Subgroups — C₁₃×#7707 #15694046

Order 3328.

500 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1817	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1749	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1272	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
9	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
10	C₂×C₁₃×#97	—	2	1	no	—
11	C₁₃×#391	C₂²	1	2	yes	—
12	C₁₃×#389	—	2	1	no	—
13	C₁₃×#389	—	2	1	no	—
14	C₁₃×#384	C₄	1	1	yes	characteristic
15	C₁₃×#380	—	2	1	no	—
16	C₁₃×#380	—	2	1	no	—
17	C₁₃×#336	—	2	1	no	—
18	C₁₃×#336	C₂²	1	2	yes	—
19	C₁₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	4	2	no	—
27	C₁₃×C₂²⋊Q₈	—	4	4	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₄²×C₁₃	—	4	2	no	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₂²⋊C₈	—	2	4	no	—
41	C₁₃×C₂²⋊C₈	—	4	2	no	—
42	C₁₃×#97	—	4	2	no	—
43	C₁₃×#95	—	2	2	no	—
44	C₁₃×#95	—	2	2	no	—
45	#7707	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Q₈×C₁₃	—	2	4	no	—
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	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	—	4	1	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₂²⋊C₄	—	4	4	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	4	1	no	—
83	C₄²×C₁₃	—	4	1	no	—
84	#2857	C₂×C₁₃	1	1	yes	characteristic
85	#1817	C₂×C₁₃	1	1	yes	characteristic
86	#1749	C₂×C₁₃	1	1	yes	characteristic
87	#1272	C₂×C₁₃	1	4	yes	complemented
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	—	2	1	no	—
90	C₂³×C₁₃	—	2	1	no	—
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	2	1	no	—
94	C₂³×C₁₃	#100	1	2	yes	—
95	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₁₃	—	8	4	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	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	#99	1	2	yes	—
120	C₂×C₄×C₁₃	#104	1	2	yes	—
121	C₂×C₄×C₁₃	#131	1	2	yes	—
122	C₂×C₄×C₁₃	#137	1	2	yes	—
123	C₂×C₄×C₁₃	#137	1	2	yes	—
124	C₈×C₁₃	—	4	4	no	—
125	C₈×C₁₃	—	8	2	no	—
126	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
127	C₂×C₂²⋊Q₈	C₄×C₁₃	1	2	yes	—
128	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₈	—	2	1	no	—
130	C₂×C₂²⋊C₈	—	2	1	no	—
131	C₂×#97	—	2	1	no	—
132	#391	C₂²×C₁₃	1	2	yes	—
133	#389	—	2	1	no	—
134	#389	—	2	1	no	—
135	#384	C₄×C₁₃	1	1	yes	characteristic
136	#380	—	2	1	no	—
137	#380	—	2	1	no	—
138	#336	—	2	1	no	—
139	#336	C₂²×C₁₃	1	2	yes	—
140	#336	C₂²×C₁₃	1	2	yes	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	#408	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#99	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
151	C₂²×C₁₃	#419	1	1	yes	characteristic
152	C₂²×C₁₃	#419	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
154	C₂²×C₁₃	#421	1	1	yes	characteristic
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	4	1	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	—	8	2	no	—
163	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
164	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
166	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₄⋊Q₈	—	4	2	no	—
170	C₂²⋊Q₈	—	4	4	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	—	4	2	no	—
179	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
183	C₂²⋊C₈	—	2	4	no	—
184	C₂²⋊C₈	—	4	2	no	—
185	#97	—	4	2	no	—
186	#95	—	2	2	no	—
187	#95	—	2	2	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	#1471	1	1	yes	characteristic
190	C₂×C₁₃	#1485	1	1	yes	characteristic
191	C₂×C₁₃	#1497	1	1	yes	characteristic
192	C₂×C₁₃	#1508	1	1	yes	characteristic
193	C₂×C₁₃	#1511	1	1	yes	characteristic
194	C₂×C₁₃	#1530	1	1	yes	characteristic
195	C₂×C₁₃	#1558	1	1	yes	characteristic
196	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₁₃	#7707	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	C₁₃×#100	1	2	yes	—
242	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
243	Q₈	—	8	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	C₁₃×#99	1	2	yes	—
267	C₂×C₄	C₁₃×#104	1	2	yes	—
268	C₂×C₄	C₁₃×#131	1	2	yes	—
269	C₂×C₄	C₁₃×#137	1	2	yes	—
270	C₂×C₄	C₁₃×#137	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₈	—	8	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	C₁₃×#408	1	1	yes	characteristic
281	C₂²	C₂×C₁₃×#99	1	1	yes	characteristic
282	C₂²	C₂×C₁₃×#100	1	1	yes	characteristic
283	C₂²	C₁₃×#419	1	1	yes	characteristic
284	C₂²	C₁₃×#419	1	1	yes	characteristic
285	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
286	C₂²	C₁₃×#421	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₂	—	4	2	no	—
296	C₂	C₁₃×#1471	1	1	yes	characteristic
297	C₂	C₁₃×#1485	1	1	yes	characteristic
298	C₂	C₁₃×#1497	1	1	yes	characteristic
299	C₂	C₁₃×#1508	1	1	yes	characteristic
300	C₂	C₁₃×#1511	1	1	yes	characteristic
301	C₂	C₁₃×#1530	1	1	yes	characteristic
302	C₂	C₁₃×#1558	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1272	C₂	semidirect
45. #7707	C₁₃	direct (N × H)
87. #1272	C₂×C₁₃	semidirect
234. C₁₃	#7707	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2857	2
C₁₃×#1817	2
C₁₃×#1749	2
C₁₃×#1272	2
#7707	13

Subgroups — C13×#7707 #15694046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7707 #15694046