Subgroups — C₁₃×#33721 #15720060

Order 3328.

794 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#3156	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2251	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2249	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2247	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#137	—	2	2	no	—
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₁₃×#536	—	2	2	no	—
13	C₁₃×#510	—	2	8	no	—
14	C₁₃×#508	—	2	4	no	—
15	C₂×C₁₃×Q₃₂	—	2	4	no	—
16	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
18	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
19	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
21	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
22	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
24	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
25	C₁₃×#497	—	2	4	no	—
26	C₁₃×#495	C₂²	1	4	yes	—
27	C₁₃×#494	C₂²	1	4	yes	—
28	C₁₃×#405	C₂²	1	1	yes	characteristic
29	C₁₃×#346	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
31	C₂×C₁₃×#41	—	2	2	no	—
32	C₂²×Q₈×C₁₃	—	2	2	no	—
33	C₁₃×#137	—	4	8	no	—
34	C₁₃×#135	—	2	4	no	—
35	C₁₃×#135	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	—	2	4	no	—
37	C₂×C₁₃×Q₁₆	—	2	4	no	—
38	C₂×C₁₃×Q₁₆	—	2	4	no	—
39	C₂×C₁₃×Q₁₆	—	2	2	no	—
40	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
42	C₂×C₁₃×SD₁₆	—	2	4	no	—
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	4	no	—
48	C₁₃×#124	—	2	4	no	—
49	C₁₃×#123	—	4	4	no	—
50	C₁₃×C₂²⋊Q₈	—	4	4	no	—
51	C₁₃×#117	D₄	1	1	yes	characteristic
52	C₁₃×Q₃₂	—	2	4	no	—
53	C₁₃×Q₃₂	—	2	4	no	—
54	C₁₃×Q₃₂	—	2	2	no	—
55	C₁₃×Q₃₂	—	2	8	no	—
56	C₁₃×SD₃₂	—	2	4	no	—
57	C₁₃×SD₃₂	—	2	8	no	—
58	C₁₃×D₁₆	—	2	2	no	—
59	C₁₃×M₃₂	D₄	1	4	yes	—
60	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×#97	D₄	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	2	yes	—
66	#33721	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	4	4	no	—
68	C₁₃×#41	—	4	4	no	—
69	C₂×Q₈×C₁₃	—	2	4	no	—
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	4	2	no	—
72	C₂×Q₈×C₁₃	—	4	4	no	—
73	C₂×Q₈×C₁₃	—	4	4	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂²×C₄×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×Q₁₆	—	2	4	no	—
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×Q₁₆	—	2	4	no	—
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	4	4	no	—
84	C₁₃×Q₁₆	—	4	8	no	—
85	C₁₃×SD₁₆	—	4	4	no	—
86	C₁₃×SD₁₆	—	4	8	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₁₃×D₈	—	2	2	no	—
89	C₁₃×M₁₆	D₈	1	4	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	4	8	no	—
96	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
97	C₁₃×C₂²⋊C₄	—	4	8	no	—
98	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
99	C₄²×C₁₃	C₂×D₄	1	2	yes	—
100	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
101	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
102	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
103	C₂×#510	C₂×C₁₃	1	2	yes	complemented
104	C₂×#508	C₂×C₁₃	1	1	yes	characteristic
105	C₂²×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#3156	C₂×C₁₃	1	2	yes	complemented
107	#2251	C₂×C₁₃	1	4	yes	complemented
108	#2249	C₂×C₁₃	1	2	yes	complemented
109	#2247	C₂×C₁₃	1	2	yes	complemented
110	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₁₃	—	4	2	no	—
114	Q₈×C₁₃	—	4	4	no	—
115	Q₈×C₁₃	—	4	4	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	8	4	no	—
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	8	4	no	—
124	C₂×C₄×C₁₃	—	8	4	no	—
125	C₂×C₄×C₁₃	#127	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
127	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₁₃	#127	1	2	yes	—
131	C₈×C₁₃	C₂×D₈	1	4	yes	—
132	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×#137	—	2	2	no	—
135	C₂×#135	C₂²×C₁₃	1	2	yes	—
136	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
137	#536	—	2	2	no	—
138	#510	—	2	8	no	—
139	#508	—	2	4	no	—
140	C₂×Q₃₂	—	2	4	no	—
141	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
142	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
143	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
145	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
