Subgroups — C₁₃×#33464 #15719803

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3414	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2957	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2179	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2178	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#359	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	2	yes	—
9	C₁₃×#568	—	2	2	no	—
10	C₁₃×#510	C₄	1	4	yes	complemented
11	C₁₃×#509	C₄	1	4	yes	complemented
12	C₁₃×#508	C₂²	1	4	yes	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#444	—	2	4	no	—
16	C₁₃×#443	C₂²	1	8	yes	complemented
17	C₂×C₁₃×#108	C₂²	1	2	yes	—
18	C₁₃×#421	—	2	2	no	—
19	C₁₃×#405	C₂²	1	1	yes	characteristic
20	C₁₃×#368	C₂²	1	4	yes	complemented
21	C₁₃×#359	C₂²	1	4	yes	complemented
22	C₁₃×#359	C₂²	1	4	yes	complemented
23	C₁₃×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
25	C₂×C₁₃×#41	—	2	2	no	—
26	C₁₃×#135	—	2	2	no	—
27	C₁₃×#135	—	2	2	no	—
28	C₁₃×#135	C₂×C₄	1	4	yes	—
29	C₁₃×#135	C₂³	1	4	yes	—
30	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×SD₁₆	—	2	2	no	—
32	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
33	C₁₃×#131	—	2	8	no	—
34	C₁₃×#131	—	4	4	no	—
35	C₂×C₁₃×M₁₆	—	2	2	no	—
36	C₂×C₁₃×M₁₆	—	4	4	no	—
37	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#117	D₄	1	1	yes	characteristic
40	C₁₃×Q₃₂	C₂×C₄	1	4	yes	—
41	C₁₃×SD₃₂	C₂×C₄	1	8	yes	—
42	C₁₃×D₁₆	C₂×C₄	1	4	yes	—
43	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
44	C₂×C₁₃×C₁₆	—	2	2	no	—
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
48	C₁₃×#108	—	2	2	no	—
49	C₁₃×#108	C₂³	1	4	yes	—
50	C₁₃×#104	—	2	8	no	—
51	C₁₃×#97	D₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂³	1	2	yes	—
53	#33464	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₁₃×#41	—	2	4	no	—
55	C₁₃×#41	—	2	4	no	—
56	C₁₃×#41	—	4	2	no	—
57	C₁₃×#41	—	4	2	no	—
58	C₂×Q₈×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₁₃×Q₁₆	—	2	2	no	—
64	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
65	C₁₃×SD₁₆	—	2	4	no	—
66	C₁₃×SD₁₆	—	2	4	no	—
67	C₁₃×D₈	—	2	2	no	—
68	C₁₃×D₈	C₂²×C₄	1	4	yes	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₁₃×M₁₆	—	4	8	no	—
72	C₁₃×M₁₆	D₈	1	4	yes	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	—	4	8	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	1	no	—
81	C₁₃×C₂²⋊C₄	—	2	1	no	—
82	C₄²×C₁₃	C₂×D₄	1	2	yes	—
83	C₁₃×C₁₆	—	2	1	no	—
84	C₁₃×C₁₆	—	2	1	no	—
85	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
86	#3414	C₂×C₁₃	1	1	yes	characteristic
87	#2957	C₂×C₁₃	1	2	yes	complemented
88	#2179	C₂×C₁₃	1	4	yes	complemented
89	#2178	C₂×C₁₃	1	4	yes	complemented
90	#2161	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂×#359	C₂×C₁₃	1	2	yes	complemented
92	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂³×C₁₃	—	4	2	no	—
94	C₂³×C₁₃	#135	1	1	yes	characteristic
95	Q₈×C₁₃	—	2	4	no	—
96	Q₈×C₁₃	—	4	2	no	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	4	4	no	—
100	D₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₁₃	—	4	4	no	—
110	C₈×C₁₃	C₄×D₄	1	2	yes	—
111	C₈×C₁₃	C₂×D₈	1	4	yes	—
112	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂×#135	C₂²×C₁₃	1	2	yes	—
115	#568	—	2	2	no	—
116	#510	C₄×C₁₃	1	4	yes	complemented
117	#509	C₄×C₁₃	1	4	yes	complemented
118	#508	C₂²×C₁₃	1	4	yes	—
119	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
120	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
121	#444	—	2	4	no	—
122	#443	C₂²×C₁₃	1	8	yes	complemented
123	C₂×#108	C₂²×C₁₃	1	2	yes	—
124	#421	—	2	2	no	—
125	#405	C₂²×C₁₃	1	1	yes	characteristic
126	#368	C₂²×C₁₃	1	4	yes	complemented
127	#359	C₂²×C₁₃	1	4	yes	complemented
128	#359	C₂²×C₁₃	1	4	yes	complemented
129	#346	C₂²×C₁₃	1	2	yes	complemented
130	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	—	4	4	no	—
