Subgroups — C₁₃×#13470 #15699809

Order 3328.

326 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
13	C₁₃×#428	—	2	1	no	—
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₁₃×#418	—	2	1	no	—
16	C₂×C₁₃×#100	—	2	1	no	—
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	C₁₃×#368	—	2	2	no	—
19	C₁₃×C₄⋊C₁₆	—	2	2	no	—
20	C₁₃×#361	—	2	2	no	—
21	C₁₃×#357	—	2	2	no	—
22	C₁₃×#343	—	2	1	no	—
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	Q₈	1	2	yes	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	D₄	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₁₃×M₃₂	—	2	1	no	—
37	C₁₃×M₃₂	—	2	1	no	—
38	C₂×C₁₃×C₁₆	—	2	1	no	—
39	C₂×C₁₃×C₁₆	—	2	1	no	—
40	C₂×C₁₃×C₁₆	—	2	1	no	—
41	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
42	C₁₃×#108	—	2	1	no	—
43	C₁₃×#108	—	2	1	no	—
44	C₁₃×#108	—	2	1	no	—
45	C₁₃×#108	D₄	1	2	yes	—
46	C₁₃×#107	—	2	1	no	—
47	C₁₃×#107	—	2	1	no	—
48	C₁₃×#107	—	4	1	no	—
49	C₁₃×#106	—	4	1	no	—
50	C₁₃×#102	—	4	1	no	—
51	C₁₃×#102	—	4	1	no	—
52	C₁₃×#100	—	4	2	no	—
53	C₁₃×#97	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	#13470	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×D₈	—	2	2	no	—
66	C₁₃×D₈	—	2	2	no	—
67	C₁₃×D₈	—	4	1	no	—
68	C₁₃×D₈	—	4	1	no	—
69	C₁₃×M₁₆	—	2	2	no	—
70	C₁₃×M₁₆	—	2	1	no	—
71	C₁₃×M₁₆	—	2	1	no	—
72	C₁₃×M₁₆	—	4	1	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₁₃×M₁₆	—	4	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	4	1	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
88	C₁₃×C₂²⋊C₄	—	2	1	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₁₃×C₁₆	—	2	2	no	—
91	C₁₃×C₁₆	—	4	1	no	—
92	C₂×#368	C₂×C₁₃	1	1	yes	characteristic, complemented
93	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
94	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
96	#1978	C₂×C₁₃	1	1	yes	characteristic
97	#1952	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#1387	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	—	8	1	no	—
101	C₂³×C₁₃	—	8	1	no	—
102	C₂³×C₁₃	—	16	1	no	—
103	C₂³×C₁₃	—	16	1	no	—
104	C₂³×C₁₃	#135	1	1	yes	characteristic
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	8	1	no	—
108	D₄×C₁₃	—	8	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	—	16	1	no	—
113	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
119	C₈×C₁₃	—	4	2	no	—
120	C₈×C₁₃	—	4	1	no	—
121	C₈×C₁₃	—	8	1	no	—
122	C₈×C₁₃	#121	1	1	yes	characteristic
123	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₁₃	#126	1	1	yes	characteristic
126	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
131	#428	—	2	1	no	—
132	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
133	#418	—	2	1	no	—
134	C₂×#100	—	2	1	no	—
135	#405	C₂²×C₁₃	1	1	yes	characteristic
136	#368	—	2	2	no	—
137	C₄⋊C₁₆	—	2	2	no	—
138	#361	—	2	2	no	—
139	#357	—	2	2	no	—
140	#343	—	2	1	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	16	2	no	—
145	C₂²×C₁₃	—	16	2	no	—
146	C₂²×C₁₃	—	16	1	no	—
147	C₂²×C₁₃	—	16	1	no	—
148	C₂²×C₁₃	#464	1	1	yes	characteristic
149	C₂²×C₁₃	#466	1	1	yes	characteristic
150	C₂²×C₁₃	#478	1	1	yes	characteristic
151	C₄×C₁₃	—	4	1	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	—	16	1	no	—
154	C₄×C₁₃	#396	1	1	yes	characteristic
155	C₄×C₁₃	#459	1	1	yes	characteristic
156	C₄×C₁₃	#468	1	1	yes	characteristic
157	C₄×C₁₃	#474	1	1	yes	characteristic
158	C₂²×D₄	—	2	1	no	—
