Subgroups — C₁₃×#12820 #15699159

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2139	C₂	1	2	yes	maximal
4	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1387	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#131	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₁₃×#443	—	2	2	no	—
10	C₁₃×#443	C₄	1	4	yes	complemented
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	—	2	1	no	—
14	C₂×C₁₃×#104	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#382	—	2	1	no	—
17	C₁₃×#364	—	2	2	no	—
18	C₁₃×#358	—	2	2	no	—
19	C₁₃×#357	—	2	2	no	—
20	C₂×C₁₃×#41	—	2	1	no	—
21	C₁₃×#135	—	2	1	no	—
22	C₁₃×#135	—	2	1	no	—
23	C₁₃×#135	C₂×C₄	1	2	yes	—
24	C₁₃×#135	Q₈	1	2	yes	—
25	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic
26	C₂×C₁₃×SD₁₆	—	2	1	no	—
27	C₂×C₁₃×D₈	D₄	1	1	yes	characteristic
28	C₁₃×#131	—	2	4	no	—
29	C₁₃×#131	—	4	2	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	4	1	no	—
32	C₂²×C₈×C₁₃	—	4	1	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₁₃×M₃₂	—	2	2	no	—
38	C₁₃×M₃₂	—	2	2	no	—
39	C₂×C₁₃×C₁₆	—	2	2	no	—
40	C₁₃×#108	—	2	1	no	—
41	C₁₃×#108	C₂×C₄	1	2	yes	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	—	2	2	no	—
44	C₁₃×#106	—	4	1	no	—
45	C₁₃×#106	—	4	1	no	—
46	C₁₃×#104	—	2	4	no	—
47	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	D₄	1	2	yes	—
49	C₁₃×#95	—	4	2	no	—
50	#12820	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₁₃×#41	—	2	2	no	—
52	C₁₃×#41	—	2	2	no	—
53	C₁₃×#41	—	4	1	no	—
54	C₁₃×#41	—	4	1	no	—
55	C₂×Q₈×C₁₃	—	2	1	no	—
56	C₂×D₄×C₁₃	—	2	1	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₁₃×Q₁₆	—	2	1	no	—
63	C₁₃×Q₁₆	C₄⋊C₄	1	2	yes	—
64	C₁₃×SD₁₆	—	2	2	no	—
65	C₁₃×SD₁₆	—	2	2	no	—
66	C₁₃×D₈	—	2	1	no	—
67	C₁₃×D₈	C₄⋊C₄	1	2	yes	—
68	C₁₃×M₁₆	—	2	2	no	—
69	C₁₃×M₁₆	—	4	1	no	—
70	C₁₃×M₁₆	—	4	4	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₂×C₈×C₁₃	—	4	4	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
75	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
76	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
87	C₁₃×C₁₆	—	4	2	no	—
88	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
89	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
90	#2139	C₂×C₁₃	1	2	yes	—
91	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
92	#1387	C₂×C₁₃	1	2	yes	complemented
93	C₂³×C₁₃	—	4	1	no	—
94	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	Q₈×C₁₃	—	2	2	no	—
96	Q₈×C₁₃	—	4	1	no	—
97	D₄×C₁₃	—	2	2	no	—
98	D₄×C₁₃	—	4	2	no	—
99	D₄×C₁₃	—	4	2	no	—
100	D₄×C₁₃	—	4	1	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	8	4	no	—
107	C₂×C₄×C₁₃	—	8	2	no	—
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
114	C₈×C₁₃	—	4	2	no	—
115	C₈×C₁₃	#102	1	2	yes	—
116	C₈×C₁₃	#102	1	2	yes	—
117	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₈×C₁₃	#117	1	1	yes	characteristic
119	C₈×C₁₃	#121	1	1	yes	characteristic
120	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
122	C₂×#131	—	2	1	no	—
123	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
124	#443	—	2	2	no	—
125	#443	C₄×C₁₃	1	4	yes	complemented
126	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×#107	C₂²×C₁₃	1	2	yes	—
128	C₂×#106	—	2	1	no	—
129	C₂×#104	—	2	1	no	—
130	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
131	#382	—	2	1	no	—
132	#364	—	2	2	no	—
133	#358	—	2	2	no	—
134	#357	—	2	2	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	4	2	no	—
137	C₂²×C₁₃	—	8	2	no	—
138	C₂²×C₁₃	#417	1	1	yes	characteristic
139	C₂²×C₁₃	#459	1	1	yes	characteristic
140	C₂²×C₁₃	#476	1	1	yes	characteristic
