Subgroups — C₁₃×#12822 #15699161

Order 3328.

400 conjugacy classes of subgroups.

The subgroup lattice has 296 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×#2143	C₂	1	2	yes	maximal
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1388	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	1	no	—
10	C₁₃×#443	—	2	1	no	—
11	C₁₃×#443	C₄	1	2	yes	—
12	C₁₃×#443	C₄	1	2	yes	—
13	C₂×C₁₃×#108	—	2	1	no	—
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#423	—	2	1	no	—
17	C₂×C₁₃×#104	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#364	—	2	1	no	—
20	C₁₃×#364	—	2	1	no	—
21	C₁₃×#359	—	2	4	no	—
22	C₂×C₁₃×#41	—	2	1	no	—
23	C₁₃×#135	—	2	1	no	—
24	C₁₃×#135	—	2	1	no	—
25	C₁₃×#135	C₂×C₄	1	2	yes	—
26	C₁₃×#135	D₄	1	2	yes	—
27	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic
28	C₂×C₁₃×SD₁₆	—	2	1	no	—
29	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic
30	C₁₃×#131	—	2	2	no	—
31	C₁₃×#131	—	2	2	no	—
32	C₁₃×#131	—	4	1	no	—
33	C₁₃×#131	—	4	1	no	—
34	C₂×C₁₃×M₁₆	—	2	1	no	—
35	C₂×C₁₃×M₁₆	—	2	2	no	—
36	C₂×C₁₃×M₁₆	—	4	1	no	—
37	C₂²×C₈×C₁₃	—	4	1	no	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₁₃×M₃₂	—	2	2	no	—
42	C₁₃×M₃₂	—	2	2	no	—
43	C₂×C₁₃×C₁₆	—	2	2	no	—
44	C₁₃×#108	—	2	1	no	—
45	C₁₃×#108	—	2	2	no	—
46	C₁₃×#108	—	2	2	no	—
47	C₁₃×#108	—	4	1	no	—
48	C₁₃×#108	—	4	1	no	—
49	C₁₃×#108	C₂×C₄	1	2	yes	—
50	C₁₃×C₄⋊C₈	—	4	2	no	—
51	C₁₃×#104	—	2	2	no	—
52	C₁₃×#104	—	2	2	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	#12822	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₁₃×#41	—	2	2	no	—
59	C₁₃×#41	—	2	2	no	—
60	C₁₃×#41	—	4	1	no	—
61	C₁₃×#41	—	4	1	no	—
62	C₂×Q₈×C₁₃	—	2	1	no	—
63	C₂×D₄×C₁₃	—	2	1	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₁₃×Q₁₆	—	2	1	no	—
69	C₁₃×Q₁₆	C₄⋊C₄	1	2	yes	—
70	C₁₃×SD₁₆	—	2	2	no	—
71	C₁₃×SD₁₆	—	2	2	no	—
72	C₁₃×D₈	—	2	1	no	—
73	C₁₃×D₈	C₄⋊C₄	1	2	yes	—
74	C₁₃×M₁₆	—	2	2	no	—
75	C₁₃×M₁₆	—	4	1	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₁₃×M₁₆	—	4	2	no	—
79	C₁₃×M₁₆	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	—	4	2	no	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
88	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
89	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
96	C₄²×C₁₃	—	2	1	no	—
97	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₁₃×C₁₆	—	4	2	no	—
100	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
101	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#2143	C₂×C₁₃	1	2	yes	—
103	#1848	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#1388	C₂×C₁₃	1	2	yes	complemented
105	C₂³×C₁₃	—	4	1	no	—
106	C₂³×C₁₃	#137	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	2	no	—
108	Q₈×C₁₃	—	4	1	no	—
109	D₄×C₁₃	—	2	2	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	D₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₈×C₁₃	—	4	2	no	—
126	C₈×C₁₃	—	8	2	no	—
127	C₈×C₁₃	#102	1	1	yes	characteristic
128	C₈×C₁₃	#102	1	1	yes	characteristic
129	C₈×C₁₃	#102	1	1	yes	characteristic
130	C₈×C₁₃	#102	1	1	yes	characteristic
131	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₈×C₁₃	#117	1	1	yes	characteristic
133	C₈×C₁₃	#121	1	1	yes	characteristic
134	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#131	—	2	1	no	—
137	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
138	#443	—	2	1	no	—
139	#443	—	2	1	no	—
140	#443	C₄×C₁₃	1	2	yes	—
141	#443	C₄×C₁₃	1	2	yes	—
142	C₂×#108	—	2	1	no	—
143	C₂×#108	C₂²×C₁₃	1	2	yes	—
144	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
145	#423	—	2	1	no	—
146	C₂×#104	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	#364	—	2	1	no	—
149	#364	—	2	1	no	—
150	#359	—	2	4	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	#417	1	1	yes	characteristic
