Subgroups — C₁₃×#10276 #15696615

Order 3328.

574 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#475	C₂	1	2	yes	maximal
3	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#332	C₂	1	2	yes	maximal
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	—	2	2	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#494	—	2	4	no	—
10	C₁₃×#475	—	2	4	no	—
11	C₁₃×#475	—	2	4	no	—
12	C₂×C₁₃×#106	—	2	2	no	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#103	—	2	2	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#336	—	2	2	no	—
18	C₁₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₁₃×Q₁₆	—	2	8	no	—
22	C₂×C₁₃×Q₁₆	—	4	4	no	—
23	C₂×C₁₃×Q₁₆	—	4	4	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	1	no	—
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₁₃×C₄⋊Q₈	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
31	C₄×Q₈×C₁₃	—	2	2	no	—
32	C₄×Q₈×C₁₃	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#106	—	2	8	no	—
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
43	C₁₃×#103	—	2	8	no	—
44	C₄×C₈×C₁₃	—	2	2	no	—
45	#10276	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×Q₈×C₁₃	—	2	1	no	—
47	C₂×Q₈×C₁₃	—	2	2	no	—
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	—	2	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	4	no	—
52	C₂×Q₈×C₁₃	—	4	4	no	—
53	C₂×Q₈×C₁₃	Q₁₆	1	2	yes	complemented
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₁₃×Q₁₆	—	4	16	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	—	4	1	no	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	2	2	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	4	2	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₄²×C₁₃	—	4	1	no	—
76	C₄²×C₁₃	—	4	1	no	—
77	C₄²×C₁₃	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×#494	C₂×C₁₃	1	1	yes	characteristic
80	C₂×#475	C₂×C₁₃	1	2	yes	—
81	#1984	C₂×C₁₃	1	1	yes	characteristic
82	C₂×#332	C₂×C₁₃	1	2	yes	—
83	#1277	C₂×C₁₃	1	1	yes	characteristic
84	C₂³×C₁₃	#121	1	1	yes	characteristic
85	Q₈×C₁₃	—	2	4	no	—
86	Q₈×C₁₃	—	4	2	no	—
87	Q₈×C₁₃	—	4	8	no	—
88	Q₈×C₁₃	—	8	2	no	—
89	Q₈×C₁₃	—	8	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	1	no	—
93	C₂×C₄×C₁₃	—	4	1	no	—
94	C₂×C₄×C₁₃	—	4	1	no	—
95	C₂×C₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	8	4	no	—
98	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#120	1	2	yes	—
100	C₂×C₄×C₁₃	#121	1	2	yes	—
101	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
105	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₁₃	#137	1	2	yes	—
107	C₈×C₁₃	—	2	4	no	—
108	C₈×C₁₃	—	4	4	no	—
109	C₈×C₁₃	—	8	1	no	—
110	C₂²×Q₁₆	—	2	2	no	—
111	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
112	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
113	#494	—	2	4	no	—
114	#475	—	2	4	no	—
115	#475	—	2	4	no	—
116	C₂×#106	—	2	2	no	—
117	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
118	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
119	C₂×#103	—	2	2	no	—
120	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
121	#336	—	2	2	no	—
122	#332	—	2	4	no	—
123	C₂²×C₁₃	#392	1	1	yes	characteristic
124	C₂²×C₁₃	#460	1	2	yes	—
125	C₂²×C₁₃	#467	1	1	yes	characteristic
126	C₂²×C₁₃	#470	1	1	yes	characteristic
127	C₂²×C₁₃	#473	1	2	yes	—
128	C₄×C₁₃	—	2	2	no	—
129	C₄×C₁₃	—	4	2	no	—
130	C₄×C₁₃	—	8	4	no	—
131	C₄×C₁₃	—	8	1	no	—
132	C₄×C₁₃	#450	1	2	yes	—
133	C₄×C₁₃	#451	1	2	yes	—
134	C₄×C₁₃	#452	1	2	yes	—
135	C₄×C₁₃	#492	1	1	yes	characteristic
136	C₄×C₁₃	#496	1	1	yes	characteristic
137	C₂²×Q₈	—	2	2	no	—
138	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
139	C₂×Q₁₆	—	2	8	no	—
140	C₂×Q₁₆	—	4	4	no	—
141	C₂×Q₁₆	—	4	4	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	D₄×C₁₃	1	2	yes	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	D₄×C₁₃	1	4	yes	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
157	#106	—	2	8	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
161	#103	—	2	8	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂×C₁₃	#1670	1	2	yes	—
164	C₂×C₁₃	#1690	1	2	yes	—
165	C₂×C₁₃	#2001	1	1	yes	characteristic
166	C₂×C₁₃	#2014	1	1	yes	characteristic
167	C₂×C₁₃	#2020	1	1	yes	characteristic
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	C₁₃×Q₁₆	1	2	yes	complemented
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
182	Q₁₆	—	4	16	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	1	no	—
198	C₄²	—	4	1	no	—
199	C₄²	C₂×D₄×C₁₃	1	2	yes	—
200	C₄²	C₂×D₄×C₁₃	1	2	yes	—
201	C₁₃	#10276	1	1	yes	characteristic, direct product factor, Sylow
202	C₂³	C₁₃×#121	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	4	8	no	—
206	Q₈	—	8	2	no	—
207	Q₈	—	8	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
217	C₂×C₄	C₁₃×#120	1	2	yes	—
218	C₂×C₄	C₁₃×#121	1	2	yes	—
219	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
220	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
221	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
222	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
223	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
224	C₂×C₄	C₁₃×#137	1	2	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	4	4	no	—
227	C₈	—	8	1	no	—
228	C₂²	C₁₃×#392	1	1	yes	characteristic
229	C₂²	C₁₃×#460	1	2	yes	—
230	C₂²	C₁₃×#467	1	1	yes	characteristic
231	C₂²	C₁₃×#470	1	1	yes	characteristic
232	C₂²	C₁₃×#473	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	4	no	—
236	C₄	—	8	1	no	—
237	C₄	C₁₃×#450	1	2	yes	—
238	C₄	C₁₃×#451	1	2	yes	—
239	C₄	C₁₃×#452	1	2	yes	—
240	C₄	C₁₃×#492	1	1	yes	characteristic
241	C₄	C₁₃×#496	1	1	yes	characteristic
242	C₂	C₁₃×#1670	1	2	yes	—
243	C₂	C₁₃×#1690	1	2	yes	—
244	C₂	C₁₃×#2001	1	1	yes	characteristic
245	C₂	C₁₃×#2014	1	1	yes	characteristic
246	C₂	C₁₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
45. #10276	C₁₃	direct (N × H)
53. C₂×Q₈×C₁₃	Q₁₆	semidirect
175. C₂×Q₈	C₁₃×Q₁₆	semidirect
201. C₁₃	#10276	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×C₁₃×#494	2
C₂×C₁₃×#475	2
C₁₃×#1984	2
C₂×C₁₃×#332	2
C₁₃×#1277	2
#10276	13

Subgroups — C13×#10276 #15696615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10276 #15696615