Subgroups — C₁₃×#33734 #15720073

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	2	yes	maximal
2	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2241	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2216	C₂	1	4	yes	maximal
6	C₄×C₁₃×Q₃₂	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	—	2	2	no	—
9	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#500	C₂²	1	2	yes	—
12	C₁₃×#494	C₂²	1	2	yes	—
13	C₁₃×#487	—	2	2	no	—
14	C₁₃×#467	—	2	2	no	—
15	C₁₃×#467	C₂²	1	4	yes	—
16	C₄×C₁₃×Q₁₆	—	2	2	no	—
17	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#366	C₂²	1	4	yes	—
20	C₁₃×#358	C₂²	1	8	yes	—
21	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
22	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
23	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
24	C₂×C₁₃×Q₁₆	—	2	2	no	—
25	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×C₄⋊Q₈	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×#125	—	4	2	no	—
31	C₁₃×C₂²⋊Q₈	—	2	4	no	—
32	C₁₃×C₂²⋊Q₈	—	4	2	no	—
33	C₄×Q₈×C₁₃	—	2	4	no	—
34	C₄×Q₈×C₁₃	—	4	2	no	—
35	C₁₃×#117	—	4	2	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₁₃×Q₃₂	—	2	8	no	—
38	C₂×C₁₃×C₁₆	—	2	4	no	—
39	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
41	C₁₃×#107	—	2	2	no	—
42	C₁₃×#107	C₂³	1	4	yes	—
43	C₁₃×#103	—	2	4	no	—
44	C₁₃×#103	—	2	4	no	—
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	#33734	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×Q₈×C₁₃	—	2	4	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₁₃×Q₁₆	—	2	8	no	—
56	C₁₃×Q₁₆	—	4	4	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	2	4	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	4	no	—
67	C₁₃×C₄⋊C₄	—	4	4	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₂²⋊C₄	—	4	4	no	—
70	C₄²×C₁₃	—	4	4	no	—
71	C₄²×C₁₃	D₈	1	1	yes	characteristic
72	C₄²×C₁₃	D₈	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₁₃×C₁₆	—	2	2	no	—
76	C₁₃×C₁₆	#41	1	4	yes	—
77	#3170	C₂×C₁₃	1	2	yes	—
78	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
79	#2247	C₂×C₁₃	1	1	yes	characteristic, complemented
80	#2241	C₂×C₁₃	1	2	yes	complemented
81	#2216	C₂×C₁₃	1	4	yes	—
82	C₄×Q₃₂	C₂×C₁₃	1	4	yes	complemented
83	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
84	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₁₃	—	4	4	no	—
86	Q₈×C₁₃	—	4	4	no	—
87	Q₈×C₁₃	—	8	4	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	Q₃₂	1	4	yes	complemented
94	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	—	2	1	no	—
102	C₈×C₁₃	#135	1	2	yes	—
103	C₈×C₁₃	C₂×#41	1	2	yes	—
104	#533	—	2	2	no	—
105	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
106	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
107	#500	C₂²×C₁₃	1	2	yes	—
108	#494	C₂²×C₁₃	1	2	yes	—
109	#487	—	2	2	no	—
110	#467	—	2	2	no	—
111	#467	C₂²×C₁₃	1	4	yes	—
112	C₄×Q₁₆	—	2	2	no	—
113	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
114	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
115	#366	C₂²×C₁₃	1	4	yes	—
116	#358	C₂²×C₁₃	1	8	yes	—
117	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
118	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
119	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
123	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	8	4	no	—
127	C₄×C₁₃	—	8	4	no	—
128	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
129	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
130	C₄×C₁₃	#508	1	2	yes	—
131	C₄×C₁₃	#532	1	1	yes	characteristic
132	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
135	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	#125	—	4	2	no	—
140	C₂²⋊Q₈	—	2	4	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₄×Q₈	—	2	4	no	—
143	C₄×Q₈	—	4	2	no	—
144	#117	—	4	2	no	—
145	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
146	Q₃₂	—	2	8	no	—
147	C₂×C₁₆	—	2	4	no	—
148	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
149	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
150	#107	—	2	2	no	—
151	#107	C₂³×C₁₃	1	4	yes	—
152	#103	—	2	4	no	—
153	#103	—	2	4	no	—
154	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
155	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
156	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
158	C₂×C₁₃	—	2	2	no	—
159	C₂×C₁₃	#3169	1	1	yes	characteristic
160	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
161	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
165	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
167	Q₁₆	—	2	8	no	—
168	Q₁₆	—	4	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
174	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	4	no	—
182	C₄²	—	4	4	no	—
183	C₄²	C₁₃×D₈	1	1	yes	characteristic
184	C₄²	C₁₃×D₈	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₁₆	—	2	2	no	—
188	C₁₆	C₁₃×#41	1	4	yes	—
189	C₁₃	#33734	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	Q₈	—	4	4	no	—
193	Q₈	—	8	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	C₁₃×Q₃₂	1	4	yes	complemented
200	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
201	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
202	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
203	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	C₁₃×#135	1	2	yes	—
209	C₈	C₂×C₁₃×#41	1	2	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×C₁₃×Q₃₂	1	2	yes	—
213	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₂×C₁₃×Q₃₂	1	2	yes	—
219	C₄	C₂×C₁₃×Q₃₂	1	2	yes	—
220	C₄	C₁₃×#508	1	2	yes	—
221	C₄	C₁₃×#532	1	1	yes	characteristic
222	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	C₁₃×#3169	1	1	yes	characteristic
225	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic
226	C₂	C₂×C₁₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
2. C₁₃×#2252	C₂	semidirect
3. C₁₃×#2247	C₂	semidirect
4. C₁₃×#2241	C₂	semidirect
6. C₄×C₁₃×Q₃₂	C₂	semidirect
49. #33734	C₁₃	direct (N × H)
78. #2252	C₂×C₁₃	semidirect
79. #2247	C₂×C₁₃	semidirect
80. #2241	C₂×C₁₃	semidirect
82. C₄×Q₃₂	C₂×C₁₃	semidirect
93. C₂×C₄×C₁₃	Q₃₂	semidirect
189. C₁₃	#33734	direct (N × H)
199. C₂×C₄	C₁₃×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3170	2
C₁₃×#2252	2
C₁₃×#2247	2
C₁₃×#2241	2
C₁₃×#2216	2
C₄×C₁₃×Q₃₂	2
C₂×C₄×C₁₃×C₁₆	2
#33734	13

Subgroups — C13×#33734 #15720073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33734 #15720073