Subgroups — C₁₃×#32851 #15719190

Order 3328.

532 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2864	C₂	1	2	yes	maximal
2	C₁₃×#2101	C₂	1	2	yes	maximal
3	C₁₃×#2101	C₂	1	2	yes	maximal
4	C₁₃×#2088	C₂	1	2	yes	maximal
5	C₁₃×#2087	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#2086	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2085	C₂	1	2	yes	maximal
8	C₁₃×#2085	C₂	1	2	yes	maximal
9	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#106	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₁₃×#389	C₂²	1	2	yes	—
16	C₁₃×#389	C₂²	1	2	yes	—
17	C₁₃×#389	C₂²	1	2	yes	—
18	C₁₃×#389	C₂²	1	2	yes	—
19	C₁₃×#387	—	2	2	no	—
20	C₁₃×#387	—	2	2	no	—
21	C₁₃×#387	—	2	2	no	—
22	C₁₃×#387	—	2	2	no	—
23	C₁₃×#382	C₂²	1	2	yes	—
24	C₁₃×#382	C₂²	1	2	yes	—
25	C₄³×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#340	C₂²	1	2	yes	—
27	C₁₃×#340	C₂²	1	2	yes	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#340	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	1	yes	characteristic
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#107	—	4	1	no	—
52	C₁₃×#107	—	4	1	no	—
53	C₁₃×#106	—	4	2	no	—
54	C₁₃×C₄⋊C₈	—	4	1	no	—
55	C₁₃×C₄⋊C₈	—	4	1	no	—
56	C₁₃×#95	—	2	2	no	—
57	C₁₃×#95	—	2	2	no	—
58	C₁₃×#95	—	2	2	no	—
59	C₁₃×#95	—	2	2	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	#32851	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₁₃×C₄⋊C₄	—	4	4	no	—
90	C₁₃×C₄⋊C₄	—	4	4	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	1	no	—
94	C₄²×C₁₃	—	2	1	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	4	1	no	—
98	C₄²×C₁₃	—	4	2	no	—
99	#2864	C₂×C₁₃	1	2	yes	—
100	#2101	C₂×C₁₃	1	2	yes	—
101	#2101	C₂×C₁₃	1	2	yes	—
102	#2088	C₂×C₁₃	1	2	yes	—
103	#2087	C₂×C₁₃	1	1	yes	characteristic
104	#2086	C₂×C₁₃	1	1	yes	characteristic
105	#2085	C₂×C₁₃	1	2	yes	—
106	#2085	C₂×C₁₃	1	2	yes	—
107	#1831	C₂×C₁₃	1	1	yes	characteristic
108	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	#135	1	2	yes	—
132	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
134	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
135	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
136	C₈×C₁₃	—	4	2	no	—
137	C₈×C₁₃	—	4	1	no	—
138	C₈×C₁₃	—	4	1	no	—
139	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×#106	C₂²×C₁₃	1	2	yes	—
142	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
144	#389	C₂²×C₁₃	1	2	yes	—
145	#389	C₂²×C₁₃	1	2	yes	—
146	#389	C₂²×C₁₃	1	2	yes	—
147	#389	C₂²×C₁₃	1	2	yes	—
148	#387	—	2	2	no	—
149	#387	—	2	2	no	—
150	#387	—	2	2	no	—
151	#387	—	2	2	no	—
152	#382	C₂²×C₁₃	1	2	yes	—
153	#382	C₂²×C₁₃	1	2	yes	—
154	C₄³	C₂²×C₁₃	1	1	yes	characteristic
155	#340	C₂²×C₁₃	1	2	yes	—
156	#340	C₂²×C₁₃	1	2	yes	—
157	#340	C₂²×C₁₃	1	2	yes	—
158	#340	C₂²×C₁₃	1	2	yes	—
159	#340	C₂²×C₁₃	1	1	yes	characteristic
160	#340	C₂²×C₁₃	1	1	yes	characteristic
161	#340	C₂²×C₁₃	1	2	yes	—
162	#336	C₂²×C₁₃	1	2	yes	—
163	#336	C₂²×C₁₃	1	2	yes	—
164	C₂²×C₁₃	#536	1	2	yes	—
165	C₂²×C₁₃	#537	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
167	C₂²×C₁₃	#575	1	2	yes	—
168	C₂²×C₁₃	#575	1	1	yes	characteristic
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	4	2	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	—	4	1	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	—	8	2	no	—
175	C₄×C₁₃	—	8	2	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	#563	1	1	yes	characteristic
178	C₄×C₁₃	#563	1	1	yes	characteristic
179	C₄×C₁₃	#564	1	2	yes	—
180	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
191	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₁₃	1	2	yes	—
193	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	#107	—	4	1	no	—
197	#107	—	4	1	no	—
198	#106	—	4	2	no	—
199	C₄⋊C₈	—	4	1	no	—
200	C₄⋊C₈	—	4	1	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₂×C₁₃	#2822	1	1	yes	characteristic
210	C₂×C₁₃	#3233	1	2	yes	—
211	C₂×C₁₃	#3240	1	2	yes	—
212	C₂×C₁₃	#3250	1	1	yes	characteristic
213	C₂×C₁₃	#3256	1	1	yes	characteristic
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₁₃	#32851	1	1	yes	characteristic, direct product factor, Sylow
249	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	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	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₁₃×#135	1	2	yes	—
273	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
275	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
276	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
277	C₈	—	4	2	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₂²	C₁₃×#536	1	2	yes	—
281	C₂²	C₁₃×#537	1	1	yes	characteristic
282	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
283	C₂²	C₁₃×#575	1	2	yes	—
284	C₂²	C₁₃×#575	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	—	8	2	no	—
293	C₄	C₁₃×#563	1	1	yes	characteristic
294	C₄	C₁₃×#563	1	1	yes	characteristic
295	C₄	C₁₃×#564	1	2	yes	—
296	C₂	C₁₃×#2822	1	1	yes	characteristic
297	C₂	C₁₃×#3233	1	2	yes	—
298	C₂	C₁₃×#3240	1	2	yes	—
299	C₂	C₁₃×#3250	1	1	yes	characteristic
300	C₂	C₁₃×#3256	1	1	yes	characteristic

	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
64. #32851	C₁₃	direct (N × H)
248. C₁₃	#32851	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×#2864	2
C₁₃×#2101	2
C₁₃×#2088	2
C₁₃×#2087	2
C₁₃×#2086	2
C₁₃×#2085	2
C₁₃×#1831	2
#32851	13

Subgroups — C13×#32851 #15719190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32851 #15719190