155	C₂²×C₁₃	#461	1	1	yes	characteristic
156	C₂²×C₁₃	#474	1	1	yes	characteristic
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	4	1	no	—
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	#380	1	1	yes	characteristic
161	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
162	C₄×C₁₃	#459	1	1	yes	characteristic
163	C₄×C₁₃	#476	1	1	yes	characteristic
164	C₂×#41	—	2	1	no	—
165	#135	—	2	1	no	—
166	#135	—	2	1	no	—
167	#135	C₂×C₄×C₁₃	1	2	yes	—
168	#135	D₄×C₁₃	1	2	yes	—
169	C₂×Q₁₆	Q₈×C₁₃	1	1	yes	characteristic
170	C₂×SD₁₆	—	2	1	no	—
171	C₂×D₈	Q₈×C₁₃	1	1	yes	characteristic
172	#131	—	2	2	no	—
173	#131	—	2	2	no	—
174	#131	—	4	1	no	—
175	#131	—	4	1	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	4	1	no	—
179	C₂²×C₈	—	4	1	no	—
180	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	#108	—	2	1	no	—
187	#108	—	2	2	no	—
188	#108	—	2	2	no	—
189	#108	—	4	1	no	—
190	#108	—	4	1	no	—
191	#108	C₂×C₄×C₁₃	1	2	yes	—
192	C₄⋊C₈	—	4	2	no	—
193	#104	—	2	2	no	—
194	#104	—	2	2	no	—
195	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
196	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
198	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	8	2	no	—
203	C₂×C₁₃	#1802	1	1	yes	characteristic
204	#41	—	2	2	no	—
205	#41	—	2	2	no	—
206	#41	—	4	1	no	—
207	#41	—	4	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	Q₁₆	—	2	1	no	—
215	Q₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
216	SD₁₆	—	2	2	no	—
217	SD₁₆	—	2	2	no	—
218	D₈	—	2	1	no	—
219	D₈	C₁₃×C₄⋊C₄	1	2	yes	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
233	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
234	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
235	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
240	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
241	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
242	C₄²	—	2	1	no	—
243	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
244	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
245	C₁₆	—	4	2	no	—
246	C₁₃	#12822	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	4	1	no	—
248	C₂³	C₁₃×#137	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	4	1	no	—
251	D₄	—	2	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
263	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
266	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₈	—	4	2	no	—
268	C₈	—	8	2	no	—
269	C₈	C₁₃×#102	1	1	yes	characteristic
270	C₈	C₁₃×#102	1	1	yes	characteristic
271	C₈	C₁₃×#102	1	1	yes	characteristic
272	C₈	C₁₃×#102	1	1	yes	characteristic
273	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
274	C₈	C₁₃×#117	1	1	yes	characteristic
275	C₈	C₁₃×#121	1	1	yes	characteristic
276	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	C₁₃×#417	1	1	yes	characteristic
281	C₂²	C₁₃×#461	1	1	yes	characteristic
282	C₂²	C₁₃×#474	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	2	no	—
286	C₄	C₁₃×#380	1	1	yes	characteristic
287	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
288	C₄	C₁₃×#459	1	1	yes	characteristic
289	C₄	C₁₃×#476	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	8	2	no	—
294	C₂	C₁₃×#1802	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1802
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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×#1848	C₂	semidirect
5. C₁₃×#1388	C₂	semidirect
57. #12822	C₁₃	direct (N × H)
101. C₂×#364	C₂×C₁₃	semidirect
103. #1848	C₂×C₁₃	semidirect
104. #1388	C₂×C₁₃	semidirect
246. C₁₃	#12822	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×#2143	2
C₁₃×#1848	2
C₁₃×#1388	2
#12822	13

Subgroups — C13×#12822 #15699161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12822 #15699161