Subgroups — C₁₃×#10278 #15696617

Order 3328.

562 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#476	C₂	1	2	yes	maximal
3	C₁₃×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#333	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#500	—	2	4	no	—
9	C₁₃×#476	—	2	4	no	—
10	C₁₃×#476	—	2	4	no	—
11	C₂×C₁₃×#107	—	2	2	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×#102	—	2	2	no	—
16	C₂×C₁₃×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#333	—	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	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	2	yes	—
25	C₁₃×C₄⋊Q₈	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#107	—	2	8	no	—
37	C₁₃×#106	—	2	8	no	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
41	C₁₃×#102	—	2	8	no	—
42	C₁₃×C₂²⋊C₈	—	4	4	no	—
43	C₄×C₈×C₁₃	—	2	2	no	—
44	#10278	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁴×C₁₃	—	2	1	no	—
46	C₂×Q₈×C₁₃	—	2	4	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂×Q₈×C₁₃	—	8	4	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	—	4	1	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
70	C₁₃×C₄⋊C₄	—	2	4	no	—
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₂²⋊C₄	—	4	2	no	—
76	C₄²×C₁₃	C₂×D₄	1	2	yes	—
77	C₄²×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
79	C₂×#476	C₂×C₁₃	1	2	yes	—
80	#1954	C₂×C₁₃	1	1	yes	characteristic
81	C₂×#333	C₂×C₁₃	1	2	yes	complemented
82	#1277	C₂×C₁₃	1	1	yes	characteristic, complemented
83	C₂³×C₁₃	—	2	2	no	—
84	C₂³×C₁₃	—	4	1	no	—
85	C₂³×C₁₃	—	4	2	no	—
86	C₂³×C₁₃	—	4	1	no	—
87	C₂³×C₁₃	—	4	2	no	—
88	C₂³×C₁₃	#121	1	1	yes	characteristic
89	Q₈×C₁₃	—	8	4	no	—
90	Q₈×C₁₃	—	8	4	no	—
91	D₄×C₁₃	—	2	4	no	—
92	D₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	2	2	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	—	8	2	no	—
99	C₂×C₄×C₁₃	—	8	1	no	—
100	C₂×C₄×C₁₃	—	8	1	no	—
101	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#120	1	2	yes	—
103	C₂×C₄×C₁₃	#121	1	2	yes	—
104	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#137	1	2	yes	—
109	C₂×C₄×C₁₃	#137	1	2	yes	—
110	C₈×C₁₃	—	2	4	no	—
111	C₈×C₁₃	—	4	4	no	—
112	C₈×C₁₃	—	8	1	no	—
113	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
114	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
115	#500	—	2	4	no	—
116	#476	—	2	4	no	—
117	#476	—	2	4	no	—
118	C₂×#107	—	2	2	no	—
119	C₂×#106	—	2	2	no	—
120	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
122	C₂×#102	—	2	2	no	—
123	C₂×C₂²⋊C₈	—	2	2	no	—
124	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
125	#333	—	2	4	no	—
126	C₂²×C₁₃	—	4	4	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	8	2	no	—
129	C₂²×C₁₃	—	8	1	no	—
130	C₂²×C₁₃	—	8	1	no	—
131	C₂²×C₁₃	#392	1	1	yes	characteristic
132	C₂²×C₁₃	#461	1	2	yes	—
133	C₂²×C₁₃	#467	1	1	yes	characteristic
134	C₂²×C₁₃	#470	1	1	yes	characteristic
135	C₂²×C₁₃	#472	1	2	yes	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	8	1	no	—
138	C₄×C₁₃	—	8	4	no	—
139	C₄×C₁₃	#448	1	2	yes	—
140	C₄×C₁₃	#451	1	2	yes	—
141	C₄×C₁₃	#452	1	2	yes	—
142	C₄×C₁₃	#492	1	1	yes	characteristic
143	C₄×C₁₃	#497	1	1	yes	characteristic
144	C₂²×Q₈	—	2	2	no	—
145	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
146	C₂³×C₄	—	2	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	D₄×C₁₃	1	2	yes	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	D₄×C₁₃	1	4	yes	—
154	C₄×D₄	—	2	2	no	—
155	C₄×D₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	—	2	1	no	—
160	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
161	#107	—	2	8	no	—
162	#106	—	2	8	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
166	#102	—	2	8	no	—
167	C₂²⋊C₈	—	4	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₂×C₁₃	—	8	2	no	—
170	C₂×C₁₃	#1675	1	2	yes	—
171	C₂×C₁₃	#1695	1	2	yes	—
172	C₂×C₁₃	#2001	1	1	yes	characteristic
173	C₂×C₁₃	#2013	1	1	yes	characteristic
174	C₂×C₁₃	#2018	1	1	yes	characteristic
175	C₂⁴	—	2	1	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	8	4	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	C₁₃×Q₁₆	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₄²	C₂×D₄×C₁₃	1	2	yes	—
207	C₄²	C₂×D₄×C₁₃	1	2	yes	—
208	C₁₃	#10278	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₁₃×#121	1	1	yes	characteristic
215	Q₈	—	8	4	no	—
216	Q₈	—	8	4	no	—
217	D₄	—	2	4	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
228	C₂×C₄	C₁₃×#120	1	2	yes	—
229	C₂×C₄	C₁₃×#121	1	2	yes	—
230	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
231	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
232	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
233	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
234	C₂×C₄	C₁₃×#137	1	2	yes	—
235	C₂×C₄	C₁₃×#137	1	2	yes	—
236	C₈	—	2	4	no	—
237	C₈	—	4	4	no	—
238	C₈	—	8	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	8	1	no	—
242	C₂²	—	8	1	no	—
243	C₂²	—	8	2	no	—
244	C₂²	C₁₃×#392	1	1	yes	characteristic
245	C₂²	C₁₃×#461	1	2	yes	—
246	C₂²	C₁₃×#467	1	1	yes	characteristic
247	C₂²	C₁₃×#470	1	1	yes	characteristic
248	C₂²	C₁₃×#472	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	8	1	no	—
251	C₄	—	8	4	no	—
252	C₄	C₁₃×#448	1	2	yes	—
253	C₄	C₁₃×#451	1	2	yes	—
254	C₄	C₁₃×#452	1	2	yes	—
255	C₄	C₁₃×#492	1	1	yes	characteristic
256	C₄	C₁₃×#497	1	1	yes	characteristic
257	C₂	—	8	2	no	—
258	C₂	C₁₃×#1675	1	2	yes	—
259	C₂	C₁₃×#1695	1	2	yes	—
260	C₂	C₁₃×#2001	1	1	yes	characteristic
261	C₂	C₁₃×#2013	1	1	yes	characteristic
262	C₂	C₁₃×#2018	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×#500	C₂	semidirect
4. C₂×C₁₃×#333	C₂	semidirect
5. C₁₃×#1277	C₂	semidirect
44. #10278	C₁₃	direct (N × H)
78. C₂×#500	C₂×C₁₃	semidirect
81. C₂×#333	C₂×C₁₃	semidirect
82. #1277	C₂×C₁₃	semidirect
208. C₁₃	#10278	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×#500	2
C₂×C₁₃×#476	2
C₁₃×#1954	2
C₂×C₁₃×#333	2
C₁₃×#1277	2
#10278	13

Subgroups — C13×#10278 #15696617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10278 #15696617