Subgroups — C₁₃×#23434 #15709773

Order 3328.

1916 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#575	C₂	1	3	yes	maximal, complemented
2	C₁₃×#2879	C₂	1	3	yes	maximal, complemented
3	C₁₃×#2017	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₁₃×#41	C₂²	1	3	yes	—
6	C₁₃×#575	—	2	12	no	—
7	C₁₃×#575	—	2	6	no	—
8	C₁₃×#575	C₂²	1	12	yes	complemented
9	C₂×C₁₃×#137	—	2	6	no	—
10	C₂×C₁₃×AGL(1,ℤ/8)	—	2	6	no	—
11	C₂×C₁₃×#135	—	2	6	no	—
12	C₂²×C₁₃×M₁₆	C₂²	1	3	yes	—
13	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×#408	—	2	6	no	—
15	C₁₃×#408	C₂²	1	12	yes	complemented
16	C₁₃×#407	—	2	6	no	—
17	C₁₃×#338	C₂²	1	4	yes	complemented
18	C₂×C₁₃×#41	—	2	3	no	—
19	C₂×C₁₃×#41	—	2	3	no	—
20	C₂×C₁₃×#41	—	2	6	no	—
21	C₂×C₁₃×#41	—	2	6	no	—
22	C₂×C₁₃×#41	—	2	6	no	—
23	C₂×C₁₃×#41	—	2	6	no	—
24	C₂×C₁₃×#41	D₄	1	6	yes	—
25	C₂×C₁₃×#41	C₂³	1	6	yes	—
26	C₂²×Q₈×C₁₃	D₄	1	3	yes	—
27	C₂²×D₄×C₁₃	—	2	3	no	—
28	C₂²×D₄×C₁₃	D₄	1	3	yes	—
29	C₂³×C₄×C₁₃	—	2	3	no	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×#137	—	2	24	no	—
32	C₁₃×#137	—	2	24	no	—
33	C₁₃×AGL(1,ℤ/8)	—	2	24	no	—
34	C₁₃×AGL(1,ℤ/8)	—	2	24	no	—
35	C₁₃×#135	—	2	24	no	—
36	C₁₃×#135	—	2	24	no	—
37	C₂×C₁₃×Q₁₆	—	2	6	no	—
38	C₂×C₁₃×SD₁₆	—	2	12	no	—
39	C₂×C₁₃×D₈	—	2	6	no	—
40	C₂×C₁₃×M₁₆	—	2	6	no	—
41	C₂×C₁₃×M₁₆	—	2	6	no	—
42	C₂×C₁₃×M₁₆	—	2	3	no	—
43	C₂×C₁₃×M₁₆	—	2	3	no	—
44	C₂×C₁₃×M₁₆	D₄	1	6	yes	—
45	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
46	C₂²×C₈×C₁₃	D₄	1	6	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
48	C₁₃×C₂²⋊C₈	—	2	24	no	—
49	#23434	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₃	—	2	3	no	—
51	C₂⁴×C₁₃	#41	1	1	yes	characteristic
52	C₁₃×#41	—	2	12	no	—
53	C₁₃×#41	—	2	12	no	—
54	C₁₃×#41	—	2	12	no	—
55	C₁₃×#41	—	2	12	no	—
56	C₁₃×#41	—	4	3	no	—
57	C₁₃×#41	—	4	3	no	—
58	C₁₃×#41	—	4	6	no	—
59	C₁₃×#41	—	4	6	no	—
60	C₁₃×#41	—	4	3	no	—
61	C₁₃×#41	—	4	3	no	—
62	C₂×Q₈×C₁₃	—	2	3	no	—
63	C₂×Q₈×C₁₃	—	2	6	no	—
64	C₂×Q₈×C₁₃	—	2	6	no	—
65	C₂×Q₈×C₁₃	C₂×D₄	1	6	yes	—
66	C₂×D₄×C₁₃	—	2	6	no	—
67	C₂×D₄×C₁₃	—	2	6	no	—
68	C₂×D₄×C₁₃	—	2	6	no	—
69	C₂×D₄×C₁₃	—	2	6	no	—
70	C₂×D₄×C₁₃	—	2	3	no	—
71	C₂×D₄×C₁₃	—	4	6	no	—
72	C₂×D₄×C₁₃	—	4	6	no	—
73	C₂×D₄×C₁₃	C₂×D₄	1	6	yes	—
