Subgroups — C₁₃×#33963 #15720302

Order 3328.

532 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3396	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2249	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2210	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2207	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×D₁₆	C₂	1	2	yes	maximal, complemented
9	C₁₃×#559	—	2	2	no	—
10	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
12	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
14	C₁₃×#495	C₂²	1	4	yes	—
15	C₁₃×#493	C₂²	1	2	yes	—
16	C₁₃×#463	—	2	4	no	—
17	C₁₃×#460	—	2	2	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₁₃×D₈	C₂²	1	4	yes	—
21	C₁₃×#366	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₁₃×#358	C₂²	1	2	yes	—
25	C₁₃×#357	C₂²	1	4	yes	—
26	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
28	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
29	C₂×C₁₃×SD₁₆	—	2	4	no	—
30	C₂×C₁₃×D₈	—	2	2	no	—
31	C₂×C₁₃×D₈	C₂³	1	4	yes	—
32	C₁₃×#127	—	2	2	no	—
33	C₁₃×#124	—	2	4	no	—
34	C₁₃×#124	—	4	2	no	—
35	C₁₃×#123	—	4	4	no	—
36	C₄×Q₈×C₁₃	—	2	2	no	—
37	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
38	C₄×D₄×C₁₃	—	2	4	no	—
39	C₄×D₄×C₁₃	—	4	2	no	—
40	C₁₃×Q₃₂	—	2	2	no	—
41	C₁₃×SD₃₂	—	4	4	no	—
42	C₁₃×D₁₆	—	2	4	no	—
43	C₁₃×D₁₆	—	4	2	no	—
44	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
47	C₁₃×#107	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
50	C₁₃×#103	—	2	2	no	—
51	C₁₃×#102	—	2	4	no	—
52	C₄×C₈×C₁₃	C₂³	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	#33963	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
57	C₂×D₄×C₁₃	—	2	4	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₁₃×Q₁₆	—	2	2	no	—
62	C₁₃×Q₁₆	—	2	2	no	—
63	C₁₃×SD₁₆	—	8	4	no	—
64	C₁₃×D₈	—	2	4	no	—
65	C₁₃×D₈	—	2	4	no	—
66	C₁₃×D₈	—	8	2	no	—
67	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	—	2	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
74	C₁₃×C₂²⋊C₄	—	4	4	no	—
75	C₁₃×C₂²⋊C₄	—	4	4	no	—
76	C₁₃×C₂²⋊C₄	—	4	4	no	—
77	C₄²×C₁₃	—	4	2	no	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₁₃×C₁₆	—	2	1	no	—
81	C₁₃×C₁₆	—	2	2	no	—
82	C₁₃×C₁₆	#41	1	2	yes	—
83	#3396	C₂×C₁₃	1	2	yes	complemented
84	#2249	C₂×C₁₃	1	2	yes	complemented
85	#2246	C₂×C₁₃	1	1	yes	characteristic
86	#2210	C₂×C₁₃	1	4	yes	complemented
87	#2207	C₂×C₁₃	1	2	yes	complemented
88	Q₈×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₄×D₁₆	C₂×C₁₃	1	2	yes	complemented
91	C₂³×C₁₃	—	4	4	no	—
92	Q₈×C₁₃	—	2	2	no	—
93	Q₈×C₁₃	—	4	2	no	—
94	Q₈×C₁₃	—	4	2	no	—
95	D₄×C₁₃	—	4	4	no	—
96	D₄×C₁₃	—	4	4	no	—
97	D₄×C₁₃	—	8	4	no	—
98	D₄×C₁₃	—	16	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	16	4	no	—
102	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₁₃	—	2	1	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
111	#559	—	2	2	no	—
112	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
114	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
115	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
116	#495	C₂²×C₁₃	1	4	yes	—
117	#493	C₂²×C₁₃	1	2	yes	—
118	#463	—	2	4	no	—
119	#460	—	2	2	no	—
120	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
122	C₄×D₈	C₂²×C₁₃	1	4	yes	—
123	#366	C₂²×C₁₃	1	1	yes	characteristic
124	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
126	#358	C₂²×C₁₃	1	2	yes	—
127	#357	C₂²×C₁₃	1	4	yes	—
128	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
129	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂²×C₁₃	—	8	4	no	—
131	C₂²×C₁₃	—	8	4	no	—
132	C₂²×C₁₃	—	8	4	no	—
