Subgroups — C₁₃×#33905 #15720244

Order 3328.

484 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×#3338	C₂	1	2	yes	maximal
2	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2236	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2227	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2216	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#366	C₂	1	2	yes	maximal, complemented
9	C₁₃×#554	—	2	2	no	—
10	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₁₃×#500	C₂²	1	2	yes	—
13	C₁₃×#484	—	2	4	no	—
14	C₁₃×#477	—	2	2	no	—
15	C₁₃×#467	C₂²	1	4	yes	—
16	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#107	C₂²	1	4	yes	—
19	C₁₃×#366	C₂²	1	2	yes	—
20	C₁₃×#366	C₂²	1	2	yes	—
21	C₁₃×#366	C₂²	1	1	yes	characteristic
22	C₁₃×#366	C₂²	1	4	yes	—
23	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×#358	C₂²	1	2	yes	—
25	C₁₃×#358	C₂²	1	4	yes	—
26	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₁₃×C₄⋊Q₈	—	2	2	no	—
31	C₁₃×#125	—	4	2	no	—
32	C₁₃×C₂²⋊Q₈	—	2	4	no	—
33	C₁₃×C₂²⋊Q₈	—	4	4	no	—
34	C₄×Q₈×C₁₃	—	2	2	no	—
35	C₄×D₄×C₁₃	—	4	2	no	—
36	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₁₃×Q₃₂	—	2	2	no	—
39	C₂×C₁₃×C₁₆	—	2	4	no	—
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₁₃×#107	—	2	4	no	—
44	C₁₃×#107	—	2	2	no	—
45	C₁₃×#107	C₂³	1	2	yes	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₁₃×#103	—	2	2	no	—
49	C₁₃×#103	—	2	4	no	—
50	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	#33905	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×Q₈×C₁₃	—	2	2	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	4	4	no	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₁₃×Q₁₆	—	2	2	no	—
59	C₁₃×Q₁₆	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	2	4	no	—
67	C₁₃×C₄⋊C₄	—	4	4	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
72	C₁₃×C₂²⋊C₄	—	4	4	no	—
73	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
74	C₄²×C₁₃	—	4	2	no	—
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×C₁₆	—	2	2	no	—
77	C₁₃×C₁₆	—	2	1	no	—
78	C₁₃×C₁₆	#41	1	2	yes	—
79	#3338	C₂×C₁₃	1	2	yes	—
80	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
81	#2236	C₂×C₁₃	1	4	yes	complemented
82	#2227	C₂×C₁₃	1	2	yes	complemented
83	#2216	C₂×C₁₃	1	2	yes	complemented
84	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
85	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
86	C₂×#366	C₂×C₁₃	1	2	yes	complemented
87	C₂³×C₁₃	C₂×D₈	1	2	yes	—
88	Q₈×C₁₃	—	4	2	no	—
89	Q₈×C₁₃	—	4	2	no	—
90	Q₈×C₁₃	—	16	2	no	—
91	D₄×C₁₃	Q₃₂	1	4	yes	complemented
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	16	4	no	—
98	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
105	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
106	#554	—	2	2	no	—
107	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
108	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
109	#500	C₂²×C₁₃	1	2	yes	—
110	#484	—	2	4	no	—
111	#477	—	2	2	no	—
112	#467	C₂²×C₁₃	1	4	yes	—
113	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
114	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
115	C₂×#107	C₂²×C₁₃	1	4	yes	—
116	#366	C₂²×C₁₃	1	1	yes	characteristic
117	#366	C₂²×C₁₃	1	2	yes	—
118	#366	C₂²×C₁₃	1	4	yes	—
119	#366	C₂²×C₁₃	1	2	yes	—
120	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
121	#358	C₂²×C₁₃	1	2	yes	—
122	#358	C₂²×C₁₃	1	4	yes	—
123	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
124	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	C₂×Q₃₂	1	4	yes	—
128	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	8	2	no	—
132	C₄×C₁₃	—	8	2	no	—
133	C₄×C₁₃	—	8	4	no	—
134	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
135	C₄×C₁₃	#547	1	1	yes	characteristic
136	C₄×C₁₃	#578	1	1	yes	characteristic
137	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
138	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	#125	—	4	2	no	—
141	C₂²⋊Q₈	—	2	4	no	—
142	C₂²⋊Q₈	—	4	4	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×D₄	—	4	2	no	—
145	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	4	no	—
147	Q₃₂	—	2	2	no	—
148	C₂×C₁₆	—	2	4	no	—
149	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
150	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
151	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
152	#107	—	2	4	no	—
153	#107	—	2	2	no	—
154	#107	C₂³×C₁₃	1	2	yes	—
155	#107	C₂³×C₁₃	1	4	yes	—
156	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
157	#103	—	2	2	no	—
158	#103	—	2	4	no	—
159	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
160	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
161	C₂×C₁₃	—	2	4	no	—
162	C₂×C₁₃	#3334	1	1	yes	characteristic
163	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
164	C₂×C₁₃	#3415	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
170	Q₁₆	—	2	2	no	—
171	Q₁₆	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
176	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
184	C₂²⋊C₄	—	4	4	no	—
185	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
186	C₄²	—	4	2	no	—
187	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
188	C₁₆	—	2	2	no	—
189	C₁₆	—	2	1	no	—
190	C₁₆	C₁₃×#41	1	2	yes	—
191	C₁₃	#33905	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	C₂×C₁₃×D₈	1	2	yes	—
193	Q₈	—	4	2	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	16	2	no	—
196	D₄	C₁₃×Q₃₂	1	4	yes	complemented
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	16	4	no	—
203	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
205	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	1	no	—
209	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
210	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	C₂×C₁₃×Q₃₂	1	4	yes	—
214	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	4	no	—
220	C₄	C₂×C₁₃×Q₃₂	1	2	yes	—
221	C₄	C₁₃×#547	1	1	yes	characteristic
222	C₄	C₁₃×#578	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₁₃×#3334	1	1	yes	characteristic
225	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic
226	C₂	C₁₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×D₄×C₁₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×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×#2236	C₂	semidirect
4. C₁₃×#2227	C₂	semidirect
5. C₁₃×#2216	C₂	semidirect
6. C₄×C₁₃×Q₃₂	C₂	semidirect
8. C₂×C₁₃×#366	C₂	semidirect
52. #33905	C₁₃	direct (N × H)
80. #2252	C₂×C₁₃	semidirect
81. #2236	C₂×C₁₃	semidirect
82. #2227	C₂×C₁₃	semidirect
83. #2216	C₂×C₁₃	semidirect
84. C₄×Q₃₂	C₂×C₁₃	semidirect
86. C₂×#366	C₂×C₁₃	semidirect
91. D₄×C₁₃	Q₃₂	semidirect
191. C₁₃	#33905	direct (N × H)
196. D₄	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×#3338	2
C₁₃×#2252	2
C₁₃×#2236	2
C₁₃×#2227	2
C₁₃×#2216	2
C₄×C₁₃×Q₃₂	2
D₄×C₁₃×C₁₆	2
C₂×C₁₃×#366	2
#33905	13

Subgroups — C13×#33905 #15720244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33905 #15720244