Subgroups — C₁₃×#17688 #15704027

Order 3328.

1148 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3500	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2971	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1536	C₂	1	3	yes	maximal, complemented
4	C₁₃×#1527	C₂	1	6	yes	maximal, complemented
5	C₁₃×#1525	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₃×2₋¹⁺⁴	C₄	1	1	yes	characteristic
7	C₂×C₁₃×H(5,𝔽₂)	C₄	1	1	yes	characteristic
8	C₁₃×#548	—	2	2	no	—
9	C₁₃×#546	—	2	6	no	—
10	C₁₃×#536	C₄	1	3	yes	—
11	C₁₃×#535	C₄	1	3	yes	—
12	C₁₃×#532	C₂²	1	3	yes	—
13	C₂×C₁₃×#124	C₂²	1	3	yes	—
14	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₃	—	2	2	no	—
16	C₁₃×#436	—	2	6	no	—
17	C₁₃×#432	C₂²	1	6	yes	complemented
18	C₄×C₁₃×M₁₆	C₂²	1	6	yes	complemented
19	C₁₃×#329	C₂²	1	1	yes	characteristic, complemented
20	C₁₃×#329	C₂²	1	6	yes	complemented
21	C₁₃×#329	C₂²	1	3	yes	complemented
22	C₁₃×#329	C₂²	1	6	yes	complemented
23	C₁₃×2₋¹⁺⁴	—	4	4	no	—
24	C₁₃×H(5,𝔽₂)	—	4	4	no	—
25	C₂×C₁₃×#41	—	2	1	no	—
26	C₂×C₁₃×#41	—	2	3	no	—
27	C₂×C₁₃×#41	C₂×C₄	1	3	yes	—
28	C₂×C₁₃×#41	C₂×C₄	1	3	yes	—
29	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₁₃	—	2	1	no	—
31	C₂²×Q₈×C₁₃	C₂×C₄	1	3	yes	—
32	C₂²×D₄×C₁₃	—	2	3	no	—
33	C₂²×D₄×C₁₃	C₂×C₄	1	3	yes	—
34	C₂×C₁₃×M₁₆	—	2	6	no	—
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	3	yes	—
36	C₁₃×#127	C₂×C₄	1	3	yes	—
37	C₁₃×#124	—	2	6	no	—
38	C₁₃×#124	C₂×C₄	1	6	yes	—
39	C₁₃×#124	C₂³	1	3	yes	—
40	C₁₃×#124	C₂³	1	1	yes	characteristic
41	C₁₃×#123	—	2	6	no	—
42	C₁₃×C₂²⋊Q₈	—	2	6	no	—
43	C₁₃×#121	—	2	6	no	—
44	C₁₃×#121	—	2	6	no	—
45	C₁₃×#120	—	2	6	no	—
46	C₄×Q₈×C₁₃	D₄	1	2	yes	—
47	C₄×D₄×C₁₃	—	2	6	no	—
48	C₄×D₄×C₁₃	D₄	1	6	yes	—
49	C₁₃×#117	C₂×C₄	1	3	yes	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	6	no	—
51	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
52	C₁₃×C₄⋊C₈	—	2	6	no	—
53	C₁₃×C₂²⋊C₈	—	2	6	no	—
54	C₁₃×#97	C₂³	1	2	yes	—
55	C₁₃×#97	C₂³	1	6	yes	—
56	C₄×C₈×C₁₃	—	2	6	no	—
57	#17688	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	2	3	no	—
59	C₁₃×#41	—	2	4	no	—
60	C₁₃×#41	—	4	6	no	—
61	C₁₃×#41	—	4	6	no	—
62	C₁₃×#41	—	4	12	no	—
63	C₁₃×#41	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	2	3	no	—
65	C₂×Q₈×C₁₃	—	4	6	no	—
66	C₂×Q₈×C₁₃	—	4	4	no	—
67	C₂×Q₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×Q₈×C₁₃	C₂²×C₄	1	3	yes	—
69	C₂×Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₃	—	2	3	no	—
71	C₂×D₄×C₁₃	—	2	3	no	—
72	C₂×D₄×C₁₃	—	2	3	no	—
73	C₂×D₄×C₁₃	—	2	6	no	—
74	C₂×D₄×C₁₃	—	4	6	no	—
75	C₂×D₄×C₁₃	—	4	12	no	—
76	C₂×D₄×C₁₃	C₂²⋊C₄	1	3	yes	—
77	C₂×D₄×C₁₃	C₂²×C₄	1	3	yes	—
78	C₂×D₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	—	2	3	no	—
80	C₂²×C₄×C₁₃	—	2	3	no	—
81	C₂²×C₄×C₁₃	—	2	3	no	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
84	C₁₃×M₁₆	—	4	12	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	6	no	—
87	C₁₃×C₄⋊C₄	—	2	6	no	—
88	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
89	C₁₃×C₂²⋊C₄	—	2	12	no	—
90	C₁₃×C₂²⋊C₄	—	2	6	no	—
91	C₁₃×C₂²⋊C₄	—	2	2	no	—
92	C₁₃×C₂²⋊C₄	C₂²⋊C₄	1	6	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
