Subgroups — C₁₃×#29227 #15715566

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2101	C₂	1	4	yes	maximal
4	C₁₃×#1992	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1992	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1989	C₂	1	4	yes	maximal
7	C₁₃×#1836	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1836	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
15	C₁₃×#389	—	2	2	no	—
16	C₁₃×#389	—	2	2	no	—
17	C₁₃×#389	C₂²	1	2	yes	—
18	C₁₃×#387	—	2	2	no	—
19	C₁₃×#384	C₂²	1	1	yes	characteristic
20	C₁₃×#384	C₂²	1	1	yes	characteristic
21	C₁₃×#382	C₂²	1	1	yes	characteristic
22	C₁₃×#382	C₂²	1	2	yes	—
23	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄³×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₁₃×#340	C₂²	1	4	yes	—
30	C₁₃×#336	C₂²	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₁₃×#125	—	2	2	no	—
34	C₁₃×#125	—	2	8	no	—
35	C₁₃×#125	Q₈	1	4	yes	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	—	2	2	no	—
47	C₂×C₄²×C₁₃	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	—	2	4	no	—
53	C₁₃×#97	—	2	2	no	—
54	C₁₃×#97	—	2	2	no	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	—	2	2	no	—
57	C₁₃×#95	—	2	2	no	—
58	C₁₃×#95	—	2	4	no	—
59	#29227	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	2	8	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	—	4	4	no	—
83	C₁₃×C₄⋊C₄	—	4	4	no	—
84	C₁₃×C₄⋊C₄	—	4	4	no	—
85	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	4	2	no	—
88	C₄²×C₁₃	—	4	2	no	—
89	C₄²×C₁₃	—	4	4	no	—
90	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₁₃	#41	1	2	yes	—
92	C₄²×C₁₃	#41	1	4	yes	—
93	#2859	C₂×C₁₃	1	1	yes	characteristic
94	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
95	#2101	C₂×C₁₃	1	4	yes	—
96	#1992	C₂×C₁₃	1	1	yes	characteristic
97	#1992	C₂×C₁₃	1	1	yes	characteristic
98	#1989	C₂×C₁₃	1	4	yes	—
99	#1836	C₂×C₁₃	1	1	yes	characteristic
100	#1836	C₂×C₁₃	1	1	yes	characteristic
101	#1767	C₂×C₁₃	1	1	yes	characteristic
102	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	2	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	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₁₃	#124	1	2	yes	—
122	C₂×C₄×C₁₃	#124	1	2	yes	—
123	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
127	C₈×C₁₃	—	4	2	no	—
128	C₈×C₁₃	—	4	2	no	—
129	C₂×#125	C₂²×C₁₃	1	2	yes	—
130	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
132	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
134	#389	—	2	2	no	—
135	#389	—	2	2	no	—
136	#389	C₂²×C₁₃	1	2	yes	—
137	#387	—	2	2	no	—
138	#384	C₂²×C₁₃	1	1	yes	characteristic
139	#384	C₂²×C₁₃	1	1	yes	characteristic
140	#382	C₂²×C₁₃	1	1	yes	characteristic
141	#382	C₂²×C₁₃	1	2	yes	—
142	C₄×C₄⋊C₄	—	2	2	no	—
143	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
144	C₄³	C₂²×C₁₃	1	1	yes	characteristic
145	#340	C₂²×C₁₃	1	4	yes	—
146	#340	C₂²×C₁₃	1	4	yes	—
147	#340	C₂²×C₁₃	1	4	yes	—
148	#340	C₂²×C₁₃	1	4	yes	—
149	#336	C₂²×C₁₃	1	2	yes	—
150	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
152	C₂²×C₁₃	#536	1	1	yes	characteristic
153	C₂²×C₁₃	#575	1	2	yes	—
154	C₂²×C₁₃	#575	1	2	yes	—
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	4	4	no	—
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	—	8	2	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	—	8	2	no	—
163	C₄×C₁₃	#556	1	2	yes	—
164	C₄×C₁₃	#557	1	2	yes	—
165	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
167	#125	—	2	2	no	—
168	#125	—	2	8	no	—
169	#125	Q₈×C₁₃	1	4	yes	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	D₄×C₁₃	1	2	yes	—
182	C₂×C₄²	D₄×C₁₃	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₄⋊C₈	—	2	4	no	—
187	#97	—	2	2	no	—
188	#97	—	2	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	4	no	—
193	C₂×C₁₃	#2568	1	1	yes	characteristic
194	C₂×C₁₃	#3080	1	2	yes	—
195	C₂×C₁₃	#3135	1	1	yes	characteristic
196	C₂×C₁₃	#3142	1	1	yes	characteristic
197	C₂×C₁₃	#3240	1	2	yes	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
224	C₄²	—	2	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
229	C₄²	C₁₃×#41	1	2	yes	—
230	C₄²	C₁₃×#41	1	4	yes	—
231	C₁₃	#29227	1	1	yes	characteristic, direct product factor, Sylow
232	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
251	C₂×C₄	C₁₃×#124	1	2	yes	—
252	C₂×C₄	C₁₃×#124	1	2	yes	—
253	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
255	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
256	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
260	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
261	C₂²	C₁₃×#536	1	1	yes	characteristic
262	C₂²	C₁₃×#575	1	2	yes	—
263	C₂²	C₁₃×#575	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	2	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	2	no	—
272	C₄	C₁₃×#556	1	2	yes	—
273	C₄	C₁₃×#557	1	2	yes	—
274	C₂	C₁₃×#2568	1	1	yes	characteristic
275	C₂	C₁₃×#3080	1	2	yes	—
276	C₂	C₁₃×#3135	1	1	yes	characteristic
277	C₂	C₁₃×#3142	1	1	yes	characteristic
278	C₂	C₁₃×#3240	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
59. #29227	C₁₃	direct (N × H)
231. C₁₃	#29227	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×#2859	2
C₄×C₁₃×#125	2
C₁₃×#2101	2
C₁₃×#1992	2
C₁₃×#1989	2
C₁₃×#1836	2
C₁₃×#1767	2
#29227	13

Subgroups — C13×#29227 #15715566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29227 #15715566