Subgroups — C₁₃×#12681 #15699020

Order 3328.

1038 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#349	C₂	1	4	yes	maximal, complemented
3	C₁₃×#2112	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₃×M₁₆	—	2	8	no	—
6	C₂²×C₁₃×M₁₆	—	2	4	no	—
7	C₂²×C₁₃×M₁₆	C₄	1	2	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	4	yes	complemented
12	C₁₃×#349	—	2	8	no	—
13	C₁₃×C₂²⋊C₁₆	—	2	4	no	—
14	C₂⁵×C₁₃	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	—	2	4	no	—
16	C₂³×C₄×C₁₃	—	2	2	no	—
17	C₂³×C₄×C₁₃	—	2	2	no	—
18	C₂³×C₄×C₁₃	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₈	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×M₁₆	—	2	32	no	—
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂×C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₁₃×M₁₆	D₄	1	8	yes	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	—	2	4	no	—
32	C₂²×C₈×C₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×M₃₂	—	2	8	no	—
38	C₂×C₁₃×C₁₆	—	2	4	no	—
39	#12681	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	—	2	4	no	—
41	C₂⁴×C₁₃	—	2	2	no	—
42	C₂⁴×C₁₃	—	4	4	no	—
43	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂⁴×C₁₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₃	—	2	8	no	—
46	C₂²×C₄×C₁₃	—	2	16	no	—
47	C₂²×C₄×C₁₃	—	2	8	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
64	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₁₃×M₁₆	—	2	16	no	—
67	C₁₃×M₁₆	—	2	16	no	—
68	C₂×C₈×C₁₃	—	2	8	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	8	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₁₃×C₁₆	—	4	4	no	—
78	C₂³×M₁₆	C₂×C₁₃	1	1	yes	characteristic
79	C₂×#349	C₂×C₁₃	1	4	yes	complemented
80	#2112	C₂×C₁₃	1	2	yes	complemented
81	C₂³×C₁₃	—	2	4	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	—	2	8	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	4	16	no	—
87	C₂³×C₁₃	—	4	4	no	—
88	C₂³×C₁₃	—	4	4	no	—
89	C₂³×C₁₃	—	4	4	no	—
90	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
91	C₂³×C₁₃	C₂²⋊C₈	1	4	yes	—
92	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	8	no	—
94	C₂×C₄×C₁₃	—	2	16	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
106	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₈×C₁₃	—	2	4	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	#120	1	4	yes	—
114	C₂⁴×C₄	C₄×C₁₃	1	1	yes	characteristic
115	C₂²×M₁₆	—	2	8	no	—
116	C₂²×M₁₆	—	2	4	no	—
117	C₂²×M₁₆	C₄×C₁₃	1	2	yes	—
118	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
119	C₂³×C₈	C₄×C₁₃	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×M₃₂	C₂²×C₁₃	1	4	yes	complemented
122	#349	—	2	8	no	—
123	C₂²⋊C₁₆	—	2	4	no	—
124	C₂²×C₁₃	—	2	8	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	4	no	—
129	C₂²×C₁₃	—	4	4	no	—
130	C₂²×C₁₃	—	4	16	no	—
131	C₂²×C₁₃	—	4	4	no	—
132	C₂²×C₁₃	—	4	4	no	—
133	C₂²×C₁₃	#349	1	4	yes	complemented
134	C₂²×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
135	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	#379	1	1	yes	characteristic
139	C₄×C₁₃	#407	1	2	yes	—
140	C₄×C₁₃	#407	1	1	yes	characteristic
141	C₂⁵	C₈×C₁₃	1	1	yes	characteristic
142	C₂³×C₄	—	2	4	no	—
143	C₂³×C₄	—	2	2	no	—
144	C₂³×C₄	—	2	2	no	—
145	C₂³×C₄	—	2	4	no	—
146	C₂³×C₄	C₈×C₁₃	1	1	yes	characteristic
147	C₂³×C₄	C₈×C₁₃	1	2	yes	—
148	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
149	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
150	C₂×M₁₆	—	2	32	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	D₄×C₁₃	1	8	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	4	no	—
160	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
161	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₁₃	1	4	yes	—
163	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
164	M₃₂	—	2	8	no	—
165	C₂×C₁₆	—	2	4	no	—
166	C₂×C₁₃	—	2	4	no	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	4	4	no	—
169	C₂×C₁₃	#1779	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#349	1	2	yes	—
171	C₂⁴	—	2	4	no	—
172	C₂⁴	—	2	2	no	—
173	C₂⁴	—	4	4	no	—
174	C₂⁴	C₂×C₈×C₁₃	1	1	yes	characteristic
175	C₂⁴	C₂×C₈×C₁₃	1	2	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	16	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
191	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
192	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
194	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
195	C₂²×C₄	C₁₃×M₁₆	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
197	M₁₆	—	2	16	no	—
198	M₁₆	—	2	16	no	—
199	C₂×C₈	—	2	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
205	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
208	C₁₆	—	4	4	no	—
209	C₁₃	#12681	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³	—	2	2	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	8	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	4	no	—
216	C₂³	—	4	16	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	C₁₃×C₂²⋊C₈	1	2	yes	—
220	C₂³	C₁₃×C₂²⋊C₈	1	4	yes	—
221	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	16	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	C₁₃×C₂²⋊C₈	1	4	yes	—
235	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
236	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
237	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
239	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	2	2	no	—
242	C₈	C₁₃×#120	1	4	yes	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	4	16	no	—
251	C₂²	—	4	4	no	—
252	C₂²	C₁₃×#349	1	4	yes	complemented
253	C₂²	C₂×C₁₃×C₂²⋊C₈	1	2	yes	—
254	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	C₁₃×#379	1	1	yes	characteristic
258	C₄	C₁₃×#407	1	2	yes	—
259	C₄	C₁₃×#407	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	2	4	no	—
262	C₂	—	4	4	no	—
263	C₂	C₁₃×#1779	1	1	yes	characteristic
264	C₂	C₂×C₁₃×#349	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁵×C₁₃	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#349	C₂	semidirect
3. C₁₃×#2112	C₂	semidirect
11. C₂×C₁₃×M₃₂	C₂²	semidirect
39. #12681	C₁₃	direct (N × H)
79. C₂×#349	C₂×C₁₃	semidirect
80. #2112	C₂×C₁₃	semidirect
121. C₂×M₃₂	C₂²×C₁₃	semidirect
133. C₂²×C₁₃	#349	semidirect
209. C₁₃	#12681	direct (N × H)
252. C₂²	C₁₃×#349	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×M₁₆	2
C₂×C₁₃×#349	2
C₁₃×#2112	2
#12681	13

Subgroups — C13×#12681 #15699020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12681 #15699020