Subgroups — C₁₃×#21278 #15707617

Order 3328.

798 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2295	C₂	1	2	yes	maximal
3	C₁₃×#1971	C₂	1	2	yes	maximal
4	C₁₃×#1943	C₂	1	2	yes	maximal
5	C₁₃×#1935	C₂	1	4	yes	maximal
6	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₃×#103	C₂	1	2	yes	maximal
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×#103	C₂²	1	4	yes	—
15	C₂×C₁₃×#103	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#391	C₂²	1	4	yes	—
21	C₁₃×#389	C₂²	1	4	yes	—
22	C₁₃×#387	—	2	4	no	—
23	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₁₃×#376	—	2	2	no	—
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₁₃×Q₁₆	—	2	4	no	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	—	2	4	no	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₃	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#107	C₂×C₄	1	4	yes	—
48	C₁₃×#106	—	2	4	no	—
49	C₁₃×#103	—	2	8	no	—
50	C₁₃×#103	C₂×C₄	1	8	yes	—
51	C₁₃×#97	—	2	4	no	—
52	C₁₃×#97	—	2	4	no	—
53	C₄×C₈×C₁₃	—	2	2	no	—
54	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
55	C₁₃×#95	—	2	4	no	—
56	C₁₃×#95	—	2	2	no	—
57	C₁₃×#95	—	2	4	no	—
58	C₁₃×#95	—	2	4	no	—
59	#21278	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	—	2	4	no	—
63	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₁₃×Q₁₆	—	2	8	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	#41	1	4	yes	—
78	C₂×C₈×C₁₃	#41	1	4	yes	—
79	C₁₃×C₄⋊C₄	—	2	8	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	4	8	no	—
84	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
85	C₄²×C₁₃	—	2	8	no	—
86	C₄²×C₁₃	—	2	4	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	4	2	no	—
90	C₄²×C₁₃	—	4	4	no	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₃	#41	1	2	yes	—
93	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
94	#2295	C₂×C₁₃	1	2	yes	—
95	#1971	C₂×C₁₃	1	2	yes	—
96	#1943	C₂×C₁₃	1	2	yes	—
97	#1935	C₂×C₁₃	1	4	yes	—
98	C₄×#106	C₂×C₁₃	1	1	yes	characteristic
99	C₄×#103	C₂×C₁₃	1	2	yes	—
100	C₄×#97	C₂×C₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₁₃	—	2	4	no	—
103	Q₈×C₁₃	—	2	4	no	—
104	Q₈×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	#137	1	4	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₁₃	—	2	2	no	—
124	C₈×C₁₃	—	2	4	no	—
125	C₈×C₁₃	#117	1	4	yes	—
126	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
128	C₄×Q₁₆	C₄×C₁₃	1	8	yes	complemented
129	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×#106	C₂²×C₁₃	1	2	yes	—
131	C₂×#103	C₂²×C₁₃	1	4	yes	—
132	C₂×#103	C₂²×C₁₃	1	2	yes	—
133	C₂×#97	C₂²×C₁₃	1	2	yes	—
134	C₂×#97	C₂²×C₁₃	1	2	yes	—
135	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	#391	C₂²×C₁₃	1	4	yes	—
138	#389	C₂²×C₁₃	1	4	yes	—
139	#387	—	2	4	no	—
140	C₄×C₄⋊C₄	—	2	2	no	—
141	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
142	#376	—	2	2	no	—
143	C₄³	C₂²×C₁₃	1	1	yes	characteristic
144	#340	C₂²×C₁₃	1	2	yes	—
145	#340	C₂²×C₁₃	1	4	yes	—
146	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₁₃	#532	1	2	yes	—
148	C₂²×C₁₃	C₂×#137	1	2	yes	—
149	C₂²×C₁₃	#575	1	2	yes	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	—	8	4	no	—
155	C₄×C₁₃	#441	1	4	yes	—
156	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
157	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
158	C₄×C₁₃	#529	1	2	yes	—
159	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
162	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
164	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
165	C₄×Q₈	—	2	4	no	—
166	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	D₄×C₁₃	1	2	yes	—
174	C₂×C₄²	D₄×C₁₃	1	2	yes	—
175	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	#107	C₂×C₄×C₁₃	1	4	yes	—
178	#106	—	2	4	no	—
179	#103	—	2	8	no	—
180	#103	C₂×C₄×C₁₃	1	8	yes	—
181	#97	—	2	4	no	—
182	#97	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
185	#95	—	2	4	no	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	#95	—	2	2	no	—
189	C₂×C₁₃	#2293	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
191	C₂×C₁₃	#2943	1	1	yes	characteristic
192	C₂×C₁₃	#3172	1	4	yes	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
201	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
204	Q₁₆	—	2	8	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
210	C₂×C₈	C₁₃×#41	1	4	yes	—
211	C₂×C₈	C₁₃×#41	1	4	yes	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	8	no	—
217	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
218	C₄²	—	2	8	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	2	no	—
223	C₄²	—	4	4	no	—
224	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
225	C₄²	C₁₃×#41	1	2	yes	—
226	C₁₃	#21278	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	2	4	no	—
230	Q₈	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
243	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
244	C₂×C₄	C₁₃×#137	1	4	yes	—
245	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
247	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
248	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	2	4	no	—
251	C₈	C₁₃×#117	1	4	yes	—
252	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
253	C₂²	C₁₃×#532	1	2	yes	—
254	C₂²	C₂×C₁₃×#137	1	2	yes	—
255	C₂²	C₁₃×#575	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	—	4	4	no	—
260	C₄	—	8	4	no	—
261	C₄	C₁₃×#441	1	4	yes	—
262	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
263	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
264	C₄	C₁₃×#529	1	2	yes	—
265	C₂	C₁₃×#2293	1	1	yes	characteristic
266	C₂	C₂×C₁₃×#441	1	1	yes	characteristic
267	C₂	C₁₃×#2943	1	1	yes	characteristic
268	C₂	C₁₃×#3172	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₄×C₁₃×Q₁₆	C₄	semidirect
59. #21278	C₁₃	direct (N × H)
128. C₄×Q₁₆	C₄×C₁₃	semidirect
226. C₁₃	#21278	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×C₄×C₁₃×Q₁₆	2
C₁₃×#2295	2
C₁₃×#1971	2
C₁₃×#1943	2
C₁₃×#1935	2
C₄×C₁₃×#106	2
C₄×C₁₃×#103	2
C₄×C₁₃×#97	2
#21278	13

Subgroups — C13×#21278 #15707617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21278 #15707617