94	C₄²×C₁₃	C₂×D₄	1	6	yes	—
95	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
96	#3500	C₂×C₁₃	1	1	yes	characteristic
97	#2971	C₂×C₁₃	1	2	yes	complemented
98	#1536	C₂×C₁₃	1	3	yes	complemented
99	#1527	C₂×C₁₃	1	6	yes	complemented
100	#1525	C₂×C₁₃	1	3	yes	complemented
101	C₂³×C₁₃	—	2	3	no	—
102	C₂³×C₁₃	—	2	1	no	—
103	C₂³×C₁₃	—	4	6	no	—
104	C₂³×C₁₃	—	8	6	no	—
105	C₂³×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
106	Q₈×C₁₃	—	4	3	no	—
107	Q₈×C₁₃	—	4	1	no	—
108	Q₈×C₁₃	—	4	6	no	—
109	Q₈×C₁₃	#100	1	4	yes	—
110	D₄×C₁₃	—	2	6	no	—
111	D₄×C₁₃	—	4	3	no	—
112	D₄×C₁₃	—	4	1	no	—
113	D₄×C₁₃	—	4	6	no	—
114	D₄×C₁₃	—	4	6	no	—
115	D₄×C₁₃	—	4	6	no	—
116	D₄×C₁₃	—	8	6	no	—
117	C₂×C₄×C₁₃	—	2	6	no	—
118	C₂×C₄×C₁₃	—	2	3	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	4	6	no	—
121	C₂×C₄×C₁₃	—	4	6	no	—
122	C₂×C₄×C₁₃	—	4	6	no	—
123	C₂×C₄×C₁₃	—	4	6	no	—
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
125	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
128	C₈×C₁₃	—	2	4	no	—
129	C₈×C₁₃	—	4	6	no	—
130	C₂×2₋¹⁺⁴	C₄×C₁₃	1	1	yes	characteristic
131	C₂×H(5,𝔽₂)	C₄×C₁₃	1	1	yes	characteristic
132	#548	—	2	2	no	—
133	#546	—	2	6	no	—
134	#536	C₄×C₁₃	1	3	yes	—
135	#535	C₄×C₁₃	1	3	yes	—
136	#532	C₂²×C₁₃	1	3	yes	—
137	C₂×#124	C₂²×C₁₃	1	3	yes	—
138	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
139	C₈×Q₈	—	2	2	no	—
140	#436	—	2	6	no	—
141	#432	C₂²×C₁₃	1	6	yes	complemented
142	C₄×M₁₆	C₂²×C₁₃	1	6	yes	complemented
143	#329	C₂²×C₁₃	1	3	yes	complemented
144	#329	C₂²×C₁₃	1	6	yes	complemented
145	#329	C₂²×C₁₃	1	1	yes	characteristic, complemented
146	#329	C₂²×C₁₃	1	6	yes	complemented
147	C₂²×C₁₃	—	2	3	no	—
148	C₂²×C₁₃	—	2	6	no	—
149	C₂²×C₁₃	—	4	1	no	—
150	C₂²×C₁₃	—	4	3	no	—
151	C₂²×C₁₃	—	4	2	no	—
152	C₂²×C₁₃	—	4	6	no	—
153	C₂²×C₁₃	—	16	6	no	—
154	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	6	no	—
158	C₄×C₁₃	—	4	1	no	—
159	C₄×C₁₃	—	4	3	no	—
160	C₄×C₁₃	C₂×#100	1	6	yes	—
161	2₋¹⁺⁴	—	4	4	no	—
162	H(5,𝔽₂)	—	4	4	no	—
163	C₂×#41	—	2	1	no	—
164	C₂×#41	—	2	3	no	—
165	C₂×#41	C₂×C₄×C₁₃	1	3	yes	—
166	C₂×#41	C₂×C₄×C₁₃	1	3	yes	—
167	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×Q₈	C₂×C₄×C₁₃	1	3	yes	—
170	C₂²×D₄	—	2	3	no	—
171	C₂²×D₄	C₂×C₄×C₁₃	1	3	yes	—
172	C₂×M₁₆	—	2	6	no	—
173	C₄⋊Q₈	C₂×C₄×C₁₃	1	3	yes	—
174	#127	C₂×C₄×C₁₃	1	3	yes	—
175	#124	—	2	6	no	—
176	#124	C₂×C₄×C₁₃	1	6	yes	—
177	#124	C₂³×C₁₃	1	3	yes	—
178	#124	C₂³×C₁₃	1	1	yes	characteristic
179	#123	—	2	6	no	—
180	C₂²⋊Q₈	—	2	6	no	—
181	#121	—	2	6	no	—
182	#121	—	2	6	no	—
183	#120	—	2	6	no	—
184	C₄×Q₈	D₄×C₁₃	1	2	yes	—
185	C₄×D₄	—	2	6	no	—
186	C₄×D₄	D₄×C₁₃	1	6	yes	—
187	#117	C₂×C₄×C₁₃	1	3	yes	—
188	C₂×C₂²⋊C₄	—	2	6	no	—
189	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
190	C₄⋊C₈	—	2	6	no	—
191	C₂²⋊C₈	—	2	6	no	—
192	#97	C₂³×C₁₃	1	2	yes	—
193	#97	C₂³×C₁₃	1	6	yes	—
194	C₄×C₈	—	2	6	no	—
195	C₂×C₁₃	—	4	3	no	—
196	C₂×C₁₃	—	8	1	no	—
197	C₂×C₁₃	—	8	3	no	—
198	C₂×C₁₃	C₂²×#100	1	1	yes	characteristic
199	C₂×C₁₃	#2901	1	2	yes	—
200	C₂⁴	—	2	3	no	—
201	#41	—	2	4	no	—
202	#41	—	4	6	no	—
203	#41	—	4	6	no	—
204	#41	—	4	12	no	—
205	#41	—	4	4	no	—
