Subgroups — C₁₃×#9847 #15696186

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1962	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1492	C₂	1	2	yes	maximal, complemented
6	C₁₃×#532	C₂²	1	2	yes	—
7	C₁₃×#500	C₄	1	1	yes	characteristic, complemented
8	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
9	C₁₃×#486	C₄	1	2	yes	complemented
10	C₁₃×#466	—	2	2	no	—
11	C₄×C₁₃×D₈	—	2	2	no	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₂×C₁₃×#104	—	2	4	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#336	—	2	2	no	—
18	C₁₃×#335	—	2	2	no	—
19	C₁₃×D₄⋊C₈	—	2	4	no	—
20	C₂×C₁₃×#41	—	2	2	no	—
21	C₂×C₁₃×D₈	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	2	yes	—
25	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	C₁₃×#127	C₂×C₄	1	2	yes	—
27	C₁₃×#124	—	2	2	no	—
28	C₁₃×#121	—	2	2	no	—
29	C₁₃×#121	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#107	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	2	no	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×#104	—	4	8	no	—
40	C₁₃×#102	—	2	4	no	—
41	C₄×C₈×C₁₃	—	2	1	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	2	yes	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	#9847	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₁₃×#41	—	4	4	no	—
48	C₁₃×#41	—	4	4	no	—
49	C₂×Q₈×C₁₃	—	2	2	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	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₁₃×D₈	—	4	4	no	—
60	C₁₃×M₁₆	—	4	4	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	4	2	no	—
63	C₂×C₈×C₁₃	—	4	2	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₁₃×C₄⋊C₄	—	2	2	no	—
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	4	2	no	—
70	C₁₃×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	4	2	no	—
73	C₄²×C₁₃	—	4	2	no	—
74	C₄²×C₁₃	D₈	1	4	yes	complemented
75	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃	#41	1	1	yes	characteristic
79	#3169	C₂×C₁₃	1	1	yes	characteristic
80	#1962	C₂×C₁₃	1	2	yes	complemented
81	#1744	C₂×C₁₃	1	1	yes	characteristic, complemented
82	#1567	C₂×C₁₃	1	1	yes	characteristic, complemented
83	#1492	C₂×C₁₃	1	2	yes	complemented
84	C₂³×C₁₃	—	4	2	no	—
85	C₂³×C₁₃	#121	1	1	yes	characteristic
86	Q₈×C₁₃	—	4	4	no	—
87	D₄×C₁₃	—	4	4	no	—
88	D₄×C₁₃	—	4	4	no	—
89	D₄×C₁₃	—	4	4	no	—
90	D₄×C₁₃	—	8	4	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	8	2	no	—
98	C₂×C₄×C₁₃	—	8	2	no	—
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	#135	1	2	yes	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	—	4	2	no	—
110	C₈×C₁₃	—	8	2	no	—
111	#532	C₂²×C₁₃	1	2	yes	—
112	#500	C₄×C₁₃	1	1	yes	characteristic, complemented
113	#493	C₄×C₁₃	1	1	yes	characteristic, complemented
114	#486	C₄×C₁₃	1	2	yes	complemented
115	#466	—	2	2	no	—
116	C₄×D₈	—	2	2	no	—
117	#423	C₂²×C₁₃	1	2	yes	—
118	C₂×#104	—	2	4	no	—
119	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₄³	C₂²×C₁₃	1	1	yes	characteristic
122	#336	—	2	2	no	—
123	#335	—	2	2	no	—
124	D₄⋊C₈	—	2	4	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	8	2	no	—
128	C₂²×C₁₃	—	8	2	no	—
129	C₂²×C₁₃	—	8	2	no	—
130	C₂²×C₁₃	#390	1	1	yes	characteristic
131	C₂²×C₁₃	#466	1	2	yes	—
132	C₄×C₁₃	—	2	1	no	—
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	4	4	no	—
135	C₄×C₁₃	—	8	2	no	—
136	C₄×C₁₃	#421	1	2	yes	—
137	C₄×C₁₃	C₄×D₈	1	2	yes	—
138	C₄×C₁₃	#444	1	2	yes	—
139	C₄×C₁₃	#493	1	1	yes	characteristic
140	C₄×C₁₃	#495	1	1	yes	characteristic
141	C₂×#41	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	D₄×C₁₃	1	2	yes	—
146	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
147	#127	C₂×C₄×C₁₃	1	2	yes	—
148	#124	—	2	2	no	—
149	#121	—	2	2	no	—
150	#121	—	2	2	no	—
151	C₄×D₄	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	D₄×C₁₃	1	2	yes	—
156	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
157	#107	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	#104	—	4	8	no	—
161	#102	—	2	4	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
164	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₄×C₈	D₄×C₁₃	1	2	yes	—
166	C₄×C₈	D₄×C₁₃	1	2	yes	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	16	2	no	—
169	C₂×C₁₃	#1963	1	1	yes	characteristic
170	C₂×C₁₃	#1970	1	1	yes	characteristic
171	C₂×C₁₃	#1975	1	1	yes	characteristic
172	#41	—	4	4	no	—
173	#41	—	4	4	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
183	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
184	D₈	—	4	4	no	—
185	M₁₆	—	4	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
191	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₁₃×D₈	1	4	yes	complemented
200	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₄²	C₁₃×#41	1	1	yes	characteristic
204	C₁₃	#9847	1	1	yes	characteristic, direct product factor, Sylow
205	C₂³	—	4	2	no	—
206	C₂³	C₁₃×#121	1	1	yes	characteristic
207	Q₈	—	4	4	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	8	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
224	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
225	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
226	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
227	C₂×C₄	C₁₃×#135	1	2	yes	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	2	no	—
231	C₈	—	8	2	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	8	2	no	—
235	C₂²	—	8	2	no	—
236	C₂²	—	8	2	no	—
237	C₂²	C₁₃×#390	1	1	yes	characteristic
238	C₂²	C₁₃×#466	1	2	yes	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	4	4	no	—
242	C₄	—	8	2	no	—
243	C₄	C₁₃×#421	1	2	yes	—
244	C₄	C₄×C₁₃×D₈	1	2	yes	—
245	C₄	C₁₃×#444	1	2	yes	—
246	C₄	C₁₃×#493	1	1	yes	characteristic
247	C₄	C₁₃×#495	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	16	2	no	—
250	C₂	C₁₃×#1963	1	1	yes	characteristic
251	C₂	C₁₃×#1970	1	1	yes	characteristic
252	C₂	C₁₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₁₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1962	C₂	semidirect
3. C₁₃×#1744	C₂	semidirect
4. C₁₃×#1567	C₂	semidirect
5. C₁₃×#1492	C₂	semidirect
7. C₁₃×#500	C₄	semidirect
8. C₁₃×#493	C₄	semidirect
9. C₁₃×#486	C₄	semidirect
46. #9847	C₁₃	direct (N × H)
74. C₄²×C₁₃	D₈	semidirect
80. #1962	C₂×C₁₃	semidirect
81. #1744	C₂×C₁₃	semidirect
82. #1567	C₂×C₁₃	semidirect
83. #1492	C₂×C₁₃	semidirect
112. #500	C₄×C₁₃	semidirect
113. #493	C₄×C₁₃	semidirect
114. #486	C₄×C₁₃	semidirect
199. C₄²	C₁₃×D₈	semidirect
204. C₁₃	#9847	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3169	2
C₁₃×#1962	2
C₁₃×#1744	2
C₁₃×#1567	2
C₁₃×#1492	2
#9847	13

Subgroups — C13×#9847 #15696186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9847 #15696186