Subgroups — C₁₃×#33733 #15720072

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2215	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₃×D₁₆	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#532	—	2	2	no	—
9	C₂×C₁₃×D₁₆	C₂²	1	4	yes	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#500	C₂²	1	2	yes	—
12	C₁₃×#493	C₂²	1	2	yes	—
13	C₁₃×#486	—	2	2	no	—
14	C₁₃×#466	—	2	2	no	—
15	C₁₃×#466	C₂²	1	4	yes	—
16	C₄×C₁₃×D₈	—	2	2	no	—
17	C₄×C₁₃×D₈	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#366	C₂²	1	4	yes	—
20	C₁₃×#357	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×#41	—	4	2	no	—
25	C₂×C₁₃×D₈	—	2	2	no	—
26	C₂×C₁₃×D₈	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×#127	—	2	2	no	—
31	C₁₃×#124	—	4	2	no	—
32	C₁₃×#121	—	2	4	no	—
33	C₁₃×#121	—	4	2	no	—
34	C₄×D₄×C₁₃	—	2	4	no	—
35	C₄×D₄×C₁₃	—	4	2	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₁₃×D₁₆	—	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×#102	—	2	4	no	—
44	C₁₃×#102	—	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	#33733	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₁₃×#41	—	8	8	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×D₄×C₁₃	—	2	4	no	—
53	C₂×D₄×C₁₃	—	4	4	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	—	4	4	no	—
57	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₁₃×D₈	—	2	8	no	—
61	C₁₃×D₈	—	4	4	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₂²⋊C₄	—	4	4	no	—
71	C₁₃×C₂²⋊C₄	—	4	4	no	—
72	C₄²×C₁₃	D₈	1	1	yes	characteristic
73	C₄²×C₁₃	D₈	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×C₁₆	—	2	2	no	—
77	C₁₃×C₁₆	#41	1	4	yes	complemented
78	#3169	C₂×C₁₃	1	2	yes	complemented
79	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
80	#2246	C₂×C₁₃	1	1	yes	characteristic, complemented
81	#2240	C₂×C₁₃	1	2	yes	complemented
82	#2215	C₂×C₁₃	1	4	yes	complemented
83	C₄×D₁₆	C₂×C₁₃	1	4	yes	complemented
84	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂³×C₁₃	—	4	4	no	—
86	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₁₃	—	8	4	no	—
88	D₄×C₁₃	—	4	4	no	—
89	D₄×C₁₃	—	4	4	no	—
90	D₄×C₁₃	—	8	8	no	—
91	D₄×C₁₃	—	8	4	no	—
92	D₄×C₁₃	—	8	4	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	8	8	no	—
98	C₂×C₄×C₁₃	D₁₆	1	4	yes	complemented
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₁₃	—	2	1	no	—
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	#135	1	2	yes	—
108	C₈×C₁₃	C₂×#41	1	2	yes	—
109	#532	—	2	2	no	—
110	C₂×D₁₆	C₂²×C₁₃	1	4	yes	—
111	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
112	#500	C₂²×C₁₃	1	2	yes	—
113	#493	C₂²×C₁₃	1	2	yes	—
114	#486	—	2	2	no	—
115	#466	—	2	2	no	—
116	#466	C₂²×C₁₃	1	4	yes	—
117	C₄×D₈	—	2	2	no	—
118	C₄×D₈	C₂²×C₁₃	1	4	yes	—
119	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
120	#366	C₂²×C₁₃	1	4	yes	—
121	#357	C₂²×C₁₃	1	8	yes	—
122	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
123	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
124	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	8	4	no	—
128	C₂²×C₁₃	—	8	4	no	—
129	C₂²×C₁₃	—	8	4	no	—
130	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	2	1	no	—
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	8	4	no	—
135	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
136	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
137	C₄×C₁₃	#508	1	2	yes	—
138	C₄×C₁₃	#532	1	1	yes	characteristic
139	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
140	C₂×#41	—	4	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	C₂³×C₁₃	1	4	yes	—
143	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
145	C₄⋊Q₈	—	2	2	no	—
146	#127	—	2	2	no	—
147	#124	—	4	2	no	—
148	#121	—	2	4	no	—
149	#121	—	4	2	no	—
150	C₄×D₄	—	2	4	no	—
151	C₄×D₄	—	4	2	no	—
152	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
153	D₁₆	—	2	8	no	—
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
156	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
157	#107	—	2	2	no	—
158	#107	C₂³×C₁₃	1	4	yes	—
159	#102	—	2	4	no	—
160	#102	—	2	4	no	—
161	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
162	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
163	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
165	C₂×C₁₃	—	2	2	no	—
166	C₂×C₁₃	—	16	4	no	—
167	C₂×C₁₃	#3169	1	1	yes	characteristic
168	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
169	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
170	#41	—	8	8	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
178	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
180	D₈	—	2	8	no	—
181	D₈	—	4	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	4	no	—
191	C₂²⋊C₄	—	4	4	no	—
192	C₄²	C₁₃×D₈	1	1	yes	characteristic
193	C₄²	C₁₃×D₈	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₁₆	C₁₃×#41	1	4	yes	complemented
198	C₁₃	#33733	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	4	4	no	—
200	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	8	8	no	—
205	D₄	—	8	4	no	—
206	D₄	—	8	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	8	no	—
212	C₂×C₄	C₁₃×D₁₆	1	4	yes	complemented
213	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
214	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
215	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
216	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
217	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	C₁₃×#135	1	2	yes	—
222	C₈	C₂×C₁₃×#41	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	8	4	no	—
226	C₂²	—	8	4	no	—
227	C₂²	—	8	4	no	—
228	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
229	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	8	4	no	—
233	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
234	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
235	C₄	C₁₃×#508	1	2	yes	—
236	C₄	C₁₃×#532	1	1	yes	characteristic
237	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	16	4	no	—
240	C₂	C₁₃×#3169	1	1	yes	characteristic
241	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
242	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
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
1. C₁₃×#3169	C₂	semidirect
2. C₁₃×#2252	C₂	semidirect
3. C₁₃×#2246	C₂	semidirect
4. C₁₃×#2240	C₂	semidirect
5. C₁₃×#2215	C₂	semidirect
6. C₄×C₁₃×D₁₆	C₂	semidirect
7. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
49. #33733	C₁₃	direct (N × H)
77. C₁₃×C₁₆	#41	semidirect
78. #3169	C₂×C₁₃	semidirect
79. #2252	C₂×C₁₃	semidirect
80. #2246	C₂×C₁₃	semidirect
81. #2240	C₂×C₁₃	semidirect
82. #2215	C₂×C₁₃	semidirect
83. C₄×D₁₆	C₂×C₁₃	semidirect
84. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
98. C₂×C₄×C₁₃	D₁₆	semidirect
197. C₁₆	C₁₃×#41	semidirect
198. C₁₃	#33733	direct (N × H)
212. C₂×C₄	C₁₃×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3169	2
C₁₃×#2252	2
C₁₃×#2246	2
C₁₃×#2240	2
C₁₃×#2215	2
C₄×C₁₃×D₁₆	2
C₂×C₄×C₁₃×C₁₆	2
#33733	13

Subgroups — C13×#33733 #15720072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33733 #15720072