146	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
147	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
150	#497	—	2	4	no	—
151	#495	C₂²×C₁₃	1	4	yes	—
152	#494	C₂²×C₁₃	1	4	yes	—
153	#405	C₂²×C₁₃	1	1	yes	characteristic
154	#346	C₂²×C₁₃	1	2	yes	—
155	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	8	4	no	—
159	C₂²×C₁₃	#493	1	2	yes	—
160	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₁₃	—	2	4	no	—
162	C₄×C₁₃	—	8	2	no	—
163	C₄×C₁₃	—	8	4	no	—
164	C₄×C₁₃	#493	1	2	yes	—
165	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
166	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×#41	—	2	2	no	—
168	C₂²×Q₈	—	2	2	no	—
169	#137	—	4	8	no	—
170	#135	—	2	4	no	—
171	#135	—	2	4	no	—
172	C₂×Q₁₆	—	2	4	no	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×Q₁₆	—	2	4	no	—
176	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
177	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×SD₁₆	—	2	4	no	—
180	C₂×D₈	C₂³×C₁₃	1	2	yes	—
181	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	4	no	—
184	#124	—	2	4	no	—
185	#123	—	4	4	no	—
186	C₂²⋊Q₈	—	4	4	no	—
187	#117	D₄×C₁₃	1	1	yes	characteristic
188	Q₃₂	—	2	4	no	—
189	Q₃₂	—	2	4	no	—
190	Q₃₂	—	2	2	no	—
191	Q₃₂	—	2	8	no	—
192	SD₃₂	—	2	4	no	—
193	SD₃₂	—	2	8	no	—
194	D₁₆	—	2	2	no	—
195	M₃₂	D₄×C₁₃	1	4	yes	—
196	C₂×C₁₆	D₄×C₁₃	1	4	yes	—
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	#97	D₄×C₁₃	1	2	yes	—
201	C₄×C₈	C₂³×C₁₃	1	2	yes	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	16	2	no	—
204	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
205	C₂×C₁₃	#3417	1	2	yes	—
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	2	no	—
220	Q₁₆	—	2	4	no	—
221	Q₁₆	—	2	2	no	—
222	Q₁₆	—	4	4	no	—
223	Q₁₆	—	4	8	no	—
224	SD₁₆	—	4	4	no	—
225	SD₁₆	—	4	8	no	—
226	D₈	—	2	2	no	—
227	D₈	—	2	2	no	—
228	M₁₆	C₁₃×D₈	1	4	yes	—
229	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
231	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
233	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
234	C₄⋊C₄	—	4	8	no	—
235	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
236	C₂²⋊C₄	—	4	8	no	—
237	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
238	C₄²	C₂×D₄×C₁₃	1	2	yes	—
239	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
240	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
241	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
242	C₁₃	#33721	1	1	yes	characteristic, direct product factor, Sylow
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
245	Q₈	—	4	2	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	4	4	no	—
248	Q₈	—	4	2	no	—
249	Q₈	—	8	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₁₃×#127	1	2	yes	—
258	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
259	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
260	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
261	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
262	C₈	C₁₃×#127	1	2	yes	—
263	C₈	C₂×C₁₃×D₈	1	4	yes	—
264	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
265	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	8	4	no	—
269	C₂²	C₁₃×#493	1	2	yes	—
270	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₁₃×#493	1	2	yes	—
275	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
276	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
280	C₂	C₁₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#510	C₂	semidirect
3. C₂²×C₁₃×Q₃₂	C₂	semidirect
4. C₁₃×#3156	C₂	semidirect
5. C₁₃×#2251	C₂	semidirect
6. C₁₃×#2249	C₂	semidirect
7. C₁₃×#2247	C₂	semidirect
8. C₁₃×#2108	C₂	semidirect
66. #33721	C₁₃	direct (N × H)
103. C₂×#510	C₂×C₁₃	semidirect
105. C₂²×Q₃₂	C₂×C₁₃	semidirect
106. #3156	C₂×C₁₃	semidirect
107. #2251	C₂×C₁₃	semidirect
108. #2249	C₂×C₁₃	semidirect
109. #2247	C₂×C₁₃	semidirect
110. #2108	C₂×C₁₃	semidirect
242. C₁₃	#33721	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#510	2
C₂×C₁₃×#508	2
C₂²×C₁₃×Q₃₂	2
C₁₃×#3156	2
C₁₃×#2251	2
C₁₃×#2249	2
C₁₃×#2247	2
C₁₃×#2108	2
#33721	13

Subgroups — C13×#33721 #15720060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33721 #15720060