Subgroups — C₁₃×#10274 #15696613

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 274 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, complemented
2	C₂×C₁₃×#474	C₂	1	2	yes	maximal
3	C₁₃×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#332	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×Q₁₆	—	2	2	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#494	—	2	4	no	—
10	C₁₃×#474	—	2	4	no	—
11	C₁₃×#474	—	2	4	no	—
12	C₂×C₁₃×#107	—	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×#102	—	2	2	no	—
16	C₂×C₁₃×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	—	2	8	no	—
23	C₂×C₁₃×Q₁₆	—	4	4	no	—
24	C₂×C₁₃×Q₁₆	—	4	4	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₁₃×C₄⋊Q₈	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×C₄⋊Q₈	—	2	2	no	—
31	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₄×D₄×C₁₃	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#107	—	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×#102	—	2	8	no	—
44	C₁₃×C₂²⋊C₈	—	4	4	no	—
45	C₄×C₈×C₁₃	—	2	2	no	—
46	#10274	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂⁴×C₁₃	—	2	1	no	—
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	4	no	—
51	C₂×Q₈×C₁₃	—	4	4	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	2	1	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	Q₁₆	1	2	yes	complemented
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	—	4	1	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₁₃×Q₁₆	—	4	16	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	D₈	1	4	yes	—
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₂²⋊C₄	—	4	2	no	—
80	C₄²×C₁₃	C₂×D₄	1	2	yes	—
81	C₄²×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
83	C₂×#474	C₂×C₁₃	1	2	yes	—
84	#1954	C₂×C₁₃	1	1	yes	characteristic
85	C₂×#332	C₂×C₁₃	1	2	yes	complemented
86	#1277	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	4	1	no	—
89	C₂³×C₁₃	—	4	2	no	—
90	C₂³×C₁₃	—	4	1	no	—
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	#121	1	1	yes	characteristic
93	Q₈×C₁₃	—	4	8	no	—
94	Q₈×C₁₃	—	8	2	no	—
95	Q₈×C₁₃	—	8	2	no	—
96	D₄×C₁₃	—	2	4	no	—
97	D₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	4	1	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	8	4	no	—
103	C₂×C₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	—	8	1	no	—
105	C₂×C₄×C₁₃	—	8	1	no	—
106	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#120	1	2	yes	—
108	C₂×C₄×C₁₃	#121	1	2	yes	—
109	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
113	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₁₃	#137	1	2	yes	—
115	C₈×C₁₃	—	2	4	no	—
116	C₈×C₁₃	—	4	4	no	—
117	C₈×C₁₃	—	8	1	no	—
118	C₂²×Q₁₆	—	2	2	no	—
119	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
120	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
121	#494	—	2	4	no	—
122	#474	—	2	4	no	—
123	#474	—	2	4	no	—
124	C₂×#107	—	2	2	no	—
125	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
127	C₂×#102	—	2	2	no	—
128	C₂×C₂²⋊C₈	—	2	2	no	—
129	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	#332	—	2	4	no	—
131	C₂²×C₁₃	—	4	4	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	—	8	2	no	—
134	C₂²×C₁₃	—	8	1	no	—
135	C₂²×C₁₃	—	8	1	no	—
136	C₂²×C₁₃	#392	1	1	yes	characteristic
137	C₂²×C₁₃	#459	1	2	yes	—
138	C₂²×C₁₃	#467	1	1	yes	characteristic
139	C₂²×C₁₃	#470	1	1	yes	characteristic
140	C₂²×C₁₃	#473	1	2	yes	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	8	1	no	—
143	C₄×C₁₃	—	8	4	no	—
144	C₄×C₁₃	#447	1	2	yes	—
145	C₄×C₁₃	#451	1	2	yes	—
146	C₄×C₁₃	#452	1	2	yes	—
147	C₄×C₁₃	#493	1	1	yes	characteristic
148	C₄×C₁₃	#496	1	1	yes	characteristic
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂×Q₁₆	—	2	8	no	—
153	C₂×Q₁₆	—	4	4	no	—
154	C₂×Q₁₆	—	4	4	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	D₄×C₁₃	1	2	yes	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	D₄×C₁₃	1	4	yes	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
169	#107	—	2	8	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
173	#102	—	2	8	no	—
174	C₂²⋊C₈	—	4	4	no	—
175	C₄×C₈	—	2	2	no	—
176	C₂×C₁₃	—	8	2	no	—
177	C₂×C₁₃	#1669	1	2	yes	—
178	C₂×C₁₃	#1689	1	2	yes	—
179	C₂×C₁₃	#2001	1	1	yes	characteristic
180	C₂×C₁₃	#2011	1	1	yes	characteristic
181	C₂×C₁₃	#2020	1	1	yes	characteristic
182	C₂⁴	—	2	1	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	C₁₃×Q₁₆	1	2	yes	complemented
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
200	Q₁₆	—	4	16	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₁₃×D₈	1	4	yes	—
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	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	C₂×D₄×C₁₃	1	2	yes	—
216	C₄²	C₂×D₄×C₁₃	1	2	yes	—
217	C₁₃	#10274	1	1	yes	characteristic, direct product factor, Sylow
218	C₂³	—	2	2	no	—
219	C₂³	—	4	1	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₁₃×#121	1	1	yes	characteristic
224	Q₈	—	4	8	no	—
225	Q₈	—	8	2	no	—
226	Q₈	—	8	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
238	C₂×C₄	C₁₃×#120	1	2	yes	—
239	C₂×C₄	C₁₃×#121	1	2	yes	—
240	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
241	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
242	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
243	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
244	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
245	C₂×C₄	C₁₃×#137	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	4	4	no	—
248	C₈	—	8	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	8	1	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	8	2	no	—
254	C₂²	C₁₃×#392	1	1	yes	characteristic
255	C₂²	C₁₃×#459	1	2	yes	—
256	C₂²	C₁₃×#467	1	1	yes	characteristic
257	C₂²	C₁₃×#470	1	1	yes	characteristic
258	C₂²	C₁₃×#473	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	8	1	no	—
261	C₄	—	8	4	no	—
262	C₄	C₁₃×#447	1	2	yes	—
263	C₄	C₁₃×#451	1	2	yes	—
264	C₄	C₁₃×#452	1	2	yes	—
265	C₄	C₁₃×#493	1	1	yes	characteristic
266	C₄	C₁₃×#496	1	1	yes	characteristic
267	C₂	—	8	2	no	—
268	C₂	C₁₃×#1669	1	2	yes	—
269	C₂	C₁₃×#1689	1	2	yes	—
270	C₂	C₁₃×#2001	1	1	yes	characteristic
271	C₂	C₁₃×#2011	1	1	yes	characteristic
272	C₂	C₁₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂²×D₄×C₁₃	D₄
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
1. C₂×C₁₃×#494	C₂	semidirect
4. C₂×C₁₃×#332	C₂	semidirect
5. C₁₃×#1277	C₂	semidirect
46. #10274	C₁₃	direct (N × H)
55. C₂×D₄×C₁₃	Q₁₆	semidirect
82. C₂×#494	C₂×C₁₃	semidirect
85. C₂×#332	C₂×C₁₃	semidirect
86. #1277	C₂×C₁₃	semidirect
190. C₂×D₄	C₁₃×Q₁₆	semidirect
217. C₁₃	#10274	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×#474	2
C₁₃×#1954	2
C₂×C₁₃×#332	2
C₁₃×#1277	2
#10274	13

Subgroups — C13×#10274 #15696613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10274 #15696613