Subgroups — C₁₃×#23845 #15710184

Order 3328.

654 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2331	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2330	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2085	C₂	1	2	yes	maximal
4	C₁₃×#2084	C₂	1	2	yes	maximal
5	C₁₃×#2059	C₂	1	2	yes	maximal
6	C₁₃×#2058	C₂	1	2	yes	maximal
7	C₁₃×#1776	C₂	1	2	yes	maximal
8	C₄×C₁₃×#106	C₂	1	2	yes	maximal
9	C₁₃×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#391	—	2	2	no	—
19	C₁₃×#389	—	2	2	no	—
20	C₁₃×#389	C₂²	1	2	yes	—
21	C₁₃×#389	C₂²	1	2	yes	—
22	C₁₃×#387	—	2	2	no	—
23	C₁₃×#387	—	2	2	no	—
24	C₁₃×#384	C₂²	1	1	yes	characteristic
25	C₁₃×#382	C₂²	1	1	yes	characteristic
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₁₃×#376	C₂²	1	1	yes	characteristic
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	—	2	1	no	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₁₃×C₄⋊Q₈	—	2	4	no	—
39	C₁₃×#125	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#107	—	2	4	no	—
55	C₁₃×#106	—	2	4	no	—
56	C₁₃×#106	—	2	4	no	—
57	C₁₃×#106	—	2	4	no	—
58	C₁₃×#97	Q₈	1	4	yes	complemented
59	C₄×C₈×C₁₃	Q₈	1	4	yes	complemented
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	1	no	—
63	C₁₃×#95	—	2	1	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	D₄	1	2	yes	—
67	C₁₃×#95	D₄	1	2	yes	—
68	#23845	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×Q₈×C₁₃	—	2	4	no	—
70	C₂×Q₈×C₁₃	—	8	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
86	C₁₃×C₄⋊C₄	—	2	8	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	8	2	no	—
95	C₁₃×C₄⋊C₄	—	8	2	no	—
96	C₁₃×C₄⋊C₄	—	8	2	no	—
97	C₁₃×C₄⋊C₄	#41	1	4	yes	—
98	C₁₃×C₄⋊C₄	#41	1	4	yes	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	8	2	no	—
102	C₄²×C₁₃	—	8	2	no	—
103	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
104	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
105	#2331	C₂×C₁₃	1	1	yes	characteristic
106	#2330	C₂×C₁₃	1	1	yes	characteristic
107	#2085	C₂×C₁₃	1	2	yes	—
108	#2084	C₂×C₁₃	1	2	yes	—
109	#2059	C₂×C₁₃	1	2	yes	—
110	#2058	C₂×C₁₃	1	2	yes	—
111	#1776	C₂×C₁₃	1	2	yes	—
112	C₄×#106	C₂×C₁₃	1	2	yes	—
113	#1746	C₂×C₁₃	1	1	yes	characteristic
114	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₁₃	—	8	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₈×C₁₃	—	2	4	no	—
143	C₈×C₁₃	—	2	4	no	—
144	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×#107	C₂²×C₁₃	1	2	yes	—
147	C₂×#106	C₂²×C₁₃	1	2	yes	—
148	C₂×#106	C₂²×C₁₃	1	2	yes	—
149	C₂×#106	C₂²×C₁₃	1	2	yes	—
150	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
152	#391	—	2	2	no	—
153	#389	—	2	2	no	—
154	#389	C₂²×C₁₃	1	2	yes	—
155	#389	C₂²×C₁₃	1	2	yes	—
156	#387	—	2	2	no	—
157	#387	—	2	2	no	—
158	#384	C₂²×C₁₃	1	1	yes	characteristic
159	#382	C₂²×C₁₃	1	1	yes	characteristic
160	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
161	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
162	#376	C₂²×C₁₃	1	1	yes	characteristic
163	#340	C₂²×C₁₃	1	2	yes	—
164	#340	C₂²×C₁₃	1	2	yes	—
165	#340	C₂²×C₁₃	1	2	yes	—
166	#340	C₂²×C₁₃	1	2	yes	—
167	#336	C₂²×C₁₃	1	2	yes	—
168	#336	C₂²×C₁₃	1	2	yes	—
169	C₂²×C₁₃	#525	1	1	yes	characteristic
170	C₂²×C₁₃	#525	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
172	C₂²×C₁₃	#577	1	2	yes	—
173	C₂²×C₁₃	#577	1	2	yes	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	—	4	1	no	—
176	C₄×C₁₃	—	4	4	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	—	4	1	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	#533	1	2	yes	—
183	C₄×C₁₃	#533	1	2	yes	—
184	C₂²×Q₈	—	2	1	no	—
185	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
187	C₄⋊Q₈	—	2	4	no	—
188	#125	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	#107	—	2	4	no	—
204	#106	—	2	4	no	—
205	#106	—	2	4	no	—
206	#106	—	2	4	no	—
207	#97	Q₈×C₁₃	1	4	yes	complemented
208	C₄×C₈	Q₈×C₁₃	1	4	yes	complemented
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	D₄×C₁₃	1	2	yes	—
216	#95	D₄×C₁₃	1	2	yes	—
217	C₂×C₁₃	#2399	1	1	yes	characteristic
218	C₂×C₁₃	#3123	1	1	yes	characteristic
219	C₂×C₁₃	#3125	1	2	yes	—
220	C₂×C₁₃	#3126	1	1	yes	characteristic
221	C₂×C₁₃	#3163	1	2	yes	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	8	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
238	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	8	2	no	—
248	C₄⋊C₄	—	8	2	no	—
249	C₄⋊C₄	—	8	2	no	—
250	C₄⋊C₄	C₁₃×#41	1	4	yes	—
251	C₄⋊C₄	C₁₃×#41	1	4	yes	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	8	2	no	—
255	C₄²	—	8	2	no	—
256	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
257	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
258	C₁₃	#23845	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
260	Q₈	—	8	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₁₃×C₄⋊Q₈	1	4	yes	—
283	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
286	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
287	C₈	—	2	4	no	—
288	C₈	—	2	4	no	—
289	C₂²	C₁₃×#525	1	1	yes	characteristic
290	C₂²	C₁₃×#525	1	1	yes	characteristic
291	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
292	C₂²	C₁₃×#577	1	2	yes	—
293	C₂²	C₁₃×#577	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	4	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₁₃×#533	1	2	yes	—
303	C₄	C₁₃×#533	1	2	yes	—
304	C₂	C₁₃×#2399	1	1	yes	characteristic
305	C₂	C₁₃×#3123	1	1	yes	characteristic
306	C₂	C₁₃×#3125	1	2	yes	—
307	C₂	C₁₃×#3126	1	1	yes	characteristic
308	C₂	C₁₃×#3163	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
58. C₁₃×#97	Q₈	semidirect
59. C₄×C₈×C₁₃	Q₈	semidirect
68. #23845	C₁₃	direct (N × H)
207. #97	Q₈×C₁₃	semidirect
208. C₄×C₈	Q₈×C₁₃	semidirect
258. C₁₃	#23845	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×#2331	2
C₁₃×#2330	2
C₁₃×#2085	2
C₁₃×#2084	2
C₁₃×#2059	2
C₁₃×#2058	2
C₁₃×#1776	2
C₄×C₁₃×#106	2
C₁₃×#1746	2
#23845	13

Subgroups — C13×#23845 #15710184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23845 #15710184