Subgroups — C₁₃×#33438 #15719777

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#504	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2169	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2168	C₂	1	4	yes	maximal, complemented
5	D₄×C₁₃×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2113	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#131	C₂²	1	2	yes	—
10	C₁₃×#532	C₄	1	1	yes	characteristic
11	C₁₃×#504	—	2	8	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
17	C₁₃×#436	C₄	1	2	yes	—
18	C₁₃×#436	C₄	1	2	yes	—
19	C₈×D₄×C₁₃	C₂²	1	2	yes	—
20	C₈×D₄×C₁₃	C₂²	1	2	yes	—
21	C₁₃×#423	C₄	1	1	yes	characteristic
22	C₁₃×#408	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₁₃×#346	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
31	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
32	C₁₃×#131	—	2	4	no	—
33	C₁₃×#131	—	2	4	no	—
34	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
40	C₁₃×#127	C₈	1	1	yes	characteristic
41	C₁₃×#124	C₈	1	2	yes	—
42	C₁₃×#121	C₈	1	2	yes	—
43	C₁₃×#121	C₈	1	2	yes	—
44	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₁₃×M₃₂	—	2	2	no	—
48	C₁₃×M₃₂	—	2	8	no	—
49	C₁₃×M₃₂	D₄	1	4	yes	complemented
50	C₂×C₁₃×C₁₆	—	2	4	no	—
51	C₂×C₁₃×C₁₆	—	2	8	no	—
52	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
53	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
58	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#33438	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₁₃×#41	—	2	4	no	—
66	C₁₃×#41	—	2	4	no	—
67	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
70	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
71	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	2	8	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
89	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
90	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
91	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
92	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₁₃×C₁₆	—	2	2	no	—
97	C₁₃×C₁₆	—	2	2	no	—
98	C₁₃×C₁₆	—	2	2	no	—
99	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
100	C₂×#504	C₂×C₁₃	1	2	yes	complemented
101	#2931	C₂×C₁₃	1	1	yes	characteristic
102	#2169	C₂×C₁₃	1	2	yes	complemented
103	#2168	C₂×C₁₃	1	4	yes	complemented
104	D₄×C₁₆	C₂×C₁₃	1	2	yes	complemented
105	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#2113	C₂×C₁₃	1	2	yes	complemented
107	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
109	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	Q₈×C₁₃	—	2	4	no	—
111	D₄×C₁₃	—	2	8	no	—
112	D₄×C₁₃	—	2	4	no	—
113	D₄×C₁₃	—	2	4	no	—
114	D₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#131	1	2	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₈×C₁₃	—	2	4	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	2	1	no	—
131	C₈×C₁₃	C₄×D₄	1	2	yes	—
132	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×#131	C₂²×C₁₃	1	2	yes	—
135	#532	C₄×C₁₃	1	1	yes	characteristic
136	#504	—	2	8	no	—
137	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
139	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
141	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
142	#436	C₄×C₁₃	1	2	yes	—
143	#436	C₄×C₁₃	1	2	yes	—
144	C₈×D₄	C₂²×C₁₃	1	2	yes	—
145	C₈×D₄	C₂²×C₁₃	1	2	yes	—
146	#423	C₄×C₁₃	1	1	yes	characteristic
147	#408	C₄×C₁₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
150	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
151	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
152	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
153	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
154	#346	C₂²×C₁₃	1	2	yes	—
155	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	4	no	—
159	C₂²×C₁₃	—	2	4	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	C₈×D₄	1	2	yes	—
162	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	4	no	—
167	C₄×C₁₃	C₈×D₄	1	2	yes	—
168	C₄×C₁₃	#504	1	2	yes	complemented
169	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
171	C₂×#41	C₂×C₄×C₁₃	1	2	yes	—
172	#131	—	2	4	no	—
173	#131	—	2	4	no	—
174	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
175	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
176	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
178	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
179	C₄⋊Q₈	C₈×C₁₃	1	1	yes	characteristic
180	#127	C₈×C₁₃	1	1	yes	characteristic
181	#124	C₈×C₁₃	1	2	yes	—
182	#121	C₈×C₁₃	1	2	yes	—
183	#121	C₈×C₁₃	1	2	yes	—
184	C₄×D₄	C₂×C₄×C₁₃	1	2	yes	—
185	C₄×D₄	C₂×C₄×C₁₃	1	2	yes	—
186	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
187	M₃₂	—	2	2	no	—
188	M₃₂	—	2	8	no	—
189	M₃₂	D₄×C₁₃	1	4	yes	complemented
190	C₂×C₁₆	—	2	4	no	—
191	C₂×C₁₆	—	2	8	no	—
192	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
193	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
194	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
195	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
196	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
197	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
198	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
199	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
200	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
201	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	4	4	no	—
206	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
208	C₂×C₁₃	#3407	1	1	yes	characteristic
209	#41	—	2	4	no	—
210	#41	—	2	4	no	—
211	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
212	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
213	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
214	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
215	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
216	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
228	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
230	C₂×C₈	C₁₃×#41	1	2	yes	—
231	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
232	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
233	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
234	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
235	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
236	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
237	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
238	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
239	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
244	C₁₃	#33438	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	C₂²×C₈×C₁₃	1	4	yes	—
246	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	D₄	—	2	8	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
257	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
261	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
263	C₂×C₄	C₁₃×#131	1	2	yes	—
264	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	C₄×D₄×C₁₃	1	2	yes	—
269	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
270	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	C₈×D₄×C₁₃	1	2	yes	—
277	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	4	no	—
282	C₄	C₈×D₄×C₁₃	1	2	yes	—
283	C₄	C₁₃×#504	1	2	yes	complemented
284	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
285	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	4	4	no	—
288	C₂	C₂×C₈×D₄×C₁₃	1	1	yes	characteristic
289	C₂	C₂×C₁₃×#504	1	1	yes	characteristic
290	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	C₁₃×#532	C₄
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
1. C₂×C₁₃×#504	C₂	semidirect
3. C₁₃×#2169	C₂	semidirect
4. C₁₃×#2168	C₂	semidirect
5. D₄×C₁₃×C₁₆	C₂	semidirect
6. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
7. C₁₃×#2113	C₂	semidirect
8. C₄×C₁₃×M₃₂	C₂	semidirect
49. C₁₃×M₃₂	D₄	semidirect
64. #33438	C₁₃	direct (N × H)
100. C₂×#504	C₂×C₁₃	semidirect
102. #2169	C₂×C₁₃	semidirect
103. #2168	C₂×C₁₃	semidirect
104. D₄×C₁₆	C₂×C₁₃	semidirect
105. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
106. #2113	C₂×C₁₃	semidirect
107. C₄×M₃₂	C₂×C₁₃	semidirect
168. C₄×C₁₃	#504	semidirect
189. M₃₂	D₄×C₁₃	semidirect
244. C₁₃	#33438	direct (N × H)
283. C₄	C₁₃×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#504	2
C₁₃×#2931	2
C₁₃×#2169	2
C₁₃×#2168	2
D₄×C₁₃×C₁₆	2
C₂×C₁₃×C₄⋊C₁₆	2
C₁₃×#2113	2
C₄×C₁₃×M₃₂	2
#33438	13

Subgroups — C13×#33438 #15719777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33438 #15719777