Subgroups — C₁₃×#8392 #15694731

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#557	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1801	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1548	C₂	1	2	yes	maximal
4	C₁₃×#1487	C₂	1	2	yes	maximal
5	C₁₃×#1461	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#557	—	2	4	no	—
7	C₁₃×#557	—	2	4	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#125	—	2	2	no	—
11	C₂×C₁₃×#125	—	2	1	no	—
12	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#431	C₂²	1	2	yes	—
17	C₁₃×#423	C₂²	1	2	yes	—
18	C₂×C₁₃×#101	—	2	1	no	—
19	C₁₃×#343	—	2	1	no	—
20	C₁₃×#333	—	2	4	no	—
21	C₁₃×Q₈⋊C₈	—	2	4	no	—
22	C₁₃×#324	—	2	2	no	—
23	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	2	4	no	—
28	C₁₃×C₄⋊Q₈	—	2	4	no	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
31	C₁₃×#125	—	2	16	no	—
32	C₁₃×#125	—	2	8	no	—
33	C₄×Q₈×C₁₃	—	2	8	no	—
34	C₄×Q₈×C₁₃	—	2	4	no	—
35	C₄×Q₈×C₁₃	—	2	4	no	—
36	C₄×Q₈×C₁₃	—	2	2	no	—
37	C₄×Q₈×C₁₃	—	2	2	no	—
38	C₄×Q₈×C₁₃	—	2	2	no	—
39	C₄×Q₈×C₁₃	D₄	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	—	2	1	no	—
51	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₈	—	2	2	no	—
54	C₁₃×C₄⋊C₈	—	2	2	no	—
55	C₁₃×C₄⋊C₈	—	2	2	no	—
56	C₁₃×#101	—	4	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	2	no	—
58	C₄×C₈×C₁₃	—	2	2	no	—
59	#8392	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₃	—	2	1	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	4	2	no	—
65	C₂×Q₈×C₁₃	—	4	4	no	—
66	C₂×Q₈×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂×Q₈×C₁₃	C₂²⋊C₄	1	4	yes	—
68	C₂×Q₈×C₁₃	SD₁₆	1	2	yes	—
69	C₂×Q₈×C₁₃	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₁₃×M₁₆	—	4	2	no	—
79	C₁₃×M₁₆	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	2	8	no	—
83	C₁₃×C₄⋊C₄	—	2	8	no	—
84	C₁₃×C₄⋊C₄	—	2	8	no	—
85	C₁₃×C₄⋊C₄	—	2	1	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₁₃	—	2	8	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	4	no	—
96	C₄²×C₁₃	C₂×D₄	1	2	yes	—
97	C₄²×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×#557	C₂×C₁₃	1	1	yes	characteristic
99	#1801	C₂×C₁₃	1	1	yes	characteristic
100	#1548	C₂×C₁₃	1	2	yes	—
101	#1487	C₂×C₁₃	1	2	yes	—
102	#1461	C₂×C₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₁₃	—	2	4	no	—
105	Q₈×C₁₃	—	2	4	no	—
106	Q₈×C₁₃	—	2	4	no	—
107	Q₈×C₁₃	—	4	2	no	—
108	Q₈×C₁₃	—	4	4	no	—
109	Q₈×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	#103	1	4	yes	—
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#120	1	2	yes	—
126	C₂×C₄×C₁₃	#120	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#137	1	2	yes	—
130	C₈×C₁₃	—	8	2	no	—
131	C₈×C₁₃	—	8	2	no	—
132	#557	—	2	4	no	—
133	#557	—	2	4	no	—
134	C₂×C₄⋊Q₈	C₄×C₁₃	1	2	yes	—
135	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#125	—	2	2	no	—
137	C₂×#125	—	2	1	no	—
138	C₂×C₄×Q₈	—	2	1	no	—
139	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
140	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
142	#431	C₂²×C₁₃	1	2	yes	—
143	#423	C₂²×C₁₃	1	2	yes	—
144	C₂×#101	—	2	1	no	—
145	#343	—	2	1	no	—
146	#333	—	2	4	no	—
147	Q₈⋊C₈	—	2	4	no	—
148	#324	—	2	2	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	#379	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
153	C₂²×C₁₃	#419	1	1	yes	characteristic
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	4	no	—
157	C₄×C₁₃	—	4	2	no	—
158	C₄×C₁₃	—	4	2	no	—
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	—	4	4	no	—
161	C₄×C₁₃	#413	1	2	yes	—
162	C₄×C₁₃	#448	1	2	yes	—
163	C₄×C₁₃	#451	1	2	yes	—
164	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
165	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₄⋊Q₈	—	2	4	no	—
169	C₄⋊Q₈	—	2	4	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	D₄×C₁₃	1	4	yes	—
172	#125	—	2	16	no	—
173	#125	—	2	8	no	—
174	C₄×Q₈	—	2	8	no	—
175	C₄×Q₈	—	2	4	no	—
176	C₄×Q₈	—	2	4	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×Q₈	—	2	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	D₄×C₁₃	1	4	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
188	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
193	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	#101	—	4	2	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	#1785	1	1	yes	characteristic
203	C₂×C₁₃	#1787	1	1	yes	characteristic
204	C₂×C₁₃	#1796	1	1	yes	characteristic
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	C₁₃×C₂²⋊C₄	1	2	yes	—
212	C₂×Q₈	C₁₃×C₂²⋊C₄	1	4	yes	—
213	C₂×Q₈	C₁₃×SD₁₆	1	2	yes	—
214	C₂×Q₈	C₁₃×Q₁₆	1	2	yes	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
219	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	4	no	—
241	C₄²	C₂×D₄×C₁₃	1	2	yes	—
242	C₄²	C₂×D₄×C₁₃	1	2	yes	—
243	C₁₃	#8392	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	4	2	no	—
250	Q₈	—	4	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₁₃×#103	1	4	yes	—
264	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₁₃×#120	1	2	yes	—
267	C₂×C₄	C₁₃×#120	1	2	yes	—
268	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
270	C₂×C₄	C₁₃×#137	1	2	yes	—
271	C₈	—	8	2	no	—
272	C₈	—	8	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	C₁₃×#379	1	1	yes	characteristic
276	C₂²	C₂×C₁₃×#103	1	1	yes	characteristic
277	C₂²	C₁₃×#419	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	C₁₃×#413	1	2	yes	—
286	C₄	C₁₃×#448	1	2	yes	—
287	C₄	C₁₃×#451	1	2	yes	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	C₁₃×#1785	1	1	yes	characteristic
291	C₂	C₁₃×#1787	1	1	yes	characteristic
292	C₂	C₁₃×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
59. #8392	C₁₃	direct (N × H)
243. C₁₃	#8392	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×#557	2
C₁₃×#1801	2
C₁₃×#1548	2
C₁₃×#1487	2
C₁₃×#1461	2
#8392	13

Subgroups — C13×#8392 #15694731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8392 #15694731