Subgroups — C₁₃×#9812 #15696151

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1965	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1545	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
10	C₁₃×#486	—	2	2	no	—
11	C₁₃×#486	—	2	2	no	—
12	C₁₃×#431	C₂²	1	1	yes	characteristic
13	C₁₃×#423	C₂²	1	1	yes	characteristic
14	C₁₃×#423	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#102	C₄	1	4	yes	—
16	C₂×C₁₃×#100	—	2	2	no	—
17	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#343	—	2	2	no	—
20	C₁₃×#335	—	2	2	no	—
21	C₁₃×#332	—	2	2	no	—
22	C₁₃×#331	—	2	2	no	—
23	C₁₃×#322	—	2	2	no	—
24	C₂²×Q₈×C₁₃	—	2	1	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×M₁₆	—	2	1	no	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₁₃×#127	—	2	1	no	—
37	C₁₃×#127	—	2	1	no	—
38	C₁₃×#127	—	2	1	no	—
39	C₁₃×#127	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊C₈	—	2	1	no	—
44	C₁₃×C₄⋊C₈	—	2	1	no	—
45	C₁₃×C₄⋊C₈	—	2	1	no	—
46	C₁₃×C₄⋊C₈	—	2	1	no	—
47	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
48	C₁₃×#102	—	2	8	no	—
49	C₁₃×#100	—	4	4	no	—
50	C₁₃×C₂²⋊C₈	—	2	1	no	—
51	C₁₃×#97	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	D₄	1	2	yes	—
55	C₄×C₈×C₁₃	D₄	1	2	yes	—
56	#9812	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	2	2	no	—
58	C₂×Q₈×C₁₃	—	4	1	no	—
59	C₂×Q₈×C₁₃	—	4	1	no	—
60	C₂×Q₈×C₁₃	—	8	2	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂×D₄×C₁₃	—	4	1	no	—
66	C₂×D₄×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	4	1	no	—
68	C₂×D₄×C₁₃	—	4	2	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	8	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
75	C₁₃×M₁₆	—	2	2	no	—
76	C₁₃×M₁₆	—	2	2	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₁₃×M₁₆	—	4	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	4	1	no	—
88	C₁₃×C₄⋊C₄	—	4	1	no	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	#41	1	1	yes	characteristic
93	C₂×#486	C₂×C₁₃	1	1	yes	characteristic
94	#1997	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#1965	C₂×C₁₃	1	1	yes	characteristic
96	#1567	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#1549	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#1545	C₂×C₁₃	1	1	yes	characteristic
99	#1451	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	—	4	4	no	—
102	C₂³×C₁₃	—	8	2	no	—
103	C₂³×C₁₃	—	8	2	no	—
104	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₁₃	—	8	2	no	—
106	Q₈×C₁₃	—	8	2	no	—
107	D₄×C₁₃	—	4	4	no	—
108	D₄×C₁₃	—	8	2	no	—
109	D₄×C₁₃	—	8	2	no	—
110	D₄×C₁₃	—	8	2	no	—
111	D₄×C₁₃	—	8	2	no	—
112	D₄×C₁₃	—	8	1	no	—
113	D₄×C₁₃	—	8	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	8	2	no	—
118	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
124	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₁₃	#137	1	2	yes	—
126	C₈×C₁₃	—	2	4	no	—
127	C₈×C₁₃	—	4	2	no	—
128	C₈×C₁₃	—	4	2	no	—
129	C₈×C₁₃	—	8	1	no	—
130	C₈×C₁₃	—	8	1	no	—
131	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
133	#486	—	2	2	no	—
134	#486	—	2	2	no	—
135	#431	C₂²×C₁₃	1	1	yes	characteristic
136	#423	C₂²×C₁₃	1	1	yes	characteristic
137	#423	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×#102	C₄×C₁₃	1	4	yes	—
139	C₂×#100	—	2	2	no	—
