Subgroups — C₁₃×#23745 #15710084

Order 3328.

902 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2296	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2100	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2099	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1736	C₂	1	2	yes	maximal
6	C₁₃×#1736	C₂	1	2	yes	maximal
7	C₁₃×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#389	C₂²	1	2	yes	—
15	C₁₃×#389	C₂²	1	2	yes	—
16	C₁₃×#385	—	2	2	no	—
17	C₁₃×#384	—	2	2	no	—
18	C₁₃×#384	C₂²	1	2	yes	—
19	C₁₃×#384	C₂²	1	2	yes	—
20	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₁₃×#340	C₂²	1	8	yes	—
22	C₁₃×#340	C₂²	1	8	yes	—
23	C₁₃×#336	C₂²	1	2	yes	—
24	C₁₃×#336	C₂²	1	2	yes	—
25	C₂³×C₄×C₁₃	—	2	2	no	—
26	C₂³×C₄×C₁₃	D₄	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₁₃×M₁₆	—	2	4	no	—
30	C₂×C₁₃×M₁₆	—	2	4	no	—
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₁₃×#117	—	2	2	no	—
33	C₁₃×#117	—	2	4	no	—
34	C₁₃×#117	Q₈	1	4	yes	—
35	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
41	C₂×C₁₃×C₄⋊C₄	—	4	8	no	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
47	C₂×C₄²×C₁₃	—	2	1	no	—
48	C₂×C₄²×C₁₃	—	2	4	no	—
49	C₂×C₄²×C₁₃	—	2	1	no	—
50	C₂×C₄²×C₁₃	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#95	—	2	2	no	—
54	C₁₃×#95	—	2	2	no	—
55	#23745	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	4	8	no	—
64	C₂²×C₄×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	4	yes	—
71	C₂²×C₄×C₁₃	#41	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×M₁₆	—	4	8	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	8	no	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	8	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	—	4	4	no	—
83	C₁₃×C₄⋊C₄	—	4	8	no	—
84	C₁₃×C₄⋊C₄	—	4	4	no	—
85	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
86	C₁₃×C₂²⋊C₄	—	2	4	no	—
87	C₁₃×C₂²⋊C₄	—	4	2	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	4	2	no	—
93	#2296	C₂×C₁₃	1	1	yes	characteristic
94	C₂×#384	C₂×C₁₃	1	1	yes	characteristic
95	#2100	C₂×C₁₃	1	4	yes	complemented
96	#2099	C₂×C₁₃	1	4	yes	complemented
97	#1736	C₂×C₁₃	1	2	yes	—
98	#1736	C₂×C₁₃	1	2	yes	—
99	#1728	C₂×C₁₃	1	1	yes	characteristic
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₁₃	#123	1	2	yes	—
104	C₂³×C₁₃	#124	1	2	yes	—
105	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	—	2	16	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	8	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	8	8	no	—
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₃	#123	1	2	yes	—
122	C₂×C₄×C₁₃	#124	1	2	yes	—
123	C₂×C₄×C₁₃	#125	1	8	yes	—
124	C₂×C₄×C₁₃	#126	1	8	yes	—
125	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
126	C₂×C₄×C₁₃	#137	1	4	yes	—
127	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₈×C₁₃	—	4	4	no	—
132	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
133	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
135	C₂²×C₄⋊C₄	—	2	2	no	—
136	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
137	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
138	#389	C₂²×C₁₃	1	2	yes	—
139	#389	C₂²×C₁₃	1	2	yes	—
140	#385	—	2	2	no	—
141	#384	—	2	2	no	—
142	#384	C₂²×C₁₃	1	2	yes	—
143	#384	C₂²×C₁₃	1	2	yes	—
144	C₄×C₄⋊C₄	—	2	2	no	—
145	#340	C₂²×C₁₃	1	8	yes	—
146	#340	C₂²×C₁₃	1	8	yes	—
147	#336	C₂²×C₁₃	1	2	yes	—
148	#336	C₂²×C₁₃	1	2	yes	—
149	C₂²×C₁₃	—	2	4	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	#400	1	4	yes	—
154	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#123	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
158	C₂²×C₁₃	C₂×#137	1	2	yes	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	—	8	2	no	—
163	C₄×C₁₃	—	8	4	no	—
164	C₄×C₁₃	#400	1	4	yes	—
165	C₄×C₁₃	C₂×#125	1	2	yes	—
166	C₄×C₁₃	C₂×#126	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂³×C₄	D₄×C₁₃	1	2	yes	—
169	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
174	#117	—	2	2	no	—
175	#117	—	2	4	no	—
176	#117	Q₈×C₁₃	1	4	yes	—
177	C₂×C₄⋊C₄	—	2	8	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	4	8	no	—
184	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	1	no	—
188	C₂×C₂²⋊C₄	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	2	4	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	D₄×C₁₃	1	2	yes	—
193	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₂×C₁₃	—	2	4	no	—
198	C₂×C₁₃	C₂×#400	1	1	yes	characteristic
199	C₂×C₁₃	#3079	1	2	yes	—
200	C₂×C₁₃	#3092	1	2	yes	—
201	C₂×C₁₃	#3138	1	1	yes	characteristic
202	C₂×C₁₃	#3139	1	1	yes	characteristic
203	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
216	C₂²×C₄	C₁₃×#41	1	2	yes	—
217	C₂²×C₄	C₁₃×#41	1	4	yes	—
218	C₂²×C₄	C₁₃×#41	1	4	yes	—
219	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
220	M₁₆	—	4	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	8	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	4	2	no	—
240	C₁₃	#23745	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
244	C₂³	C₁₃×#123	1	2	yes	—
245	C₂³	C₁₃×#124	1	2	yes	—
246	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₄	—	2	16	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	8	no	—
261	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
262	C₂×C₄	C₁₃×#123	1	2	yes	—
263	C₂×C₄	C₁₃×#124	1	2	yes	—
264	C₂×C₄	C₁₃×#125	1	8	yes	—
265	C₂×C₄	C₁₃×#126	1	8	yes	—
266	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
267	C₂×C₄	C₁₃×#137	1	4	yes	—
268	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
270	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
271	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
272	C₈	—	4	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	C₁₃×#400	1	4	yes	—
278	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂²	C₂×C₁₃×#123	1	1	yes	characteristic
280	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
281	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
282	C₂²	C₂×C₁₃×#137	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	2	4	no	—
285	C₄	—	4	4	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	4	no	—
288	C₄	C₁₃×#400	1	4	yes	—
289	C₄	C₂×C₁₃×#125	1	2	yes	—
290	C₄	C₂×C₁₃×#126	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	C₂×C₁₃×#400	1	1	yes	characteristic
293	C₂	C₁₃×#3079	1	2	yes	—
294	C₂	C₁₃×#3092	1	2	yes	—
295	C₂	C₁₃×#3138	1	1	yes	characteristic
296	C₂	C₁₃×#3139	1	1	yes	characteristic

	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 ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2100	C₂	semidirect
4. C₁₃×#2099	C₂	semidirect
55. #23745	C₁₃	direct (N × H)
95. #2100	C₂×C₁₃	semidirect
96. #2099	C₂×C₁₃	semidirect
240. C₁₃	#23745	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×#2296	2
C₂×C₁₃×#384	2
C₁₃×#2100	2
C₁₃×#2099	2
C₁₃×#1736	2
C₁₃×#1728	2
#23745	13

Subgroups — C13×#23745 #15710084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23745 #15710084