Subgroups — C₁₃×#33491 #15719830

Order 3328.

584 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2168	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₃×C₁₆	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2164	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2113	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#525	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	complemented
11	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
12	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	complemented
13	C₁₃×#435	C₄	1	4	yes	—
14	C₈×D₄×C₁₃	C₂²	1	4	yes	—
15	C₁₃×#424	C₄	1	1	yes	characteristic
16	C₁₃×#408	C₄	1	1	yes	characteristic
17	C₁₃×#407	C₄	1	1	yes	characteristic
18	C₁₃×#405	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
21	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
24	C₁₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
26	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₁₃×#131	—	4	2	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	—	4	2	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₁₃×#123	C₈	1	2	yes	—
39	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
40	C₁₃×#121	C₈	1	2	yes	—
41	C₁₃×#120	C₈	1	2	yes	—
42	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
43	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
44	C₁₃×M₃₂	—	4	2	no	—
45	C₂×C₁₃×C₁₆	—	4	4	no	—
46	C₂×C₁₃×C₁₆	—	4	2	no	—
47	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
48	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₃×#97	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	#33491	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₁₃×#41	—	4	1	no	—
57	C₁₃×#41	—	4	1	no	—
58	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
60	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₁₃×M₁₆	—	4	1	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₁₃×M₁₆	—	4	1	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
84	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
85	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
86	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
87	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₁₃×C₁₆	—	2	4	no	—
90	C₁₃×C₁₆	—	2	4	no	—
91	#2929	C₂×C₁₃	1	1	yes	characteristic
92	#2168	C₂×C₁₃	1	4	yes	complemented
93	D₄×C₁₆	C₂×C₁₃	1	4	yes	complemented
94	#2164	C₂×C₁₃	1	2	yes	complemented
95	#2113	C₂×C₁₃	1	2	yes	complemented
96	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	2	yes	complemented
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	—	4	1	no	—
99	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
100	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
101	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
102	Q₈×C₁₃	—	4	1	no	—
103	D₄×C₁₃	—	2	8	no	—
104	D₄×C₁₃	—	4	1	no	—
105	D₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	—	2	2	no	—
123	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
124	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
126	#525	C₄×C₁₃	1	1	yes	characteristic
127	C₂×M₃₂	C₂²×C₁₃	1	2	yes	complemented
128	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
129	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	complemented
130	#435	C₄×C₁₃	1	4	yes	—
131	C₈×D₄	C₂²×C₁₃	1	4	yes	—
132	#424	C₄×C₁₃	1	1	yes	characteristic
133	#408	C₄×C₁₃	1	1	yes	characteristic
134	#407	C₄×C₁₃	1	1	yes	characteristic
135	#405	C₂²×C₁₃	1	1	yes	characteristic
136	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
137	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
138	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
139	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
140	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
141	#346	C₂²×C₁₃	1	2	yes	complemented
142	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	4	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	4	4	no	—
150	C₂²×C₁₃	#504	1	4	yes	—
151	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	#518	1	1	yes	characteristic
157	C₄×C₁₃	#568	1	1	yes	characteristic
158	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
159	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
160	#131	—	4	2	no	—
161	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
163	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	—	4	2	no	—
165	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
170	#123	C₈×C₁₃	1	2	yes	—
171	C₂²⋊Q₈	C₈×C₁₃	1	2	yes	—
172	#121	C₈×C₁₃	1	2	yes	—
173	#120	C₈×C₁₃	1	2	yes	—
174	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
175	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
176	M₃₂	—	4	2	no	—
177	C₂×C₁₆	—	4	4	no	—
178	C₂×C₁₆	—	4	2	no	—
179	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
180	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
181	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
182	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
183	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
184	#97	C₂×C₄×C₁₃	1	2	yes	—
185	C₄×C₈	C₂³×C₁₃	1	2	yes	—
186	C₂×C₁₃	—	2	4	no	—
187	C₂×C₁₃	—	4	2	no	—
188	C₂×C₁₃	—	4	1	no	—
189	C₂×C₁₃	#2972	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#504	1	2	yes	—
191	C₂⁴	C₂×C₈×C₁₃	1	1	yes	characteristic
192	#41	—	4	1	no	—
193	#41	—	4	1	no	—
194	C₂×Q₈	C₂×C₈×C₁₃	1	1	yes	characteristic
195	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
196	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
197	C₂×D₄	C₂×C₈×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
201	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
205	M₁₆	—	4	2	no	—
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	1	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
215	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
218	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
219	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
220	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
221	C₂²⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
222	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
223	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
224	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
225	C₁₆	—	2	4	no	—
226	C₁₆	—	2	4	no	—
227	C₁₃	#33491	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	—	2	4	no	—
229	C₂³	—	4	1	no	—
230	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
231	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
232	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
233	Q₈	—	4	1	no	—
234	D₄	—	2	8	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
246	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
247	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
248	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₁₃×#504	1	4	yes	—
263	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	C₁₃×#518	1	1	yes	characteristic
269	C₄	C₁₃×#568	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	2	no	—
273	C₂	C₁₃×#2972	1	1	yes	characteristic
274	C₂	C₂×C₁₃×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₁₃×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2168	C₂	semidirect
3. D₄×C₁₃×C₁₆	C₂	semidirect
4. C₁₃×#2164	C₂	semidirect
5. C₁₃×#2113	C₂	semidirect
6. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
10. C₂×C₁₃×M₃₂	C₂²	semidirect
12. C₂²×C₁₃×C₁₆	C₂²	semidirect
19. C₁₃×C₄⋊C₁₆	C₂²	semidirect
23. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
24. C₁₃×#346	C₂²	semidirect
25. C₄×C₁₃×C₁₆	C₂²	semidirect
54. #33491	C₁₃	direct (N × H)
92. #2168	C₂×C₁₃	semidirect
93. D₄×C₁₆	C₂×C₁₃	semidirect
94. #2164	C₂×C₁₃	semidirect
95. #2113	C₂×C₁₃	semidirect
96. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
127. C₂×M₃₂	C₂²×C₁₃	semidirect
129. C₂²×C₁₆	C₂²×C₁₃	semidirect
136. C₄⋊C₁₆	C₂²×C₁₃	semidirect
140. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
141. #346	C₂²×C₁₃	semidirect
142. C₄×C₁₆	C₂²×C₁₃	semidirect
227. C₁₃	#33491	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×#2929	2
C₁₃×#2168	2
D₄×C₁₃×C₁₆	2
C₁₃×#2164	2
C₁₃×#2113	2
C₂×C₁₃×C₂²⋊C₁₆	2
#33491	13

Subgroups — C13×#33491 #15719830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33491 #15719830