74	C₂²×C₄×C₁₃	—	2	3	no	—
75	C₂²×C₄×C₁₃	—	2	6	no	—
76	C₂²×C₄×C₁₃	—	2	6	no	—
77	C₂²×C₄×C₁₃	—	4	6	no	—
78	C₂²×C₄×C₁₃	—	4	6	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×Q₁₆	—	2	24	no	—
86	C₁₃×SD₁₆	—	2	48	no	—
87	C₁₃×D₈	—	2	24	no	—
88	C₁₃×M₁₆	—	2	12	no	—
89	C₁₃×M₁₆	—	2	12	no	—
90	C₁₃×M₁₆	—	4	3	no	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₂×C₈×C₁₃	—	2	6	no	—
93	C₂×C₈×C₁₃	—	2	6	no	—
94	C₂×C₈×C₁₃	—	2	6	no	—
95	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
96	C₂×#575	C₂×C₁₃	1	3	yes	complemented
97	#2879	C₂×C₁₃	1	3	yes	complemented
98	#2017	C₂×C₁₃	1	8	yes	complemented
99	C₂×#338	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂³×C₁₃	—	2	6	no	—
101	C₂³×C₁₃	—	4	6	no	—
102	C₂³×C₁₃	—	8	1	no	—
103	C₂³×C₁₃	—	8	6	no	—
104	C₂³×C₁₃	#121	1	3	yes	—
105	C₂³×C₁₃	#121	1	3	yes	—
106	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	12	no	—
108	Q₈×C₁₃	—	4	3	no	—
109	Q₈×C₁₃	—	4	3	no	—
110	D₄×C₁₃	—	2	12	no	—
111	D₄×C₁₃	—	4	12	no	—
112	D₄×C₁₃	—	4	6	no	—
113	D₄×C₁₃	—	4	6	no	—
114	D₄×C₁₃	—	4	3	no	—
115	D₄×C₁₃	—	4	3	no	—
116	C₂×C₄×C₁₃	—	2	6	no	—
117	C₂×C₄×C₁₃	—	4	6	no	—
118	C₂×C₄×C₁₃	—	4	6	no	—
119	C₂×C₄×C₁₃	—	4	6	no	—
120	C₂×C₄×C₁₃	—	4	12	no	—
121	C₂×C₄×C₁₃	#120	1	3	yes	—
122	C₂×C₄×C₁₃	#120	1	3	yes	—
123	C₂×C₄×C₁₃	#120	1	6	yes	—
124	C₂×C₄×C₁₃	#121	1	3	yes	—
125	C₂×C₄×C₁₃	#121	1	3	yes	—
126	C₂×C₄×C₁₃	#127	1	3	yes	—
127	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
130	C₈×C₁₃	—	2	12	no	—
131	C₈×C₁₃	—	4	2	no	—
132	C₂²×#41	C₂²×C₁₃	1	3	yes	—
133	#575	—	2	12	no	—
134	#575	—	2	6	no	—
135	#575	C₂²×C₁₃	1	12	yes	complemented
136	C₂×#137	—	2	6	no	—
137	C₂×AGL(1,ℤ/8)	—	2	6	no	—
138	C₂×#135	—	2	6	no	—
139	C₂²×M₁₆	C₂²×C₁₃	1	3	yes	—
140	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
141	#408	—	2	6	no	—
142	#408	C₂²×C₁₃	1	12	yes	complemented
143	#407	—	2	6	no	—
144	#338	C₂²×C₁₃	1	4	yes	complemented
145	C₂²×C₁₃	—	4	3	no	—
146	C₂²×C₁₃	—	4	3	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	—	4	12	no	—
149	C₂²×C₁₃	—	4	6	no	—
150	C₂²×C₁₃	—	8	12	no	—
151	C₂²×C₁₃	#392	1	3	yes	—
152	C₂²×C₁₃	#392	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#121	1	3	yes	—
154	C₄×C₁₃	—	4	6	no	—
155	C₄×C₁₃	#392	1	3	yes	—
