Subgroups — C₁₃×#33487 #15719826

Order 3328.

618 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2168	C₂	1	8	yes	maximal, complemented
3	C₁₃×#2163	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2112	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	4	yes	complemented
10	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	complemented
11	C₈×D₄×C₁₃	C₄	1	4	yes	—
12	C₈×D₄×C₁₃	C₂²	1	4	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
18	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
19	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
20	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
21	C₁₃×#346	C₂²	1	4	yes	complemented
22	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	4	2	no	—
26	C₂²×C₈×C₁₃	—	4	2	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₄×D₄×C₁₃	C₈	1	4	yes	—
36	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
38	C₂×C₁₃×C₂²⋊C₄	C₈	1	1	yes	characteristic
39	C₂×C₁₃×C₂²⋊C₄	C₈	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₁₃×M₃₂	—	4	4	no	—
42	C₂×C₁₃×C₁₆	—	4	4	no	—
43	C₂×C₁₃×C₁₆	C₂³	1	8	yes	—
44	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	#33487	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂×D₄×C₁₃	—	2	4	no	—
53	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	—	4	1	no	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	1	no	—
73	C₂×C₈×C₁₃	—	4	1	no	—
74	C₂×C₈×C₁₃	—	4	1	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
81	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
82	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
83	C₄²×C₁₃	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
85	C₁₃×C₁₆	—	2	8	no	—
86	C₂×C₈×D₄	C₂×C₁₃	1	1	yes	characteristic
87	#2168	C₂×C₁₃	1	8	yes	complemented
88	#2163	C₂×C₁₃	1	2	yes	complemented
89	#2112	C₂×C₁₃	1	2	yes	complemented
90	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	2	yes	complemented
91	C₂³×C₁₃	—	2	2	no	—
92	C₂³×C₁₃	—	4	1	no	—
93	C₂³×C₁₃	—	4	1	no	—
94	C₂³×C₁₃	—	4	1	no	—
95	C₂³×C₁₃	—	4	1	no	—
96	C₂³×C₁₃	M₃₂	1	4	yes	complemented
97	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
98	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
99	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
100	D₄×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	2	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	2	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	—	2	2	no	—
123	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
124	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×C₄×D₄	C₄×C₁₃	1	1	yes	characteristic
127	C₂×M₃₂	C₂²×C₁₃	1	4	yes	complemented
128	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	complemented
129	C₈×D₄	C₄×C₁₃	1	4	yes	—
130	C₈×D₄	C₂²×C₁₃	1	4	yes	—
131	C₂×C₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
132	C₂×C₂²⋊C₈	C₄×C₁₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₈	C₄×C₁₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
136	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
137	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
138	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
139	#346	C₂²×C₁₃	1	4	yes	complemented
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	4	4	no	—
148	C₂²×C₁₃	—	4	4	no	—
149	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
150	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
151	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	#518	1	1	yes	characteristic
158	C₄×C₁₃	#568	1	1	yes	characteristic
159	C₂²×D₄	C₈×C₁₃	1	1	yes	characteristic
160	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	4	2	no	—
163	C₂²×C₈	—	4	2	no	—
164	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
168	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
171	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
172	C₄×D₄	C₈×C₁₃	1	4	yes	—
173	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
174	C₂×C₄⋊C₄	C₈×C₁₃	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₈×C₁₃	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	C₈×C₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
178	M₃₂	—	4	4	no	—
179	C₂×C₁₆	—	4	4	no	—
180	C₂×C₁₆	C₂³×C₁₃	1	8	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	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
185	C₄×C₈	C₂³×C₁₃	1	2	yes	—
186	C₂×C₁₃	—	2	2	no	—
187	C₂×C₁₃	—	2	4	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	#2972	1	1	yes	characteristic
190	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
191	C₂×C₁₃	#3407	1	1	yes	characteristic
192	C₂⁴	C₂×C₈×C₁₃	1	1	yes	characteristic
193	C₂⁴	C₂×C₈×C₁₃	1	1	yes	characteristic
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
205	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
221	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
222	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
223	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
224	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
225	C₄²	C₂×C₈×C₁₃	1	2	yes	—
226	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
227	C₁₆	—	2	8	no	—
228	C₁₃	#33487	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	2	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	1	no	—
232	C₂³	—	4	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	C₁₃×M₃₂	1	4	yes	complemented
235	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
236	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
237	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
238	D₄	—	2	8	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	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
253	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
254	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
255	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
256	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×C₁₃×M₃₂	1	2	yes	—
272	C₂²	C₂×C₁₃×M₃₂	1	4	yes	—
273	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	C₁₃×#518	1	1	yes	characteristic
280	C₄	C₁₃×#568	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	—	4	2	no	—
284	C₂	C₁₃×#2972	1	1	yes	characteristic
285	C₂	C₂²×C₁₃×M₃₂	1	1	yes	characteristic
286	C₂	C₁₃×#3407	1	1	yes	characteristic

	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	C₂²×D₄×C₁₃	C₈
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. C₁₃×#2163	C₂	semidirect
4. C₁₃×#2112	C₂	semidirect
5. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
9. C₂×C₁₃×M₃₂	C₂²	semidirect
10. C₂²×C₁₃×C₁₆	C₂²	semidirect
17. C₁₃×C₄⋊C₁₆	C₂²	semidirect
19. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
20. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
21. C₁₃×#346	C₂²	semidirect
49. #33487	C₁₃	direct (N × H)
87. #2168	C₂×C₁₃	semidirect
88. #2163	C₂×C₁₃	semidirect
89. #2112	C₂×C₁₃	semidirect
90. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
96. C₂³×C₁₃	M₃₂	semidirect
127. C₂×M₃₂	C₂²×C₁₃	semidirect
128. C₂²×C₁₆	C₂²×C₁₃	semidirect
135. C₄⋊C₁₆	C₂²×C₁₃	semidirect
137. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
138. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
139. #346	C₂²×C₁₃	semidirect
228. C₁₃	#33487	direct (N × H)
234. C₂³	C₁₃×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₃	2
C₁₃×#2168	2
C₁₃×#2163	2
C₁₃×#2112	2
C₂×C₁₃×C₂²⋊C₁₆	2
#33487	13

Subgroups — C13×#33487 #15719826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33487 #15719826