Subgroups — C₁₃×#9874 #15696213

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1279	C₂	1	4	yes	maximal
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
7	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
11	C₁₃×#391	—	2	1	no	—
12	C₁₃×#389	—	2	1	no	—
13	C₁₃×#389	C₄	1	2	yes	—
14	C₁₃×#384	C₄	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
16	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#336	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	4	no	—
24	C₄×Q₈×C₁₃	—	4	2	no	—
25	C₄×Q₈×C₁₃	—	4	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	—	2	4	no	—
41	C₁₃×C₄⋊C₈	—	4	2	no	—
42	C₁₃×C₄⋊C₈	—	4	2	no	—
43	C₄×C₈×C₁₃	—	2	2	no	—
44	C₄×C₈×C₁₃	Q₈	1	4	yes	complemented
45	C₁₃×#95	—	2	2	no	—
46	C₁₃×#95	—	2	2	no	—
47	#9874	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	4	2	no	—
62	C₂×C₈×C₁₃	—	4	2	no	—
63	C₂×C₈×C₁₃	—	4	2	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	2	4	no	—
67	C₁₃×C₄⋊C₄	—	2	2	no	—
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
73	C₄²×C₁₃	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	4	1	no	—
76	C₄²×C₁₃	—	4	1	no	—
77	C₄²×C₁₃	—	4	2	no	—
78	C₄²×C₁₃	—	4	2	no	—
79	C₄²×C₁₃	D₈	1	2	yes	—
80	C₄²×C₁₃	SD₁₆	1	2	yes	—
81	C₄²×C₁₃	SD₁₆	1	2	yes	—
82	C₄²×C₁₃	Q₁₆	1	2	yes	—
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₃	#41	1	2	yes	—
85	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
86	#1842	C₂×C₁₃	1	1	yes	characteristic
87	#1744	C₂×C₁₃	1	1	yes	characteristic
88	#1279	C₂×C₁₃	1	4	yes	—
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₁₃	—	4	8	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	#102	1	2	yes	—
108	C₂×C₄×C₁₃	#102	1	2	yes	—
109	C₂×C₄×C₁₃	#103	1	2	yes	—
110	C₂×C₄×C₁₃	#103	1	2	yes	—
111	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₁₃	#131	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
121	C₈×C₁₃	—	4	4	no	—
122	C₈×C₁₃	—	8	2	no	—
123	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
124	C₂×C₄×Q₈	—	2	1	no	—
125	C₂×C₄⋊C₈	—	2	1	no	—
126	C₂×C₄⋊C₈	—	2	1	no	—
127	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
129	#391	—	2	1	no	—
130	#389	—	2	1	no	—
131	#389	C₄×C₁₃	1	2	yes	—
132	#384	C₄×C₁₃	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	2	1	no	—
134	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
135	C₄³	C₂²×C₁₃	1	1	yes	characteristic
136	#336	—	2	1	no	—
137	#336	—	2	1	no	—
138	C₂²×C₁₃	#391	1	1	yes	characteristic
139	C₂²×C₁₃	#407	1	1	yes	characteristic
140	C₂²×C₁₃	#408	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
143	C₂²×C₁₃	#421	1	1	yes	characteristic
144	C₂²×C₁₃	#421	1	1	yes	characteristic
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	4	4	no	—
151	C₄×C₁₃	—	8	2	no	—
152	C₄×C₁₃	#446	1	4	yes	—
153	C₄×C₁₃	#486	1	1	yes	characteristic
154	C₄×C₁₃	#487	1	1	yes	characteristic
155	C₄×C₁₃	#498	1	1	yes	characteristic
156	C₄×C₁₃	#500	1	1	yes	characteristic
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	4	no	—
161	C₄×Q₈	—	4	2	no	—
162	C₄×Q₈	—	4	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
172	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
173	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	—	4	2	no	—
179	C₄⋊C₈	—	4	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	Q₈×C₁₃	1	4	yes	complemented
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	C₂×C₁₃	#1480	1	1	yes	characteristic
185	C₂×C₁₃	#1481	1	1	yes	characteristic
186	C₂×C₁₃	#1486	1	1	yes	characteristic
187	C₂×C₁₃	#1487	1	1	yes	characteristic
188	C₂×C₁₃	#1985	1	1	yes	characteristic
189	C₂×C₁₃	#1990	1	1	yes	characteristic
190	C₂×C₁₃	#1995	1	1	yes	characteristic
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	C₁₃×M₁₆	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	C₁₃×D₈	1	2	yes	—
223	C₄²	C₁₃×SD₁₆	1	2	yes	—
224	C₄²	C₁₃×SD₁₆	1	2	yes	—
225	C₄²	C₁₃×Q₁₆	1	2	yes	—
226	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
227	C₄²	C₁₃×#41	1	2	yes	—
228	C₁₃	#9874	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₁₃×#102	1	2	yes	—
248	C₂×C₄	C₁₃×#102	1	2	yes	—
249	C₂×C₄	C₁₃×#103	1	2	yes	—
250	C₂×C₄	C₁₃×#103	1	2	yes	—
251	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
252	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
253	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
254	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
256	C₂×C₄	C₁₃×#131	1	2	yes	—
257	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
258	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
259	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
260	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
261	C₈	—	4	4	no	—
262	C₈	—	8	2	no	—
263	C₂²	C₁₃×#391	1	1	yes	characteristic
264	C₂²	C₁₃×#407	1	1	yes	characteristic
265	C₂²	C₁₃×#408	1	1	yes	characteristic
266	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
267	C₂²	C₂×C₁₃×#103	1	1	yes	characteristic
268	C₂²	C₁₃×#421	1	1	yes	characteristic
269	C₂²	C₁₃×#421	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₁₃×#446	1	4	yes	—
278	C₄	C₁₃×#486	1	1	yes	characteristic
279	C₄	C₁₃×#487	1	1	yes	characteristic
280	C₄	C₁₃×#498	1	1	yes	characteristic
281	C₄	C₁₃×#500	1	1	yes	characteristic
282	C₂	C₁₃×#1480	1	1	yes	characteristic
283	C₂	C₁₃×#1481	1	1	yes	characteristic
284	C₂	C₁₃×#1486	1	1	yes	characteristic
285	C₂	C₁₃×#1487	1	1	yes	characteristic
286	C₂	C₁₃×#1985	1	1	yes	characteristic
287	C₂	C₁₃×#1990	1	1	yes	characteristic
288	C₂	C₁₃×#1995	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
44. C₄×C₈×C₁₃	Q₈	semidirect
47. #9874	C₁₃	direct (N × H)
181. C₄×C₈	Q₈×C₁₃	semidirect
228. C₁₃	#9874	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#1842	2
C₁₃×#1744	2
C₁₃×#1279	2
#9874	13

Subgroups — C13×#9874 #15696213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9874 #15696213