156	C₄×C₁₃	#392	1	1	yes	characteristic
157	C₄×C₁₃	C₂×#120	1	3	yes	—
158	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
159	C₂×#41	—	2	3	no	—
160	C₂×#41	—	2	3	no	—
161	C₂×#41	—	2	6	no	—
162	C₂×#41	—	2	6	no	—
163	C₂×#41	—	2	6	no	—
164	C₂×#41	—	2	6	no	—
165	C₂×#41	D₄×C₁₃	1	6	yes	—
166	C₂×#41	C₂³×C₁₃	1	6	yes	—
167	C₂²×Q₈	D₄×C₁₃	1	3	yes	—
168	C₂²×D₄	—	2	3	no	—
169	C₂²×D₄	D₄×C₁₃	1	3	yes	—
170	C₂³×C₄	—	2	3	no	—
171	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
172	#137	—	2	24	no	—
173	#137	—	2	24	no	—
174	AGL(1,ℤ/8)	—	2	24	no	—
175	AGL(1,ℤ/8)	—	2	24	no	—
176	#135	—	2	24	no	—
177	#135	—	2	24	no	—
178	C₂×Q₁₆	—	2	6	no	—
179	C₂×SD₁₆	—	2	12	no	—
180	C₂×D₈	—	2	6	no	—
181	C₂×M₁₆	—	2	6	no	—
182	C₂×M₁₆	—	2	6	no	—
183	C₂×M₁₆	—	2	3	no	—
184	C₂×M₁₆	—	2	3	no	—
185	C₂×M₁₆	D₄×C₁₃	1	6	yes	—
186	C₂×M₁₆	C₂³×C₁₃	1	6	yes	—
187	C₂²×C₈	D₄×C₁₃	1	6	yes	—
188	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
189	C₂²⋊C₈	—	2	24	no	—
190	C₂×C₁₃	—	2	3	no	—
191	C₂×C₁₃	—	2	3	no	—
192	C₂×C₁₃	—	2	1	no	—
193	C₂×C₁₃	—	8	6	no	—
194	C₂×C₁₃	C₂×#392	1	1	yes	characteristic
195	C₂⁴	—	2	3	no	—
196	C₂⁴	C₁₃×#41	1	1	yes	characteristic
197	#41	—	2	12	no	—
198	#41	—	2	12	no	—
199	#41	—	2	12	no	—
200	#41	—	2	12	no	—
201	#41	—	4	3	no	—
202	#41	—	4	3	no	—
203	#41	—	4	6	no	—
204	#41	—	4	6	no	—
205	#41	—	4	3	no	—
206	#41	—	4	3	no	—
207	C₂×Q₈	—	2	3	no	—
208	C₂×Q₈	—	2	6	no	—
209	C₂×Q₈	—	2	6	no	—
210	C₂×Q₈	C₂×D₄×C₁₃	1	6	yes	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	—	2	6	no	—
214	C₂×D₄	—	2	6	no	—
215	C₂×D₄	—	2	3	no	—
216	C₂×D₄	—	4	6	no	—
217	C₂×D₄	—	4	6	no	—
218	C₂×D₄	C₂×D₄×C₁₃	1	6	yes	—
219	C₂²×C₄	—	2	3	no	—
220	C₂²×C₄	—	2	6	no	—
221	C₂²×C₄	—	2	6	no	—
222	C₂²×C₄	—	4	6	no	—
223	C₂²×C₄	—	4	6	no	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
227	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
228	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
230	Q₁₆	—	2	24	no	—
231	SD₁₆	—	2	48	no	—
232	D₈	—	2	24	no	—
233	M₁₆	—	2	12	no	—
234	M₁₆	—	2	12	no	—
235	M₁₆	—	4	3	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	6	no	—
238	C₂×C₈	—	2	6	no	—
239	C₂×C₈	—	2	6	no	—
240	C₂×C₈	C₂×D₄×C₁₃	1	12	yes	—
241	C₁₃	#23434	1	1	yes	characteristic, direct product factor, Sylow
