Subgroups — C₁₃×#12813 #15699152

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#357	C₂	1	2	yes	maximal
4	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₄×C₁₃×D₈	—	2	2	no	—
10	C₄×C₁₃×D₈	C₄	1	4	yes	complemented
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	—	2	1	no	—
14	C₂×C₁₃×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#382	—	2	1	no	—
17	C₁₃×C₄⋊C₁₆	—	2	2	no	—
18	C₁₃×#357	—	2	4	no	—
19	C₂²×D₄×C₁₃	—	2	1	no	—
20	C₂³×C₄×C₁₃	—	4	1	no	—
21	C₂×C₁₃×D₈	—	2	1	no	—
22	C₂×C₁₃×D₈	—	2	1	no	—
23	C₂×C₁₃×D₈	—	2	1	no	—
24	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×D₈	D₄	1	2	yes	—
26	C₂×C₁₃×D₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₁₃	—	2	4	no	—
30	C₄×D₄×C₁₃	—	4	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	—	2	2	no	—
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₁₃×#107	—	2	1	no	—
39	C₁₃×#107	—	2	2	no	—
40	C₁₃×#107	—	2	2	no	—
41	C₁₃×#107	C₂×C₄	1	2	yes	—
42	C₁₃×#106	—	4	2	no	—
43	C₁₃×#102	—	2	4	no	—
44	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₁₃×#95	—	4	1	no	—
47	C₁₃×#95	—	4	1	no	—
48	#12813	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	4	1	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	4	1	no	—
54	C₂×D₄×C₁₃	—	4	1	no	—
55	C₂×D₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	—	4	4	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	8	4	no	—
60	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₁₃×D₈	—	2	2	no	—
64	C₁₃×D₈	—	2	2	no	—
65	C₁₃×D₈	—	2	2	no	—
66	C₁₃×D₈	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	4	1	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₂²⋊C₄	—	4	4	no	—
81	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₃	SD₁₆	1	2	yes	—
83	C₁₃×C₁₆	—	4	2	no	—
84	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
85	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
86	C₂×#357	C₂×C₁₃	1	2	yes	—
87	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
88	#1380	C₂×C₁₃	1	2	yes	complemented
89	C₂³×C₁₃	—	4	2	no	—
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	—	8	2	no	—
93	C₂³×C₁₃	—	8	2	no	—
94	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
95	D₄×C₁₃	—	2	4	no	—
96	D₄×C₁₃	—	4	2	no	—
97	D₄×C₁₃	—	4	2	no	—
98	D₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	8	8	no	—
103	C₂×C₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	—	8	2	no	—
107	C₂×C₄×C₁₃	#102	1	2	yes	—
108	C₂×C₄×C₁₃	#102	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₈×C₁₃	#117	1	1	yes	characteristic
118	C₈×C₁₃	#121	1	1	yes	characteristic
119	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×C₄×D₄	—	2	1	no	—
122	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
123	C₄×D₈	—	2	2	no	—
124	C₄×D₈	C₄×C₁₃	1	4	yes	complemented
125	C₂×#107	C₂²×C₁₃	1	2	yes	—
126	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×#106	—	2	1	no	—
128	C₂×#102	—	2	1	no	—
129	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	#382	—	2	1	no	—
131	C₄⋊C₁₆	—	2	2	no	—
132	#357	—	2	4	no	—
133	C₂²×C₁₃	—	4	4	no	—
134	C₂²×C₁₃	—	8	2	no	—
135	C₂²×C₁₃	—	8	2	no	—
136	C₂²×C₁₃	—	8	4	no	—
137	C₂²×C₁₃	—	8	4	no	—
138	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
139	C₂²×C₁₃	#459	1	1	yes	characteristic
140	C₂²×C₁₃	#474	1	1	yes	characteristic
141	C₂²×C₁₃	#508	1	2	yes	—
142	C₂²×C₁₃	#509	1	2	yes	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	8	2	no	—
146	C₄×C₁₃	—	16	2	no	—
