Subgroups — C₁₃×#32682 #15719021

Order 3328.

688 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2546	C₂	1	2	yes	maximal
2	C₁₃×#2086	C₂	1	2	yes	maximal
3	C₁₃×#2057	C₂	1	4	yes	maximal
4	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1942	C₂	1	2	yes	maximal
6	C₁₃×#1918	C₂	1	4	yes	maximal
7	C₂×C₁₃×C₄⋊Q₈	—	2	2	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#107	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#398	—	2	2	no	—
16	C₁₃×#398	—	2	2	no	—
17	C₁₃×#391	—	2	2	no	—
18	C₁₃×#389	C₂²	1	4	yes	—
19	C₁₃×#389	C₂²	1	4	yes	—
20	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₁₃×#340	C₂²	1	4	yes	—
22	C₁₃×#340	C₂²	1	4	yes	—
23	C₁₃×#340	C₂²	1	2	yes	—
24	C₁₃×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₃	—	2	2	no	—
26	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₁₃×C₄⋊Q₈	—	4	2	no	—
30	C₁₃×C₄⋊Q₈	—	4	2	no	—
31	C₁₃×C₄⋊Q₈	—	4	8	no	—
32	C₄×Q₈×C₁₃	—	4	1	no	—
33	C₄×Q₈×C₁₃	—	4	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#107	—	4	2	no	—
45	C₁₃×#107	—	4	2	no	—
46	C₁₃×#107	Q₈	1	8	yes	complemented
47	C₁₃×C₄⋊C₈	—	4	1	no	—
48	C₄×C₈×C₁₃	—	4	1	no	—
49	C₁₃×#95	—	2	2	no	—
50	C₁₃×#95	—	2	4	no	—
51	C₁₃×#95	—	2	4	no	—
52	#32682	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	1	no	—
55	C₂×Q₈×C₁₃	—	4	4	no	—
56	C₂×Q₈×C₁₃	—	4	4	no	—
57	C₂×Q₈×C₁₃	—	4	4	no	—
58	C₂×Q₈×C₁₃	D₈	1	4	yes	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	2	4	no	—
74	C₁₃×C₄⋊C₄	—	4	1	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₄⋊C₄	—	4	4	no	—
77	C₁₃×C₄⋊C₄	—	4	8	no	—
78	C₁₃×C₄⋊C₄	—	4	8	no	—
79	C₁₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	—	2	1	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	4	4	no	—
85	#2546	C₂×C₁₃	1	2	yes	—
86	#2086	C₂×C₁₃	1	2	yes	—
87	#2057	C₂×C₁₃	1	4	yes	—
88	#1984	C₂×C₁₃	1	1	yes	characteristic
89	#1942	C₂×C₁₃	1	2	yes	—
90	#1918	C₂×C₁₃	1	4	yes	—
91	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₁₃	—	4	4	no	—
93	Q₈×C₁₃	—	8	8	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
109	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
111	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
114	C₈×C₁₃	—	2	4	no	—
115	C₈×C₁₃	—	4	2	no	—
116	C₂×C₄⋊Q₈	—	2	2	no	—
117	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
118	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
119	C₂×#107	C₂²×C₁₃	1	2	yes	—
120	C₂×#107	C₂²×C₁₃	1	2	yes	—
121	C₂×#107	C₂²×C₁₃	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
124	#398	—	2	2	no	—
125	#398	—	2	2	no	—
126	#391	—	2	2	no	—
127	#389	C₂²×C₁₃	1	4	yes	—
128	#389	C₂²×C₁₃	1	4	yes	—
129	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
130	#340	C₂²×C₁₃	1	4	yes	—
131	#340	C₂²×C₁₃	1	4	yes	—
132	#340	C₂²×C₁₃	1	2	yes	—
133	#336	C₂²×C₁₃	1	4	yes	—
134	C₂²×C₁₃	#534	1	1	yes	characteristic
135	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
136	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
137	C₂²×C₁₃	#575	1	1	yes	characteristic
138	C₂²×C₁₃	#578	1	2	yes	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	4	8	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	8	4	no	—
145	C₄×C₁₃	#544	1	2	yes	—
146	C₄×C₁₃	Q₈²	1	1	yes	characteristic
147	C₄×C₁₃	#563	1	1	yes	characteristic
148	C₂²×Q₈	—	2	2	no	—
149	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
151	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
152	C₄⋊Q₈	—	4	2	no	—
153	C₄⋊Q₈	—	4	2	no	—
154	C₄⋊Q₈	—	4	8	no	—
155	C₄×Q₈	—	4	1	no	—
156	C₄×Q₈	—	4	1	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
161	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
163	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
166	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
167	#107	—	4	2	no	—
168	#107	—	4	2	no	—
169	#107	Q₈×C₁₃	1	8	yes	complemented
170	C₄⋊C₈	—	4	1	no	—
171	C₄×C₈	—	4	1	no	—
172	#95	—	2	2	no	—
173	#95	—	2	4	no	—
174	#95	—	2	4	no	—
175	C₂×C₁₃	#2734	1	1	yes	characteristic
176	C₂×C₁₃	#3186	1	1	yes	characteristic
177	C₂×C₁₃	#3202	1	1	yes	characteristic
178	C₂×C₁₃	Q₈×D₈	1	2	yes	—
179	C₂×C₁₃	#3390	1	2	yes	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	C₁₃×D₈	1	4	yes	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	8	no	—
205	C₄⋊C₄	—	4	8	no	—
206	C₄⋊C₄	C₂×Q₈×C₁₃	1	8	yes	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	4	no	—
212	C₁₃	#32682	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	Q₈	—	8	8	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
231	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
232	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
233	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
236	C₈	—	2	4	no	—
237	C₈	—	4	2	no	—
238	C₂²	C₁₃×#534	1	1	yes	characteristic
239	C₂²	D₄×Q₈×C₁₃	1	2	yes	—
240	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
241	C₂²	C₁₃×#575	1	1	yes	characteristic
242	C₂²	C₁₃×#578	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	8	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	8	4	no	—
249	C₄	C₁₃×#544	1	2	yes	—
250	C₄	Q₈²×C₁₃	1	1	yes	characteristic
251	C₄	C₁₃×#563	1	1	yes	characteristic
252	C₂	C₁₃×#2734	1	1	yes	characteristic
253	C₂	C₁₃×#3186	1	1	yes	characteristic
254	C₂	C₁₃×#3202	1	1	yes	characteristic
255	C₂	Q₈×C₁₃×D₈	1	2	yes	—
256	C₂	C₁₃×#3390	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
46. C₁₃×#107	Q₈	semidirect
52. #32682	C₁₃	direct (N × H)
169. #107	Q₈×C₁₃	semidirect
212. C₁₃	#32682	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×#2546	2
C₁₃×#2086	2
C₁₃×#2057	2
C₁₃×#1984	2
C₁₃×#1942	2
C₁₃×#1918	2
#32682	13

Subgroups — C13×#32682 #15719021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32682 #15719021