159	C₂×D₈	—	2	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	Q₈×C₁₃	1	2	yes	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
169	#117	D₄×C₁₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	M₃₂	—	2	1	no	—
172	M₃₂	—	2	1	no	—
173	C₂×C₁₆	—	2	1	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
177	#108	—	2	1	no	—
178	#108	—	2	1	no	—
179	#108	—	2	1	no	—
180	#108	D₄×C₁₃	1	2	yes	—
181	#107	—	2	1	no	—
182	#107	—	2	1	no	—
183	#107	—	4	1	no	—
184	#106	—	4	1	no	—
185	#102	—	4	1	no	—
186	#102	—	4	1	no	—
187	#100	—	4	2	no	—
188	#97	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₂×C₁₃	—	2	1	no	—
191	C₂×C₁₃	—	2	1	no	—
192	C₂×C₁₃	—	16	2	no	—
193	C₂×C₁₃	#2067	1	1	yes	characteristic
194	C₂×C₁₃	#2221	1	1	yes	characteristic
195	C₂×C₁₃	#2223	1	1	yes	characteristic
196	C₂⁴	—	4	1	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	D₈	—	2	2	no	—
206	D₈	—	2	2	no	—
207	D₈	—	4	1	no	—
208	D₈	—	4	1	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
221	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
222	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
223	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
227	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
228	C₂²⋊C₄	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₁₆	—	2	2	no	—
231	C₁₆	—	4	1	no	—
232	C₁₃	#13470	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	4	2	no	—
234	C₂³	—	8	1	no	—
235	C₂³	—	8	1	no	—
236	C₂³	—	16	1	no	—
237	C₂³	—	16	1	no	—
238	C₂³	C₁₃×#135	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	8	1	no	—
242	D₄	—	8	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	8	1	no	—
246	C₂×C₄	—	16	1	no	—
247	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
248	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
249	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
251	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
253	C₈	—	4	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	8	1	no	—
256	C₈	C₁₃×#121	1	1	yes	characteristic
257	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
258	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
259	C₈	C₁₃×#126	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	16	2	no	—
264	C₂²	—	16	2	no	—
265	C₂²	—	16	1	no	—
266	C₂²	—	16	1	no	—
267	C₂²	C₁₃×#464	1	1	yes	characteristic
268	C₂²	C₁₃×#466	1	1	yes	characteristic
269	C₂²	C₁₃×#478	1	1	yes	characteristic
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	16	1	no	—
273	C₄	C₁₃×#396	1	1	yes	characteristic
274	C₄	C₁₃×#459	1	1	yes	characteristic
275	C₄	C₁₃×#468	1	1	yes	characteristic
276	C₄	C₁₃×#474	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	16	2	no	—
280	C₂	C₁₃×#2067	1	1	yes	characteristic
281	C₂	C₁₃×#2221	1	1	yes	characteristic
282	C₂	C₁₃×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#368	C₂	semidirect
2. C₁₃×#2151	C₂	semidirect
6. C₁₃×#1952	C₂	semidirect
7. C₁₃×#1387	C₂	semidirect
55. #13470	C₁₃	direct (N × H)
92. C₂×#368	C₂×C₁₃	semidirect
93. #2151	C₂×C₁₃	semidirect
97. #1952	C₂×C₁₃	semidirect
98. #1387	C₂×C₁₃	semidirect
232. C₁₃	#13470	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×#368	2
C₁₃×#2151	2
C₂×C₁₃×#361	2
C₂×C₁₃×#357	2
C₁₃×#1978	2
C₁₃×#1952	2
C₁₃×#1387	2
#13470	13

Subgroups — C13×#13470 #15699809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13470 #15699809