C₂²⋊C₄	—	2	6	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₄²	—	2	12	no	—
180	C₄²	—	2	6	no	—
181	C₄²	C₂×D₄×C₁₃	1	12	yes	—
182	C₁₃	#17352	1	1	yes	characteristic, direct product factor, Sylow
183	C₂³	—	2	1	no	—
184	C₂³	—	2	1	no	—
185	C₂³	—	4	1	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
188	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
189	Q₈	—	2	24	no	—
190	Q₈	C₁₃×C₂²⋊C₈	1	16	yes	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	24	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	3	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	6	yes	—
201	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	6	yes	—
202	C₂×C₄	C₂²×C₈×C₁₃	1	6	yes	—
203	C₂×C₄	C₂²×C₈×C₁₃	1	6	yes	—
204	C₂×C₄	C₂×C₁₃×M₁₆	1	6	yes	—
205	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
206	C₈	—	4	8	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
216	C₂²	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
217	C₄	—	2	6	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	6	no	—
220	C₄	—	4	3	no	—
221	C₄	—	4	1	no	—
222	C₄	C₂×C₁₃×C₂²⋊C₈	1	12	yes	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
227	C₂	C₁₃×#2884	1	1	yes	characteristic
228	C₂	C₁₃×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2982	C₂	semidirect
3. C₁₃×#1461	C₂	semidirect
19. C₂²×Q₈×C₁₃	C₈	semidirect
41. #17352	C₁₃	direct (N × H)
70. #2982	C₂×C₁₃	semidirect
71. #1461	C₂×C₁₃	semidirect
127. C₂²×Q₈	C₈×C₁₃	semidirect
182. C₁₃	#17352	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×C₁₃×#519	2
C₁₃×#2982	2
C₁₃×#1461	2
C₁₃×#1458	2
#17352	13

Subgroups — C13×#17352 #15703691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17352 #15703691