Subgroups — C₁₃×#21276 #15707615

Order 3328.

822 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	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2295	C₂	1	2	yes	maximal
3	C₁₃×#1969	C₂	1	2	yes	maximal
4	C₁₃×#1942	C₂	1	2	yes	maximal
5	C₁₃×#1933	C₂	1	4	yes	maximal
6	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₃×#103	C₂	1	2	yes	maximal
8	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#103	C₂²	1	4	yes	—
15	C₂×C₁₃×#103	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#391	C₂²	1	4	yes	—
21	C₁₃×#389	C₂²	1	4	yes	—
22	C₁₃×#387	—	2	4	no	—
23	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₁₃×#376	—	2	2	no	—
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₁₃×Q₁₆	—	2	4	no	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	—	2	4	no	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₃	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×#107	—	2	4	no	—
49	C₁₃×#107	C₂×C₄	1	4	yes	—
50	C₁₃×#103	—	2	8	no	—
51	C₁₃×#103	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₁₃	—	2	4	no	—
53	C₄×C₈×C₁₃	—	2	4	no	—
54	C₄×C₈×C₁₃	—	2	2	no	—
55	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
56	C₁₃×#95	—	2	4	no	—
57	C₁₃×#95	—	2	2	no	—
58	C₁₃×#95	—	2	4	no	—
59	C₁₃×#95	—	2	4	no	—
60	#21276	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	—	2	4	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×Q₁₆	—	2	8	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	#41	1	4	yes	—
80	C₂×C₈×C₁₃	#41	1	4	yes	—
81	C₁₃×C₄⋊C₄	—	2	8	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	4	8	no	—
86	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
87	C₄²×C₁₃	—	2	8	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	4	4	no	—
93	C₄²×C₁₃	Q₁₆	1	4	yes	complemented
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	#41	1	2	yes	—
96	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
97	#2295	C₂×C₁₃	1	2	yes	—
98	#1969	C₂×C₁₃	1	2	yes	—
99	#1942	C₂×C₁₃	1	2	yes	—
100	#1933	C₂×C₁₃	1	4	yes	—
101	C₄×#107	C₂×C₁₃	1	1	yes	characteristic
102	C₄×#103	C₂×C₁₃	1	2	yes	—
103	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₁₃	—	2	4	no	—
106	Q₈×C₁₃	—	2	4	no	—
107	Q₈×C₁₃	—	4	4	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	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
123	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₃	#135	1	4	yes	—
125	C₂×C₄×C₁₃	#135	1	2	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	—	2	4	no	—
132	C₈×C₁₃	#117	1	4	yes	—
133	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
135	C₄×Q₁₆	C₄×C₁₃	1	8	yes	complemented
136	C₂×#107	C₂²×C₁₃	1	2	yes	—
137	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×#103	C₂²×C₁₃	1	4	yes	—
139	C₂×#103	C₂²×C₁₃	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
144	#391	C₂²×C₁₃	1	4	yes	—
145	#389	C₂²×C₁₃	1	4	yes	—
146	#387	—	2	4	no	—
147	C₄×C₄⋊C₄	—	2	2	no	—
148	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
149	#376	—	2	2	no	—
150	C₄³	C₂²×C₁₃	1	1	yes	characteristic
151	#340	C₂²×C₁₃	1	2	yes	—
152	#340	C₂²×C₁₃	1	4	yes	—
153	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂²×C₁₃	#532	1	2	yes	—
155	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#135	1	2	yes	—
158	C₄×C₁₃	—	2	4	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	—	8	4	no	—
164	C₄×C₁₃	C₄×Q₁₆	1	4	yes	—
165	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
166	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
167	C₄×C₁₃	#529	1	2	yes	—
168	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
171	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
173	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₄×Q₈	—	2	4	no	—
175	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₁₃	1	2	yes	—
184	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
186	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
187	#107	—	2	4	no	—
188	#107	C₂×C₄×C₁₃	1	4	yes	—
189	#103	—	2	8	no	—
190	#103	C₂×C₄×C₁₃	1	8	yes	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
195	#95	—	2	4	no	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	#95	—	2	2	no	—
199	C₂×C₁₃	#2293	1	1	yes	characteristic
200	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
201	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
202	C₂×C₁₃	#3168	1	2	yes	—
203	C₂×C₁₃	#3170	1	2	yes	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
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	C₂⁴×C₁₃	1	1	yes	characteristic
216	Q₁₆	—	2	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
221	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
222	C₂×C₈	C₁₃×#41	1	4	yes	—
223	C₂×C₈	C₁₃×#41	1	4	yes	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	8	no	—
229	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
230	C₄²	—	2	8	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	4	4	no	—
236	C₄²	C₁₃×Q₁₆	1	4	yes	complemented
237	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
238	C₄²	C₁₃×#41	1	2	yes	—
239	C₁₃	#21276	1	1	yes	characteristic, direct product factor, Sylow
240	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
257	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
258	C₂×C₄	C₂×C₁₃×Q₁₆	1	4	yes	—
259	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
260	C₂×C₄	C₁₃×#135	1	4	yes	—
261	C₂×C₄	C₁₃×#135	1	2	yes	—
262	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
264	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
265	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	4	no	—
268	C₈	C₁₃×#117	1	4	yes	—
269	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
270	C₂²	C₁₃×#532	1	2	yes	—
271	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
272	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
273	C₂²	C₂×C₁₃×#135	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	4	no	—
280	C₄	C₄×C₁₃×Q₁₆	1	4	yes	—
281	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
282	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
283	C₄	C₁₃×#529	1	2	yes	—
284	C₂	C₁₃×#2293	1	1	yes	characteristic
285	C₂	C₂×C₄×C₁₃×Q₁₆	1	1	yes	characteristic
286	C₂	C₄×C₁₃×#135	1	1	yes	characteristic
287	C₂	C₁₃×#3168	1	2	yes	—
288	C₂	C₁₃×#3170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₂×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
11. C₄×C₁₃×Q₁₆	C₄	semidirect
60. #21276	C₁₃	direct (N × H)
93. C₄²×C₁₃	Q₁₆	semidirect
135. C₄×Q₁₆	C₄×C₁₃	semidirect
236. C₄²	C₁₃×Q₁₆	semidirect
239. C₁₃	#21276	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×C₄×C₁₃×Q₁₆	2
C₁₃×#2295	2
C₁₃×#1969	2
C₁₃×#1942	2
C₁₃×#1933	2
C₄×C₁₃×#107	2
C₄×C₁₃×#103	2
C₄²×C₈×C₁₃	2
#21276	13

Subgroups — C13×#21276 #15707615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21276 #15707615