Subgroups — C₁₃×#33816 #15720155

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3168	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2229	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2211	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2210	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#529	—	2	2	no	—
11	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#499	C₂²	1	2	yes	—
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₁₃×#488	—	2	2	no	—
19	C₁₃×#467	C₂²	1	2	yes	—
20	C₁₃×#466	C₂²	1	2	yes	—
21	C₁₃×#465	—	2	2	no	—
22	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
23	C₄×C₁₃×SD₁₆	—	2	2	no	—
24	C₄×C₁₃×D₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#367	C₂²	1	2	yes	—
27	C₁₃×#366	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
36	C₂×C₁₃×SD₁₆	—	2	2	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×#126	—	4	2	no	—
41	C₁₃×#125	—	2	2	no	—
42	C₁₃×#124	—	2	2	no	—
43	C₁₃×#123	—	4	2	no	—
44	C₁₃×C₂²⋊Q₈	—	2	2	no	—
45	C₁₃×#121	—	2	2	no	—
46	C₄×Q₈×C₁₃	—	2	2	no	—
47	C₄×D₄×C₁₃	—	2	2	no	—
48	C₄×D₄×C₁₃	—	4	2	no	—
49	C₁₃×#117	—	4	2	no	—
50	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×Q₃₂	—	4	1	no	—
52	C₁₃×SD₃₂	—	4	2	no	—
53	C₁₃×D₁₆	—	4	1	no	—
54	C₂×C₁₃×C₁₆	—	4	1	no	—
55	C₂×C₁₃×C₁₆	—	4	1	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#107	C₂³	1	2	yes	—
61	C₁₃×#106	—	2	2	no	—
62	C₁₃×#103	—	2	2	no	—
63	C₁₃×#103	—	2	2	no	—
64	C₁₃×#102	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₄×C₈×C₁₃	D₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#33816	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×Q₁₆	—	2	2	no	—
77	C₁₃×Q₁₆	—	2	2	no	—
78	C₁₃×SD₁₆	—	4	4	no	—
79	C₁₃×D₈	—	2	2	no	—
80	C₁₃×D₈	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₂²⋊C₄	—	4	2	no	—
94	C₁₃×C₂²⋊C₄	—	4	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₁₃×C₂²⋊C₄	—	4	2	no	—
98	C₄²×C₁₃	—	4	2	no	—
99	C₄²×C₁₃	D₈	1	2	yes	—
100	C₄²×C₁₃	C₂×D₄	1	2	yes	—
101	C₁₃×C₁₆	—	2	1	no	—
102	C₁₃×C₁₆	—	2	2	no	—
103	C₁₃×C₁₆	—	2	1	no	—
104	#3168	C₂×C₁₃	1	2	yes	complemented
105	#2229	C₂×C₁₃	1	2	yes	complemented
106	#2228	C₂×C₁₃	1	2	yes	complemented
107	#2217	C₂×C₁₃	1	2	yes	complemented
108	#2216	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#2215	C₂×C₁₃	1	1	yes	characteristic
110	#2211	C₂×C₁₃	1	2	yes	complemented
111	#2210	C₂×C₁₃	1	2	yes	complemented
112	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	—	4	2	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	8	4	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
136	#529	—	2	2	no	—
137	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
139	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
141	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
142	#499	C₂²×C₁₃	1	2	yes	—
143	#495	C₂²×C₁₃	1	2	yes	—
144	#488	—	2	2	no	—
145	#467	C₂²×C₁₃	1	2	yes	—
146	#466	C₂²×C₁₃	1	2	yes	—
147	#465	—	2	2	no	—
148	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
149	C₄×SD₁₆	—	2	2	no	—
150	C₄×D₈	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
152	#367	C₂²×C₁₃	1	2	yes	—
153	#366	C₂²×C₁₃	1	1	yes	characteristic
154	#366	C₂²×C₁₃	1	1	yes	characteristic
155	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
156	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
157	#358	C₂²×C₁₃	1	2	yes	—
158	#358	C₂²×C₁₃	1	2	yes	—
159	#357	C₂²×C₁₃	1	2	yes	—
160	#357	C₂²×C₁₃	1	2	yes	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	8	2	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	#508	1	2	yes	—
167	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	8	2	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	#537	1	1	yes	characteristic
175	C₄×C₁₃	#575	1	1	yes	characteristic
176	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
177	C₂×SD₁₆	—	2	2	no	—
178	C₂×D₈	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	#126	—	4	2	no	—
182	#125	—	2	2	no	—
183	#124	—	2	2	no	—
184	#123	—	4	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	#121	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	—	4	2	no	—
190	#117	—	4	2	no	—
191	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
192	Q₃₂	—	4	1	no	—
193	SD₃₂	—	4	2	no	—
194	D₁₆	—	4	1	no	—
195	C₂×C₁₆	—	4	1	no	—
196	C₂×C₁₆	—	4	1	no	—
197	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
198	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
199	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
200	#107	C₂³×C₁₃	1	2	yes	—
201	#107	C₂³×C₁₃	1	2	yes	—
202	#106	—	2	2	no	—
203	#103	—	2	2	no	—
204	#103	—	2	2	no	—
205	#102	—	2	2	no	—
206	#102	—	2	2	no	—
207	C₄×C₈	D₄×C₁₃	1	2	yes	—
208	C₄×C₈	C₂³×C₁₃	1	2	yes	—
209	C₂×C₁₃	—	2	2	no	—
210	C₂×C₁₃	—	16	2	no	—
211	C₂×C₁₃	#3245	1	1	yes	characteristic
212	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
213	C₂×C₁₃	#3414	1	1	yes	characteristic
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
219	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
221	Q₁₆	—	2	2	no	—
222	Q₁₆	—	2	2	no	—
223	SD₁₆	—	4	4	no	—
224	D₈	—	2	2	no	—
225	D₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
231	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	C₁₃×D₈	1	2	yes	—
245	C₄²	C₂×D₄×C₁₃	1	2	yes	—
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	2	1	no	—
249	C₁₃	#33816	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	4	2	no	—
251	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
266	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
267	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	2	no	—
271	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
272	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	C₁₃×#508	1	2	yes	—
279	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₁₃×#537	1	1	yes	characteristic
287	C₄	C₁₃×#575	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	C₁₃×#3245	1	1	yes	characteristic
291	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
292	C₂	C₁₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#3168	C₂	semidirect
2. C₁₃×#2229	C₂	semidirect
3. C₁₃×#2228	C₂	semidirect
4. C₁₃×#2217	C₂	semidirect
5. C₁₃×#2216	C₂	semidirect
7. C₁₃×#2211	C₂	semidirect
8. C₁₃×#2210	C₂	semidirect
9. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
68. #33816	C₁₃	direct (N × H)
104. #3168	C₂×C₁₃	semidirect
105. #2229	C₂×C₁₃	semidirect
106. #2228	C₂×C₁₃	semidirect
107. #2217	C₂×C₁₃	semidirect
108. #2216	C₂×C₁₃	semidirect
110. #2211	C₂×C₁₃	semidirect
111. #2210	C₂×C₁₃	semidirect
112. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
249. C₁₃	#33816	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×#3168	2
C₁₃×#2229	2
C₁₃×#2228	2
C₁₃×#2217	2
C₁₃×#2216	2
C₁₃×#2215	2
C₁₃×#2211	2
C₁₃×#2210	2
C₂×C₁₃×C₄⋊C₁₆	2
#33816	13

Subgroups — C13×#33816 #15720155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33816 #15720155