133	C₂²×C₁₃	—	16	2	no	—
134	C₂²×C₁₃	C₄×D₈	1	2	yes	—
135	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
136	C₄×C₁₃	—	4	2	no	—
137	C₄×C₁₃	—	4	4	no	—
138	C₄×C₁₃	C₄×D₈	1	2	yes	—
139	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×#41	—	2	2	no	—
142	#135	—	2	2	no	—
143	#135	—	2	2	no	—
144	#135	C₂×C₄×C₁₃	1	4	yes	—
145	#135	C₂³×C₁₃	1	4	yes	—
146	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
147	C₂×SD₁₆	—	2	2	no	—
148	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
149	#131	—	2	8	no	—
150	#131	—	4	4	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	4	4	no	—
153	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
155	#117	D₄×C₁₃	1	1	yes	characteristic
156	Q₃₂	C₂×C₄×C₁₃	1	4	yes	—
157	SD₃₂	C₂×C₄×C₁₃	1	8	yes	—
158	D₁₆	C₂×C₄×C₁₃	1	4	yes	—
159	M₃₂	C₂×C₄×C₁₃	1	4	yes	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
163	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
164	#108	—	2	2	no	—
165	#108	C₂³×C₁₃	1	4	yes	—
166	#104	—	2	8	no	—
167	#97	D₄×C₁₃	1	2	yes	—
168	C₄×C₈	C₂³×C₁₃	1	2	yes	—
169	C₂×C₁₃	—	2	1	no	—
170	C₂×C₁₃	—	2	2	no	—
171	C₂×C₁₃	—	8	4	no	—
172	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
173	#41	—	2	4	no	—
174	#41	—	2	4	no	—
175	#41	—	4	2	no	—
176	#41	—	4	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
182	Q₁₆	—	2	2	no	—
183	Q₁₆	C₂²×C₄×C₁₃	1	4	yes	—
184	SD₁₆	—	2	4	no	—
185	SD₁₆	—	2	4	no	—
186	D₈	—	2	2	no	—
187	D₈	C₂²×C₄×C₁₃	1	4	yes	—
188	M₁₆	—	2	4	no	—
189	M₁₆	—	4	2	no	—
190	M₁₆	—	4	8	no	—
191	M₁₆	C₁₃×D₈	1	4	yes	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	8	no	—
194	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
197	C₂×C₈	C₁₃×#41	1	2	yes	—
198	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₄²	C₂×D₄×C₁₃	1	2	yes	—
202	C₁₆	—	2	1	no	—
203	C₁₆	—	2	1	no	—
204	C₁₆	C₂²×C₄×C₁₃	1	4	yes	—
205	C₁₃	#33464	1	1	yes	characteristic, direct product factor, Sylow
206	C₂³	—	4	2	no	—
207	C₂³	C₁₃×#135	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
219	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
220	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
222	C₈	—	4	4	no	—
223	C₈	C₄×D₄×C₁₃	1	2	yes	—
224	C₈	C₂×C₁₃×D₈	1	4	yes	—
225	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
226	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	C₄×C₁₃×D₈	1	2	yes	—
231	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	4	4	no	—
234	C₄	C₄×C₁₃×D₈	1	2	yes	—
235	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
236	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×C₄×C₁₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	C₁₃×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2957	C₂	semidirect
3. C₁₃×#2179	C₂	semidirect
4. C₁₃×#2178	C₂	semidirect
5. C₁₃×#2161	C₂	semidirect
6. C₂×C₁₃×#359	C₂	semidirect
7. C₁₃×#2108	C₂	semidirect
10. C₁₃×#510	C₄	semidirect
11. C₁₃×#509	C₄	semidirect
16. C₁₃×#443	C₂²	semidirect
20. C₁₃×#368	C₂²	semidirect
21. C₁₃×#359	C₂²	semidirect
22. C₁₃×#359	C₂²	semidirect
23. C₁₃×#346	C₂²	semidirect
24. C₄×C₁₃×C₁₆	C₂²	semidirect
53. #33464	C₁₃	direct (N × H)
87. #2957	C₂×C₁₃	semidirect
88. #2179	C₂×C₁₃	semidirect
89. #2178	C₂×C₁₃	semidirect
90. #2161	C₂×C₁₃	semidirect
91. C₂×#359	C₂×C₁₃	semidirect
92. #2108	C₂×C₁₃	semidirect
116. #510	C₄×C₁₃	semidirect
117. #509	C₄×C₁₃	semidirect
122. #443	C₂²×C₁₃	semidirect
126. #368	C₂²×C₁₃	semidirect
127. #359	C₂²×C₁₃	semidirect
128. #359	C₂²×C₁₃	semidirect
129. #346	C₂²×C₁₃	semidirect
130. C₄×C₁₆	C₂²×C₁₃	semidirect
205. C₁₃	#33464	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×#3414	2
C₁₃×#2957	2
C₁₃×#2179	2
C₁₃×#2178	2
C₁₃×#2161	2
C₂×C₁₃×#359	2
C₁₃×#2108	2
#33464	13

Subgroups — C13×#33464 #15719803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33464 #15719803