242	C₂³	—	2	6	no	—
243	C₂³	—	4	6	no	—
244	C₂³	—	8	1	no	—
245	C₂³	—	8	6	no	—
246	C₂³	C₁₃×#121	1	3	yes	—
247	C₂³	C₁₃×#121	1	3	yes	—
248	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
249	Q₈	—	2	12	no	—
250	Q₈	—	4	3	no	—
251	Q₈	—	4	3	no	—
252	D₄	—	2	12	no	—
253	D₄	—	4	6	no	—
254	D₄	—	4	6	no	—
255	D₄	—	4	12	no	—
256	D₄	—	4	3	no	—
257	D₄	—	4	3	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	4	6	no	—
260	C₂×C₄	—	4	12	no	—
261	C₂×C₄	—	4	6	no	—
262	C₂×C₄	—	4	6	no	—
263	C₂×C₄	C₁₃×#120	1	3	yes	—
264	C₂×C₄	C₁₃×#120	1	3	yes	—
265	C₂×C₄	C₁₃×#120	1	6	yes	—
266	C₂×C₄	C₁₃×#121	1	3	yes	—
267	C₂×C₄	C₁₃×#121	1	3	yes	—
268	C₂×C₄	C₁₃×#127	1	3	yes	—
269	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
270	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
271	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
272	C₈	—	2	12	no	—
273	C₈	—	4	2	no	—
274	C₂²	—	4	3	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	12	no	—
278	C₂²	—	4	6	no	—
279	C₂²	—	8	12	no	—
280	C₂²	C₁₃×#392	1	3	yes	—
281	C₂²	C₁₃×#392	1	1	yes	characteristic
282	C₂²	C₂×C₁₃×#121	1	3	yes	—
283	C₄	—	4	6	no	—
284	C₄	C₁₃×#392	1	3	yes	—
285	C₄	C₁₃×#392	1	1	yes	characteristic
286	C₄	C₂×C₁₃×#120	1	3	yes	—
287	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
288	C₂	—	2	3	no	—
289	C₂	—	2	3	no	—
290	C₂	—	2	1	no	—
291	C₂	—	8	6	no	—
292	C₂	C₂×C₁₃×#392	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₂×#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×#392
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#575	C₂	semidirect
2. C₁₃×#2879	C₂	semidirect
3. C₁₃×#2017	C₂	semidirect
4. C₂×C₁₃×#338	C₂	semidirect
8. C₁₃×#575	C₂²	semidirect
15. C₁₃×#408	C₂²	semidirect
17. C₁₃×#338	C₂²	semidirect
49. #23434	C₁₃	direct (N × H)
96. C₂×#575	C₂×C₁₃	semidirect
97. #2879	C₂×C₁₃	semidirect
98. #2017	C₂×C₁₃	semidirect
99. C₂×#338	C₂×C₁₃	semidirect
135. #575	C₂²×C₁₃	semidirect
142. #408	C₂²×C₁₃	semidirect
144. #338	C₂²×C₁₃	semidirect
241. C₁₃	#23434	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×C₁₃×#575	2
C₁₃×#2879	2
C₁₃×#2017	2
C₂×C₁₃×#338	2
#23434	13

Subgroups — C13×#23434 #15709773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23434 #15709773