140	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	#343	—	2	2	no	—
143	#335	—	2	2	no	—
144	#332	—	2	2	no	—
145	#331	—	2	2	no	—
146	#322	—	2	2	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	8	2	no	—
150	C₂²×C₁₃	—	8	2	no	—
151	C₂²×C₁₃	—	8	2	no	—
152	C₂²×C₁₃	—	16	4	no	—
153	C₂²×C₁₃	#390	1	1	yes	characteristic
154	C₂²×C₁₃	#440	1	2	yes	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	8	2	no	—
158	C₄×C₁₃	#413	1	2	yes	—
159	C₄×C₁₃	#468	1	2	yes	—
160	C₄×C₁₃	#472	1	2	yes	—
161	C₄×C₁₃	#496	1	1	yes	characteristic
162	C₄×C₁₃	#497	1	1	yes	characteristic
163	C₂²×Q₈	—	2	1	no	—
164	C₂²×D₄	—	2	1	no	—
165	C₂²×D₄	—	2	1	no	—
166	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
170	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
171	C₄⋊Q₈	—	2	1	no	—
172	C₄⋊Q₈	—	2	1	no	—
173	C₄⋊Q₈	—	2	1	no	—
174	C₄⋊Q₈	—	2	1	no	—
175	#127	—	2	1	no	—
176	#127	—	2	1	no	—
177	#127	—	2	1	no	—
178	#127	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
181	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
187	#102	—	2	8	no	—
188	#100	—	4	4	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	D₄×C₁₃	1	2	yes	—
194	C₄×C₈	D₄×C₁₃	1	2	yes	—
195	C₂×C₁₃	—	2	2	no	—
196	C₂×C₁₃	—	16	2	no	—
197	C₂×C₁₃	#1967	1	1	yes	characteristic
198	C₂×C₁₃	#1973	1	1	yes	characteristic
199	C₂×C₁₃	#1978	1	1	yes	characteristic
200	C₂⁴	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	8	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	8	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₄²	C₁₃×#41	1	1	yes	characteristic
236	C₁₃	#9812	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	—	4	2	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
242	Q₈	—	8	2	no	—
243	Q₈	—	8	2	no	—
244	D₄	—	4	4	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	D₄	—	8	2	no	—
249	D₄	—	8	1	no	—
250	D₄	—	8	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
259	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
260	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
261	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
262	C₂×C₄	C₁₃×#137	1	2	yes	—
263	C₈	—	2	4	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	8	1	no	—
267	C₈	—	8	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	16	4	no	—
274	C₂²	C₁₃×#390	1	1	yes	characteristic
275	C₂²	C₁₃×#440	1	2	yes	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	C₁₃×#413	1	2	yes	—
280	C₄	C₁₃×#468	1	2	yes	—
281	C₄	C₁₃×#472	1	2	yes	—
282	C₄	C₁₃×#496	1	1	yes	characteristic
283	C₄	C₁₃×#497	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	16	2	no	—
286	C₂	C₁₃×#1967	1	1	yes	characteristic
287	C₂	C₁₃×#1973	1	1	yes	characteristic
288	C₂	C₁₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1997	C₂	semidirect
4. C₁₃×#1567	C₂	semidirect
5. C₁₃×#1549	C₂	semidirect
7. C₁₃×#1451	C₂	semidirect
56. #9812	C₁₃	direct (N × H)
94. #1997	C₂×C₁₃	semidirect
96. #1567	C₂×C₁₃	semidirect
97. #1549	C₂×C₁₃	semidirect
99. #1451	C₂×C₁₃	semidirect
236. C₁₃	#9812	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×#486	2
C₁₃×#1997	2
C₁₃×#1965	2
C₁₃×#1567	2
C₁₃×#1549	2
C₁₃×#1545	2
C₁₃×#1451	2
#9812	13

Subgroups — C13×#9812 #15696151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9812 #15696151