Subgroups — C₁₃×#33960 #15720299

Order 3328.

508 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3393	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2249	C₂	1	2	yes	maximal
3	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2211	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×Q₃₂	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#556	—	2	2	no	—
10	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
11	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
13	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×#495	C₂²	1	4	yes	—
15	C₁₃×#494	C₂²	1	2	yes	—
16	C₁₃×#464	—	2	4	no	—
17	C₁₃×#461	—	2	2	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
20	C₄×C₁₃×D₈	C₂²	1	2	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	4	yes	—
25	C₁₃×#357	C₂²	1	2	yes	—
26	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
28	C₂×C₁₃×Q₁₆	—	2	2	no	—
29	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
30	C₂×C₁₃×SD₁₆	—	2	4	no	—
31	C₂×C₁₃×D₈	C₂³	1	2	yes	—
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×#126	—	4	4	no	—
34	C₁₃×#124	—	2	4	no	—
35	C₁₃×C₂²⋊Q₈	—	4	2	no	—
36	C₄×Q₈×C₁₃	—	2	4	no	—
37	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
38	C₄×D₄×C₁₃	—	2	2	no	—
39	C₁₃×#117	—	4	2	no	—
40	C₁₃×Q₃₂	—	2	4	no	—
41	C₁₃×Q₃₂	—	4	2	no	—
42	C₁₃×SD₃₂	—	4	4	no	—
43	C₁₃×D₁₆	—	2	2	no	—
44	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
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	4	no	—
51	C₁₃×#102	—	2	2	no	—
52	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	#33960	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×Q₈×C₁₃	—	2	4	no	—
56	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₁₃×Q₁₆	—	2	4	no	—
60	C₁₃×Q₁₆	—	2	4	no	—
61	C₁₃×Q₁₆	—	8	2	no	—
62	C₁₃×SD₁₆	—	8	4	no	—
63	C₁₃×D₈	—	2	2	no	—
64	C₁₃×D₈	—	2	2	no	—
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	4	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	2	no	—
76	C₁₃×C₂²⋊C₄	—	4	4	no	—
77	C₄²×C₁₃	—	4	4	no	—
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	C₁₃×C₁₆	—	2	2	no	—
81	C₁₃×C₁₆	—	2	1	no	—
82	C₁₃×C₁₆	#41	1	2	yes	—
83	#3393	C₂×C₁₃	1	2	yes	complemented
84	#2249	C₂×C₁₃	1	2	yes	—
85	#2247	C₂×C₁₃	1	1	yes	characteristic, complemented
86	#2211	C₂×C₁₃	1	4	yes	complemented
87	#2208	C₂×C₁₃	1	2	yes	complemented
88	Q₈×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₄×Q₃₂	C₂×C₁₃	1	2	yes	complemented
90	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic
91	C₂³×C₁₃	—	4	2	no	—
92	Q₈×C₁₃	—	2	2	no	—
93	Q₈×C₁₃	—	4	4	no	—
94	Q₈×C₁₃	—	4	4	no	—
95	D₄×C₁₃	—	4	2	no	—
96	D₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
108	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
109	#556	—	2	2	no	—
110	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
111	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
112	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
113	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
114	#495	C₂²×C₁₃	1	4	yes	—
115	#494	C₂²×C₁₃	1	2	yes	—
116	#464	—	2	4	no	—
117	#461	—	2	2	no	—
118	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
119	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
120	C₄×D₈	C₂²×C₁₃	1	2	yes	—
121	#366	C₂²×C₁₃	1	1	yes	characteristic
122	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
123	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
124	#358	C₂²×C₁₃	1	4	yes	—
125	#357	C₂²×C₁₃	1	2	yes	—
126	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
127	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
128	C₂²×C₁₃	—	8	2	no	—
129	C₂²×C₁₃	—	8	2	no	—
130	C₂²×C₁₃	—	8	2	no	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	8	4	no	—
136	C₄×C₁₃	—	8	2	no	—
137	C₄×C₁₃	#508	1	2	yes	—
138	C₄×C₁₃	#550	1	1	yes	characteristic
139	C₄×C₁₃	#576	1	1	yes	characteristic
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
142	C₂×SD₁₆	—	2	4	no	—
143	C₂×D₈	C₂³×C₁₃	1	2	yes	—
144	C₄⋊Q₈	—	2	2	no	—
145	#126	—	4	4	no	—
146	#124	—	2	4	no	—
147	C₂²⋊Q₈	—	4	2	no	—
148	C₄×Q₈	—	2	4	no	—
149	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
150	C₄×D₄	—	2	2	no	—
151	#117	—	4	2	no	—
152	Q₃₂	—	2	4	no	—
153	Q₃₂	—	4	2	no	—
154	SD₃₂	—	4	4	no	—
155	D₁₆	—	2	2	no	—
156	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
157	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
158	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
159	#107	C₂³×C₁₃	1	2	yes	—
160	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
161	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
162	#103	—	2	4	no	—
163	#102	—	2	2	no	—
164	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
165	C₄×C₈	C₂³×C₁₃	1	2	yes	—
166	C₂×C₁₃	—	16	2	no	—
167	C₂×C₁₃	#3391	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
169	C₂×C₁₃	#3417	1	1	yes	characteristic
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
172	C₂×D₄	—	2	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	Q₁₆	—	2	4	no	—
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	8	2	no	—
177	SD₁₆	—	8	4	no	—
178	D₈	—	2	2	no	—
179	D₈	—	2	2	no	—
180	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
182	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
188	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
189	C₂²⋊C₄	—	4	4	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	4	no	—
192	C₄²	—	4	4	no	—
193	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₁₃	1	2	yes	—
195	C₁₆	—	2	2	no	—
196	C₁₆	—	2	1	no	—
197	C₁₆	C₁₃×#41	1	2	yes	—
198	C₁₃	#33960	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	4	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	D₄	—	4	2	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
209	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
210	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
211	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	1	no	—
215	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
216	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²	—	8	2	no	—
218	C₂²	—	8	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	—	8	2	no	—
226	C₄	C₁₃×#508	1	2	yes	—
227	C₄	C₁₃×#550	1	1	yes	characteristic
228	C₄	C₁₃×#576	1	1	yes	characteristic
229	C₂	—	16	2	no	—
230	C₂	C₁₃×#3391	1	1	yes	characteristic
231	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
232	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×D₁₆	C₂²
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×#3393	C₂	semidirect
3. C₁₃×#2247	C₂	semidirect
4. C₁₃×#2211	C₂	semidirect
5. C₁₃×#2208	C₂	semidirect
6. Q₈×C₁₃×C₁₆	C₂	semidirect
7. C₄×C₁₃×Q₃₂	C₂	semidirect
54. #33960	C₁₃	direct (N × H)
83. #3393	C₂×C₁₃	semidirect
85. #2247	C₂×C₁₃	semidirect
86. #2211	C₂×C₁₃	semidirect
87. #2208	C₂×C₁₃	semidirect
88. Q₈×C₁₆	C₂×C₁₃	semidirect
89. C₄×Q₃₂	C₂×C₁₃	semidirect
198. C₁₃	#33960	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×#3393	2
C₁₃×#2249	2
C₁₃×#2247	2
C₁₃×#2211	2
C₁₃×#2208	2
Q₈×C₁₃×C₁₆	2
C₄×C₁₃×Q₃₂	2
C₄×C₁₃×D₁₆	2
#33960	13

Subgroups — C13×#33960 #15720299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33960 #15720299