Subgroups — C₁₃×#12848 #15699187

Order 3328.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1379	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₁₃×#405	—	2	1	no	—
11	C₁₃×#405	—	2	1	no	—
12	C₁₃×#405	C₄	1	2	yes	—
13	C₁₃×#405	C₂²	1	2	yes	—
14	C₂×C₁₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
19	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	1	no	—
22	C₂×C₁₃×M₁₆	—	2	4	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	—	2	1	no	—
25	C₂²×C₈×C₁₃	—	2	4	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#117	—	2	1	no	—
33	C₁₃×#117	—	2	1	no	—
34	C₁₃×#117	C₈	1	2	yes	—
35	C₁₃×#117	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
38	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₁₃	—	2	1	no	—
40	C₁₃×M₃₂	—	2	4	no	—
41	C₂×C₁₃×C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₂×C₁₃×C₁₆	—	2	4	no	—
44	C₁₃×#97	—	2	2	no	—
45	C₁₃×#97	—	2	2	no	—
46	C₄×C₈×C₁₃	—	2	2	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	#12848	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₁₃×M₁₆	—	2	4	no	—
61	C₁₃×M₁₆	—	2	2	no	—
62	C₁₃×M₁₆	M₁₆	1	4	yes	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	#41	1	4	yes	—
75	C₁₃×C₄⋊C₄	—	2	1	no	—
76	C₁₃×C₄⋊C₄	—	2	1	no	—
77	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
78	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₁₃×C₂²⋊C₄	—	2	2	no	—
80	C₁₃×C₂²⋊C₄	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₁₃×C₁₆	—	4	2	no	—
84	C₁₃×C₁₆	—	4	2	no	—
85	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
86	#2112	C₂×C₁₃	1	1	yes	characteristic
87	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
88	#1379	C₂×C₁₃	1	4	yes	complemented
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	—	4	1	no	—
92	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
93	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
94	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	4	no	—
115	C₈×C₁₃	#123	1	2	yes	—
116	C₈×C₁₃	#123	1	2	yes	—
117	C₂²×M₁₆	C₄×C₁₃	1	1	yes	characteristic
118	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
119	C₂×#117	C₄×C₁₃	1	1	yes	characteristic
120	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
121	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
122	#405	—	2	1	no	—
123	#405	—	2	1	no	—
124	#405	C₄×C₁₃	1	2	yes	—
125	#405	C₂²×C₁₃	1	2	yes	—
126	C₂×#97	—	2	1	no	—
127	C₂×C₄×C₈	—	2	1	no	—
128	C₂²⋊C₁₆	—	2	1	no	—
129	C₂²⋊C₁₆	—	2	1	no	—
130	C₂²⋊C₁₆	—	2	1	no	—
131	C₂²⋊C₁₆	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	4	4	no	—
140	C₂²×C₁₃	#350	1	2	yes	—
141	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₂²×C₁₃	#432	1	2	yes	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	4	2	no	—
146	C₄×C₁₃	—	4	2	no	—
147	C₄×C₁₃	#381	1	1	yes	characteristic
148	C₄×C₁₃	#407	1	1	yes	characteristic
149	C₄×C₁₃	#431	1	2	yes	—
150	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
157	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
162	#117	—	2	1	no	—
163	#117	—	2	1	no	—
164	#117	C₈×C₁₃	1	2	yes	—
165	#117	C₂×C₄×C₁₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₈×C₁₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₈×C₁₃	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	—	2	1	no	—
169	C₂×C₄²	—	2	1	no	—
170	M₃₂	—	2	4	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	4	no	—
174	#97	—	2	2	no	—
175	#97	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₂×C₁₃	—	2	2	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	—	4	1	no	—
181	C₂×C₁₃	#1817	1	1	yes	characteristic
182	C₂×C₁₃	C₂×#350	1	1	yes	characteristic
183	C₂×C₁₃	#2117	1	1	yes	characteristic
184	C₂⁴	C₁₃×M₁₆	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	2	no	—
197	M₁₆	C₁₃×M₁₆	1	4	yes	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₈	C₁₃×#41	1	4	yes	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
213	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₁₆	—	4	2	no	—
219	C₁₆	—	4	2	no	—
220	C₁₃	#12848	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
225	C₂³	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
226	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
239	C₂×C₄	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₁₃×#117	1	2	yes	—
242	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	4	no	—
247	C₈	C₁₃×#123	1	2	yes	—
248	C₈	C₁₃×#123	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₁₃×#350	1	2	yes	—
258	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
259	C₂²	C₁₃×#432	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	C₁₃×#381	1	1	yes	characteristic
265	C₄	C₁₃×#407	1	1	yes	characteristic
266	C₄	C₁₃×#431	1	2	yes	—
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	C₁₃×#1817	1	1	yes	characteristic
271	C₂	C₂×C₁₃×#350	1	1	yes	characteristic
272	C₂	C₁₃×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1379	C₂	semidirect
48. #12848	C₁₃	direct (N × H)
88. #1379	C₂×C₁₃	semidirect
220. C₁₃	#12848	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×#405	2
C₁₃×#2112	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₁₃×#1379	2
#12848	13

Subgroups — C13×#12848 #15699187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12848 #15699187