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	—	16	2	no	—
144	C₄×C₁₃	#380	1	1	yes	characteristic
145	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
146	C₄×C₁₃	#461	1	1	yes	characteristic
147	C₄×C₁₃	#474	1	1	yes	characteristic
148	C₂×#41	—	2	1	no	—
149	#135	—	2	1	no	—
150	#135	—	2	1	no	—
151	#135	C₂×C₄×C₁₃	1	2	yes	—
152	#135	Q₈×C₁₃	1	2	yes	—
153	C₂×Q₁₆	D₄×C₁₃	1	1	yes	characteristic
154	C₂×SD₁₆	—	2	1	no	—
155	C₂×D₈	D₄×C₁₃	1	1	yes	characteristic
156	#131	—	2	4	no	—
157	#131	—	4	2	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	4	1	no	—
160	C₂²×C₈	—	4	1	no	—
161	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
165	M₃₂	—	2	2	no	—
166	M₃₂	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#108	—	2	1	no	—
169	#108	C₂×C₄×C₁₃	1	2	yes	—
170	#107	—	2	2	no	—
171	#107	—	2	2	no	—
172	#106	—	4	1	no	—
173	#106	—	4	1	no	—
174	#104	—	2	4	no	—
175	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
176	C₄×C₈	D₄×C₁₃	1	2	yes	—
177	#95	—	4	2	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	2	1	no	—
180	C₂×C₁₃	—	8	2	no	—
181	C₂×C₁₃	#1802	1	1	yes	characteristic
182	C₂×C₁₃	#2212	1	2	yes	—
183	#41	—	2	2	no	—
184	#41	—	2	2	no	—
185	#41	—	4	1	no	—
186	#41	—	4	1	no	—
187	C₂×Q₈	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
194	Q₁₆	—	2	1	no	—
195	Q₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
196	SD₁₆	—	2	2	no	—
197	SD₁₆	—	2	2	no	—
198	D₈	—	2	1	no	—
199	D₈	C₁₃×C₄⋊C₄	1	2	yes	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	4	1	no	—
202	M₁₆	—	4	4	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
206	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
207	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
208	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
213	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
214	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
219	C₁₆	—	4	2	no	—
220	C₁₃	#12820	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³	—	4	1	no	—
222	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	4	1	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
238	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
241	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
242	C₈	—	4	2	no	—
243	C₈	C₁₃×#102	1	2	yes	—
244	C₈	C₁₃×#102	1	2	yes	—
245	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
246	C₈	C₁₃×#117	1	1	yes	characteristic
247	C₈	C₁₃×#121	1	1	yes	characteristic
248	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	C₁₃×#417	1	1	yes	characteristic
253	C₂²	C₁₃×#459	1	1	yes	characteristic
254	C₂²	C₁₃×#476	1	1	yes	characteristic
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	16	2	no	—
258	C₄	C₁₃×#380	1	1	yes	characteristic
259	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
260	C₄	C₁₃×#461	1	1	yes	characteristic
261	C₄	C₁₃×#474	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	8	2	no	—
265	C₂	C₁₃×#1802	1	1	yes	characteristic
266	C₂	C₁₃×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×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×#364	C₂	semidirect
4. C₁₃×#1846	C₂	semidirect
5. C₁₃×#1387	C₂	semidirect
10. C₁₃×#443	C₄	semidirect
50. #12820	C₁₃	direct (N × H)
89. C₂×#364	C₂×C₁₃	semidirect
91. #1846	C₂×C₁₃	semidirect
92. #1387	C₂×C₁₃	semidirect
125. #443	C₄×C₁₃	semidirect
220. C₁₃	#12820	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×#443	2
C₂×C₁₃×#364	2
C₁₃×#2139	2
C₁₃×#1846	2
C₁₃×#1387	2
#12820	13

Subgroups — C13×#12820 #15699159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12820 #15699159