Subgroups — C₁₃×#23542 #15709881

Order 3328.

1362 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#419	C₂	1	2	yes	maximal
4	C₁₃×#2310	C₂	1	2	yes	maximal
5	C₁₃×#2056	C₂	1	8	yes	maximal, complemented
6	C₁₃×#1735	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂²×C₁₃×Q₁₆	—	2	8	no	—
9	C₂²×C₁₃×Q₁₆	—	2	4	no	—
10	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
11	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#117	C₂²	1	2	yes	—
14	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
15	C₁₃×#424	—	2	4	no	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
17	C₁₃×#419	—	2	8	no	—
18	C₂×C₁₃×#103	C₂²	1	8	yes	—
19	C₁₃×#391	—	2	4	no	—
20	C₁₃×#391	C₂²	1	8	yes	—
21	C₁₃×#381	—	2	4	no	—
22	C₁₃×#340	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃	—	2	2	no	—
24	C₂²×Q₈×C₁₃	—	2	4	no	—
25	C₂²×Q₈×C₁₃	—	2	4	no	—
26	C₂²×Q₈×C₁₃	—	2	4	no	—
27	C₂²×Q₈×C₁₃	—	2	4	no	—
28	C₂²×Q₈×C₁₃	—	2	4	no	—
29	C₂²×Q₈×C₁₃	D₄	1	8	yes	—
30	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
31	C₂³×C₄×C₁₃	—	2	2	no	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×Q₁₆	—	2	32	no	—
34	C₂×C₁₃×Q₁₆	—	2	16	no	—
35	C₂×C₁₃×Q₁₆	—	2	8	no	—
36	C₂×C₁₃×M₁₆	—	2	4	no	—
37	C₂×C₁₃×M₁₆	—	2	2	no	—
38	C₂²×C₈×C₁₃	—	2	4	no	—
39	C₂²×C₈×C₁₃	D₄	1	4	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₁₃×#117	—	2	2	no	—
43	C₁₃×#117	—	2	2	no	—
44	C₁₃×#117	—	2	4	no	—
45	C₁₃×#117	—	2	4	no	—
46	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
52	C₁₃×C₄⋊C₈	—	2	8	no	—
53	C₁₃×#103	—	2	16	no	—
54	C₁₃×#95	—	2	16	no	—
55	#23542	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₁₃	—	2	8	no	—
58	C₂×Q₈×C₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₃	—	2	16	no	—
60	C₂×Q₈×C₁₃	—	2	16	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	—	2	4	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	—	2	4	no	—
65	C₂×Q₈×C₁₃	—	4	4	no	—
66	C₂×Q₈×C₁₃	—	4	4	no	—
67	C₂×Q₈×C₁₃	—	4	8	no	—
68	C₂×Q₈×C₁₃	—	4	8	no	—
69	C₂×Q₈×C₁₃	C₂×D₄	1	8	yes	—
70	C₂²×C₄×C₁₃	—	2	8	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₃	—	4	4	no	—
74	C₂²×C₄×C₁₃	—	4	4	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	#41	1	2	yes	—
80	C₂²×C₄×C₁₃	#41	1	4	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×Q₁₆	—	2	32	no	—
83	C₁₃×M₁₆	—	4	4	no	—
84	C₂×C₈×C₁₃	—	2	8	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₂²⋊C₄	—	2	4	no	—
93	C₁₃×C₂²⋊C₄	—	4	4	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	4	4	no	—
97	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
98	C₂×#424	C₂×C₁₃	1	1	yes	characteristic
99	C₂×#419	C₂×C₁₃	1	2	yes	—
100	#2310	C₂×C₁₃	1	2	yes	—
101	#2056	C₂×C₁₃	1	8	yes	complemented
102	#1735	C₂×C₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	—	2	4	no	—
104	C₂³×C₁₃	#121	1	2	yes	—
105	C₂³×C₁₃	#121	1	4	yes	—
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	8	no	—
108	Q₈×C₁₃	—	2	8	no	—
109	Q₈×C₁₃	—	4	4	no	—
110	Q₈×C₁₃	—	4	4	no	—
111	Q₈×C₁₃	—	4	16	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	8	no	—
114	C₂×C₄×C₁₃	—	2	8	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	16	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	8	no	—
120	C₂×C₄×C₁₃	#120	1	4	yes	—
121	C₂×C₄×C₁₃	#121	1	4	yes	—
122	C₂×C₄×C₁₃	#121	1	2	yes	—
123	C₂×C₄×C₁₃	#123	1	8	yes	—
124	C₂×C₄×C₁₃	#124	1	4	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	—	4	2	no	—
131	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
132	C₂²×Q₁₆	—	2	8	no	—
133	C₂²×Q₁₆	—	2	4	no	—
134	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
135	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
136	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×#117	C₂²×C₁₃	1	2	yes	—
138	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
139	#424	—	2	4	no	—