206	C₂×Q₈	—	2	3	no	—
207	C₂×Q₈	—	4	6	no	—
208	C₂×Q₈	—	4	4	no	—
209	C₂×Q₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×Q₈	C₂²×C₄×C₁₃	1	3	yes	—
211	C₂×Q₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	C₂×D₄	—	2	3	no	—
213	C₂×D₄	—	2	3	no	—
214	C₂×D₄	—	2	3	no	—
215	C₂×D₄	—	2	6	no	—
216	C₂×D₄	—	4	6	no	—
217	C₂×D₄	—	4	12	no	—
218	C₂×D₄	C₁₃×C₂²⋊C₄	1	3	yes	—
219	C₂×D₄	C₂²×C₄×C₁₃	1	3	yes	—
220	C₂×D₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	—	2	3	no	—
223	C₂²×C₄	—	2	3	no	—
224	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	6	yes	—
226	M₁₆	—	4	12	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	6	no	—
229	C₄⋊C₄	—	2	6	no	—
230	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	6	yes	—
231	C₂²⋊C₄	—	2	12	no	—
232	C₂²⋊C₄	—	2	6	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	C₁₃×C₂²⋊C₄	1	6	yes	—
235	C₄²	C₂²×C₄×C₁₃	1	3	yes	—
236	C₄²	C₂×D₄×C₁₃	1	6	yes	—
237	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
238	C₁₃	#17688	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³	—	2	3	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	4	6	no	—
242	C₂³	—	8	6	no	—
243	C₂³	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
244	Q₈	—	4	3	no	—
245	Q₈	—	4	1	no	—
246	Q₈	—	4	6	no	—
247	Q₈	C₁₃×#100	1	4	yes	—
248	D₄	—	2	6	no	—
249	D₄	—	4	3	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	6	no	—
252	D₄	—	4	6	no	—
253	D₄	—	4	6	no	—
254	D₄	—	8	6	no	—
255	C₂×C₄	—	2	6	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	6	no	—
259	C₂×C₄	—	4	6	no	—
260	C₂×C₄	—	4	6	no	—
261	C₂×C₄	—	4	6	no	—
262	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	6	yes	—
263	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
264	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	—	4	6	no	—
268	C₂²	—	2	3	no	—
269	C₂²	—	2	6	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	3	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	6	no	—
274	C₂²	—	16	6	no	—
275	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	6	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	3	no	—
281	C₄	C₂×C₁₃×#100	1	6	yes	—
282	C₂	—	4	3	no	—
283	C₂	—	8	3	no	—
284	C₂	—	8	1	no	—
285	C₂	C₂²×C₁₃×#100	1	1	yes	characteristic
286	C₂	C₁₃×#2901	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2971	C₂	semidirect
3. C₁₃×#1536	C₂	semidirect
4. C₁₃×#1527	C₂	semidirect
5. C₁₃×#1525	C₂	semidirect
17. C₁₃×#432	C₂²	semidirect
18. C₄×C₁₃×M₁₆	C₂²	semidirect
19. C₁₃×#329	C₂²	semidirect
20. C₁₃×#329	C₂²	semidirect
21. C₁₃×#329	C₂²	semidirect
22. C₁₃×#329	C₂²	semidirect
57. #17688	C₁₃	direct (N × H)
97. #2971	C₂×C₁₃	semidirect
98. #1536	C₂×C₁₃	semidirect
99. #1527	C₂×C₁₃	semidirect
100. #1525	C₂×C₁₃	semidirect
141. #432	C₂²×C₁₃	semidirect
142. C₄×M₁₆	C₂²×C₁₃	semidirect
143. #329	C₂²×C₁₃	semidirect
144. #329	C₂²×C₁₃	semidirect
145. #329	C₂²×C₁₃	semidirect
146. #329	C₂²×C₁₃	semidirect
238. C₁₃	#17688	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×#3500	2
C₁₃×#2971	2
C₁₃×#1536	2
C₁₃×#1527	2
C₁₃×#1525	2
#17688	13

Subgroups — C13×#17688 #15704027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17688 #15704027