Subgroups — C₁₃×#32670 #15719009

Order 3328.

712 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2568	C₂	1	2	yes	maximal
2	C₁₃×#2056	C₂	1	2	yes	maximal
3	C₁₃×#2016	C₂	1	4	yes	maximal
4	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1969	C₂	1	2	yes	maximal
6	C₁₃×#1928	C₂	1	4	yes	maximal
7	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#103	C₂²	1	4	yes	—
14	C₂×C₁₃×#103	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#400	—	2	2	no	—
17	C₁₃×#400	—	2	2	no	—
18	C₁₃×#391	—	2	2	no	—
19	C₁₃×#391	C₂²	1	4	yes	—
20	C₁₃×#391	C₂²	1	4	yes	—
21	C₁₃×#382	—	2	2	no	—
22	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₁₃×#340	C₂²	1	2	yes	—
24	C₁₃×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
27	C₂×C₁₃×Q₁₆	—	2	4	no	—
28	C₂×C₁₃×Q₁₆	—	4	2	no	—
29	C₂×C₁₃×Q₁₆	—	4	2	no	—
30	C₂×C₁₃×Q₁₆	—	4	4	no	—
31	C₂×C₁₃×Q₁₆	—	4	2	no	—
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₁₃×C₄⋊Q₈	—	4	2	no	—
35	C₁₃×C₄⋊Q₈	—	4	2	no	—
36	C₄×Q₈×C₁₃	—	4	1	no	—
37	C₄×Q₈×C₁₃	—	4	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₃	—	2	2	no	—
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#107	—	4	2	no	—
47	C₁₃×C₄⋊C₈	—	4	1	no	—
48	C₁₃×#103	—	2	8	no	—
49	C₁₃×#103	—	4	4	no	—
50	C₄×C₈×C₁₃	—	4	1	no	—
51	C₁₃×#95	—	2	2	no	—
52	C₁₃×#95	—	2	4	no	—
53	C₁₃×#95	—	2	4	no	—
54	C₁₃×#95	—	2	4	no	—
55	C₁₃×#95	—	2	4	no	—
56	#32670	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₃	—	2	4	no	—
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	—	2	4	no	—
63	C₂×Q₈×C₁₃	—	4	1	no	—
64	C₂×Q₈×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	complemented
66	C₂²×C₄×C₁₃	—	2	4	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	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×Q₁₆	—	4	8	no	—
74	C₁₃×Q₁₆	—	4	8	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	#41	1	4	yes	—
80	C₁₃×C₄⋊C₄	—	2	4	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	C₁₃×C₄⋊C₄	—	4	4	no	—
85	C₁₃×C₄⋊C₄	—	4	4	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	—	2	1	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	4	4	no	—
91	#2568	C₂×C₁₃	1	2	yes	—
92	#2056	C₂×C₁₃	1	2	yes	—
93	#2016	C₂×C₁₃	1	4	yes	—
94	#1984	C₂×C₁₃	1	1	yes	characteristic
95	#1969	C₂×C₁₃	1	2	yes	—
96	#1928	C₂×C₁₃	1	4	yes	—
97	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₁₃	—	4	4	no	—
99	Q₈×C₁₃	—	4	4	no	—
100	Q₈×C₁₃	—	4	4	no	—
101	Q₈×C₁₃	—	4	4	no	—
102	Q₈×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
119	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₈×C₁₃	—	2	4	no	—
124	C₈×C₁₃	—	4	2	no	—
125	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
126	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
127	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×#107	C₂²×C₁₃	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×#103	C₂²×C₁₃	1	4	yes	—
132	C₂×#103	C₂²×C₁₃	1	4	yes	—
133	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
134	#400	—	2	2	no	—
135	#400	—	2	2	no	—
136	#391	—	2	2	no	—
137	#391	C₂²×C₁₃	1	4	yes	—
138	#391	C₂²×C₁₃	1	4	yes	—
139	#382	—	2	2	no	—
140	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
141	#340	C₂²×C₁₃	1	2	yes	—
142	#336	C₂²×C₁₃	1	4	yes	—
143	C₂²×C₁₃	#534	1	1	yes	characteristic
144	C₂²×C₁₃	#550	1	2	yes	—
145	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
146	C₂²×C₁₃	#575	1	1	yes	characteristic
147	C₂²×C₁₃	#578	1	2	yes	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	—	4	1	no	—
152	C₄×C₁₃	—	8	4	no	—
153	C₄×C₁₃	—	8	4	no	—
154	C₄×C₁₃	#538	1	2	yes	—
155	C₄×C₁₃	#555	1	1	yes	characteristic
156	C₄×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
157	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
158	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	4	2	no	—
161	C₂×Q₁₆	—	4	2	no	—
162	C₂×Q₁₆	—	4	4	no	—
163	C₂×Q₁₆	—	4	2	no	—
164	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
165	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
166	C₄⋊Q₈	—	4	2	no	—
167	C₄⋊Q₈	—	4	2	no	—
168	C₄×Q₈	—	4	1	no	—
169	C₄×Q₈	—	4	1	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
173	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
178	#107	—	4	2	no	—
179	C₄⋊C₈	—	4	1	no	—
180	#103	—	2	8	no	—
181	#103	—	4	4	no	—
182	C₄×C₈	—	4	1	no	—
183	#95	—	2	2	no	—
184	#95	—	2	4	no	—
185	#95	—	2	4	no	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	C₂×C₁₃	#2733	1	1	yes	characteristic
189	C₂×C₁₃	#3189	1	1	yes	characteristic
190	C₂×C₁₃	#3202	1	1	yes	characteristic
191	C₂×C₁₃	#3395	1	2	yes	—
192	C₂×C₁₃	#3402	1	2	yes	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	1	no	—
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	C₁₃×Q₁₆	1	4	yes	complemented
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
208	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
209	Q₁₆	—	4	8	no	—
210	Q₁₆	—	4	8	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₁₃×#41	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	4	no	—
227	C₁₃	#32670	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	4	4	no	—
232	Q₈	—	4	4	no	—
233	Q₈	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
249	C₂×C₄	C₂×C₁₃×Q₁₆	1	4	yes	—
250	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
251	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
253	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
254	C₈	—	2	4	no	—
255	C₈	—	4	2	no	—
256	C₂²	C₁₃×#534	1	1	yes	characteristic
257	C₂²	C₁₃×#550	1	2	yes	—
258	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
259	C₂²	C₁₃×#575	1	1	yes	characteristic
260	C₂²	C₁₃×#578	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	4	no	—
266	C₄	—	8	4	no	—
267	C₄	C₁₃×#538	1	2	yes	—
268	C₄	C₁₃×#555	1	1	yes	characteristic
269	C₄	C₁₃×H(3,𝔽₄)	1	1	yes	characteristic
270	C₂	C₁₃×#2733	1	1	yes	characteristic
271	C₂	C₁₃×#3189	1	1	yes	characteristic
272	C₂	C₁₃×#3202	1	1	yes	characteristic
273	C₂	C₁₃×#3395	1	2	yes	—
274	C₂	C₁₃×#3402	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 ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
56. #32670	C₁₃	direct (N × H)
65. C₂×Q₈×C₁₃	Q₁₆	semidirect
201. C₂×Q₈	C₁₃×Q₁₆	semidirect
227. C₁₃	#32670	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×#2568	2
C₁₃×#2056	2
C₁₃×#2016	2
C₁₃×#1984	2
C₁₃×#1969	2
C₁₃×#1928	2
#32670	13

Subgroups — C13×#32670 #15719009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32670 #15719009