140	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
141	#419	—	2	8	no	—
142	C₂×#103	C₂²×C₁₃	1	8	yes	—
143	#391	—	2	4	no	—
144	#391	C₂²×C₁₃	1	8	yes	—
145	#381	—	2	4	no	—
146	#340	C₂²×C₁₃	1	4	yes	—
147	C₂²×C₁₃	—	2	8	no	—
148	C₂²×C₁₃	—	2	4	no	—
149	C₂²×C₁₃	#394	1	4	yes	—
150	C₂²×C₁₃	C₂×#121	1	2	yes	—
151	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
152	C₂²×C₁₃	#578	1	4	yes	—
153	C₄×C₁₃	—	4	8	no	—
154	C₄×C₁₃	—	4	4	no	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	#394	1	4	yes	—
157	C₄×C₁₃	C₂×#120	1	1	yes	characteristic
158	C₄×C₁₃	C₂×#123	1	2	yes	—
159	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×Q₈	—	2	4	no	—
162	C₂²×Q₈	—	2	4	no	—
163	C₂²×Q₈	—	2	4	no	—
164	C₂²×Q₈	—	2	4	no	—
165	C₂²×Q₈	—	2	4	no	—
166	C₂²×Q₈	D₄×C₁₃	1	8	yes	—
167	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
168	C₂³×C₄	—	2	2	no	—
169	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂×Q₁₆	—	2	32	no	—
171	C₂×Q₁₆	—	2	16	no	—
172	C₂×Q₁₆	—	2	8	no	—
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	D₄×C₁₃	1	4	yes	—
177	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
178	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	4	no	—
182	#117	—	2	4	no	—
183	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
189	C₄⋊C₈	—	2	8	no	—
190	#103	—	2	16	no	—
191	#95	—	2	16	no	—
192	C₂×C₁₃	—	2	4	no	—
193	C₂×C₁₃	C₂×#394	1	1	yes	characteristic
194	C₂×C₁₃	#3044	1	4	yes	—
195	C₂×C₁₃	#3059	1	2	yes	—
196	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₂×Q₈	—	2	8	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	16	no	—
200	C₂×Q₈	—	2	16	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	4	4	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	4	8	no	—
208	C₂×Q₈	—	4	8	no	—
209	C₂×Q₈	C₂×D₄×C₁₃	1	8	yes	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
216	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
217	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₁₃×#41	1	2	yes	—
220	C₂²×C₄	C₁₃×#41	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
222	Q₁₆	—	2	32	no	—
223	M₁₆	—	4	4	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	4	no	—
237	C₁₃	#23542	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	2	4	no	—
239	C₂³	C₁₃×#121	1	2	yes	—
240	C₂³	C₁₃×#121	1	4	yes	—
241	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
242	Q₈	—	2	8	no	—
243	Q₈	—	2	8	no	—
244	Q₈	—	4	16	no	—
245	Q₈	—	4	4	no	—
246	Q₈	—	4	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	16	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	C₁₃×#120	1	4	yes	—
256	C₂×C₄	C₁₃×#121	1	4	yes	—
257	C₂×C₄	C₁₃×#121	1	2	yes	—
258	C₂×C₄	C₁₃×#123	1	8	yes	—
259	C₂×C₄	C₁₃×#124	1	4	yes	—
260	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
261	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
263	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	4	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	4	no	—
268	C₂²	C₁₃×#394	1	4	yes	—
269	C₂²	C₂×C₁₃×#121	1	2	yes	—
270	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
271	C₂²	C₁₃×#578	1	4	yes	—
272	C₄	—	4	4	no	—
273	C₄	—	4	8	no	—
274	C₄	—	8	4	no	—
275	C₄	C₁₃×#394	1	4	yes	—
276	C₄	C₂×C₁₃×#120	1	1	yes	characteristic
277	C₄	C₂×C₁₃×#123	1	2	yes	—
278	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	C₂×C₁₃×#394	1	1	yes	characteristic
281	C₂	C₁₃×#3044	1	4	yes	—
282	C₂	C₁₃×#3059	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₃×#2056	C₂	semidirect
55. #23542	C₁₃	direct (N × H)
101. #2056	C₂×C₁₃	semidirect
237. C₁₃	#23542	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂×C₁₃×#424	2
C₂×C₁₃×#419	2
C₁₃×#2310	2
C₁₃×#2056	2
C₁₃×#1735	2
#23542	13

Subgroups — C13×#23542 #15709881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23542 #15709881