133	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	8	2	no	—
138	C₄×C₁₃	—	8	2	no	—
139	C₄×C₁₃	#508	1	2	yes	—
140	C₄×C₁₃	#550	1	1	yes	characteristic
141	C₄×C₁₃	#576	1	1	yes	characteristic
142	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
143	C₂×SD₁₆	—	2	4	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	C₂³×C₁₃	1	4	yes	—
146	#127	—	2	2	no	—
147	#124	—	2	4	no	—
148	#124	—	4	2	no	—
149	#123	—	4	4	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
152	C₄×D₄	—	2	4	no	—
153	C₄×D₄	—	4	2	no	—
154	Q₃₂	—	2	2	no	—
155	SD₃₂	—	4	4	no	—
156	D₁₆	—	2	4	no	—
157	D₁₆	—	4	2	no	—
158	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
159	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
160	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
161	#107	C₂³×C₁₃	1	2	yes	—
162	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
163	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
164	#103	—	2	2	no	—
165	#102	—	2	4	no	—
166	C₄×C₈	C₂³×C₁₃	1	2	yes	—
167	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×C₁₃	—	16	4	no	—
169	C₂×C₁₃	#3391	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
171	C₂×C₁₃	#3415	1	1	yes	characteristic
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	4	no	—
178	Q₁₆	—	2	2	no	—
179	Q₁₆	—	2	2	no	—
180	SD₁₆	—	8	4	no	—
181	D₈	—	2	4	no	—
182	D₈	—	2	4	no	—
183	D₈	—	8	2	no	—
184	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
185	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
190	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
191	C₂²⋊C₄	—	4	4	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	4	4	no	—
194	C₄²	—	4	2	no	—
195	C₄²	C₂×D₄×C₁₃	1	2	yes	—
196	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₁₆	—	2	1	no	—
198	C₁₆	—	2	2	no	—
199	C₁₆	C₁₃×#41	1	2	yes	—
200	C₁₃	#33963	1	1	yes	characteristic, direct product factor, Sylow
201	C₂³	—	4	4	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	8	4	no	—
208	D₄	—	16	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	16	4	no	—
212	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
213	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
214	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
215	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
220	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	4	no	—
224	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	1	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	2	no	—
230	C₄	C₁₃×#508	1	2	yes	—
231	C₄	C₁₃×#550	1	1	yes	characteristic
232	C₄	C₁₃×#576	1	1	yes	characteristic
233	C₂	—	16	4	no	—
234	C₂	C₁₃×#3391	1	1	yes	characteristic
235	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
236	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	C₁₃×#2246	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3396	C₂	semidirect
2. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2210	C₂	semidirect
5. C₁₃×#2207	C₂	semidirect
6. Q₈×C₁₃×C₁₆	C₂	semidirect
7. C₄×C₁₃×Q₃₂	C₂	semidirect
8. C₄×C₁₃×D₁₆	C₂	semidirect
54. #33963	C₁₃	direct (N × H)
83. #3396	C₂×C₁₃	semidirect
84. #2249	C₂×C₁₃	semidirect
86. #2210	C₂×C₁₃	semidirect
87. #2207	C₂×C₁₃	semidirect
88. Q₈×C₁₆	C₂×C₁₃	semidirect
89. C₄×Q₃₂	C₂×C₁₃	semidirect
90. C₄×D₁₆	C₂×C₁₃	semidirect
200. C₁₃	#33963	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×#3396	2
C₁₃×#2249	2
C₁₃×#2246	2
C₁₃×#2210	2
C₁₃×#2207	2
Q₈×C₁₃×C₁₆	2
C₄×C₁₃×Q₃₂	2
C₄×C₁₃×D₁₆	2
#33963	13

Subgroups — C13×#33963 #15720302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33963 #15720302