147	C₄×C₁₃	#380	1	1	yes	characteristic
148	C₄×C₁₃	#417	1	1	yes	characteristic
149	C₄×C₁₃	#461	1	1	yes	characteristic
150	C₄×C₁₃	#476	1	1	yes	characteristic
151	C₂²×D₄	—	2	1	no	—
152	C₂³×C₄	—	4	1	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
157	C₂×D₈	D₄×C₁₃	1	2	yes	—
158	C₂×D₈	Q₈×C₁₃	1	2	yes	—
159	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
161	C₄×D₄	—	2	4	no	—
162	C₄×D₄	—	4	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₂²⋊C₄	—	4	1	no	—
166	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	#107	—	2	1	no	—
171	#107	—	2	2	no	—
172	#107	—	2	2	no	—
173	#107	C₂×C₄×C₁₃	1	2	yes	—
174	#106	—	4	2	no	—
175	#102	—	2	4	no	—
176	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
177	C₄×C₈	D₄×C₁₃	1	2	yes	—
178	#95	—	4	1	no	—
179	#95	—	4	1	no	—
180	C₂×C₁₃	—	8	4	no	—
181	C₂×C₁₃	#1802	1	1	yes	characteristic
182	C₂×C₁₃	#2138	1	1	yes	characteristic
183	C₂×C₁₃	#2140	1	1	yes	characteristic
184	C₂×C₁₃	#2210	1	2	yes	—
185	C₂×C₁₃	#2228	1	2	yes	—
186	C₂⁴	—	4	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	8	4	no	—
197	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
198	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
200	D₈	—	2	2	no	—
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	D₈	C₁₃×C₄⋊C₄	1	4	yes	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
207	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
211	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
212	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
219	C₄²	C₁₃×SD₁₆	1	2	yes	—
220	C₁₆	—	4	2	no	—
221	C₁₃	#12813	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	8	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	8	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₁₃×#102	1	2	yes	—
241	C₂×C₄	C₁₃×#102	1	2	yes	—
242	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
244	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
246	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
250	C₈	C₁₃×#117	1	1	yes	characteristic
251	C₈	C₁₃×#121	1	1	yes	characteristic
252	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂²	—	4	4	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	4	no	—
258	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
259	C₂²	C₁₃×#459	1	1	yes	characteristic
260	C₂²	C₁₃×#474	1	1	yes	characteristic
261	C₂²	C₁₃×#508	1	2	yes	—
262	C₂²	C₁₃×#509	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	2	no	—
266	C₄	—	16	2	no	—
267	C₄	C₁₃×#380	1	1	yes	characteristic
268	C₄	C₁₃×#417	1	1	yes	characteristic
269	C₄	C₁₃×#461	1	1	yes	characteristic
270	C₄	C₁₃×#476	1	1	yes	characteristic
271	C₂	—	8	4	no	—
272	C₂	C₁₃×#1802	1	1	yes	characteristic
273	C₂	C₁₃×#2138	1	1	yes	characteristic
274	C₂	C₁₃×#2140	1	1	yes	characteristic
275	C₂	C₁₃×#2210	1	2	yes	—
276	C₂	C₁₃×#2228	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
4. C₁₃×#1846	C₂	semidirect
5. C₁₃×#1380	C₂	semidirect
10. C₄×C₁₃×D₈	C₄	semidirect
48. #12813	C₁₃	direct (N × H)
85. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
87. #1846	C₂×C₁₃	semidirect
88. #1380	C₂×C₁₃	semidirect
124. C₄×D₈	C₄×C₁₃	semidirect
221. C₁₃	#12813	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×C₄×C₁₃×D₈	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#357	2
C₁₃×#1846	2
C₁₃×#1380	2
#12813	13

Subgroups — C13×#12813 #15699152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12813 #15699152