Subgroups — C₁₃×#9844 #15696183

Order 3328.

554 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₄²×C₈×C₁₃	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	—	2	2	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
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	8	no	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₄×C₈×C₁₃	D₄	1	2	yes	—
47	#9844	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₁₃×#41	—	4	4	no	—
49	C₁₃×#41	—	4	4	no	—
50	C₂×Q₈×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₁₃×D₈	—	4	4	no	—
61	C₁₃×M₁₆	—	4	4	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₄⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	4	2	no	—
71	C₁₃×C₂²⋊C₄	—	4	2	no	—
72	C₄²×C₁₃	—	2	4	no	—
73	C₄²×C₁₃	—	2	4	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	D₈	1	4	yes	complemented
76	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	#41	1	1	yes	characteristic
80	#3169	C₂×C₁₃	1	1	yes	characteristic
81	#1962	C₂×C₁₃	1	2	yes	complemented
82	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
83	#1567	C₂×C₁₃	1	1	yes	characteristic, complemented
84	#1492	C₂×C₁₃	1	2	yes	complemented
85	C₂³×C₁₃	—	4	2	no	—
86	C₂³×C₁₃	#121	1	1	yes	characteristic
87	Q₈×C₁₃	—	4	4	no	—
88	D₄×C₁₃	—	4	4	no	—
89	D₄×C₁₃	—	4	4	no	—
90	D₄×C₁₃	—	4	4	no	—
91	D₄×C₁₃	—	8	4	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	8	2	no	—
99	C₂×C₄×C₁₃	—	8	2	no	—
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₁₃	#135	1	2	yes	—
108	C₈×C₁₃	—	2	4	no	—
109	C₈×C₁₃	—	2	2	no	—
110	C₈×C₁₃	—	8	2	no	—
111	C₈×C₁₃	#104	1	4	yes	complemented
112	#532	C₂²×C₁₃	1	2	yes	—
113	#500	C₄×C₁₃	1	1	yes	characteristic, complemented
114	#493	C₄×C₁₃	1	1	yes	characteristic, complemented
115	#486	C₄×C₁₃	1	2	yes	complemented
116	#466	—	2	2	no	—
117	C₄×D₈	—	2	2	no	—
118	#423	C₂²×C₁₃	1	2	yes	—
119	C₂×#104	—	2	4	no	—
120	C₂×C₄×C₈	—	2	2	no	—
121	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₄³	C₂²×C₁₃	1	1	yes	characteristic
124	#335	—	2	2	no	—
125	D₄⋊C₈	—	2	4	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	8	2	no	—
129	C₂²×C₁₃	—	8	2	no	—
130	C₂²×C₁₃	—	8	2	no	—
131	C₂²×C₁₃	#390	1	1	yes	characteristic
132	C₂²×C₁₃	#466	1	2	yes	—
133	C₄×C₁₃	—	2	8	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	—	8	2	no	—
137	C₄×C₁₃	C₂×#104	1	2	yes	—
138	C₄×C₁₃	C₄×D₈	1	2	yes	—
139	C₄×C₁₃	#443	1	2	yes	—
140	C₄×C₁₃	#493	1	1	yes	characteristic
141	C₄×C₁₃	#495	1	1	yes	characteristic
142	C₂×#41	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	D₄×C₁₃	1	2	yes	—
147	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
148	#127	C₂×C₄×C₁₃	1	2	yes	—
149	#124	—	2	2	no	—
150	#121	—	2	2	no	—
151	#121	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	D₄×C₁₃	1	2	yes	—
157	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
158	#107	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	#104	—	4	8	no	—
162	#102	—	2	4	no	—
163	C₄×C₈	—	2	8	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₄×C₈	D₄×C₁₃	1	2	yes	—
168	C₄×C₈	D₄×C₁₃	1	2	yes	—
169	C₂×C₁₃	—	2	2	no	—
170	C₂×C₁₃	—	16	2	no	—
171	C₂×C₁₃	#1962	1	1	yes	characteristic
172	C₂×C₁₃	#1970	1	1	yes	characteristic
173	C₂×C₁₃	#1974	1	1	yes	characteristic
174	#41	—	4	4	no	—
175	#41	—	4	4	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
186	D₈	—	4	4	no	—
187	M₁₆	—	4	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
193	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₁₃×D₈	1	4	yes	complemented
202	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
203	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₄²	C₁₃×#41	1	1	yes	characteristic
206	C₁₃	#9844	1	1	yes	characteristic, direct product factor, Sylow
207	C₂³	—	4	2	no	—
208	C₂³	C₁₃×#121	1	1	yes	characteristic
209	Q₈	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	8	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
226	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
227	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
228	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
229	C₂×C₄	C₁₃×#135	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	2	2	no	—
232	C₈	—	8	2	no	—
233	C₈	C₁₃×#104	1	4	yes	complemented
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	8	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	C₁₃×#390	1	1	yes	characteristic
240	C₂²	C₁₃×#466	1	2	yes	—
241	C₄	—	2	8	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	8	2	no	—
245	C₄	C₂×C₁₃×#104	1	2	yes	—
246	C₄	C₄×C₁₃×D₈	1	2	yes	—
247	C₄	C₁₃×#443	1	2	yes	—
248	C₄	C₁₃×#493	1	1	yes	characteristic
249	C₄	C₁₃×#495	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	16	2	no	—
252	C₂	C₁₃×#1962	1	1	yes	characteristic
253	C₂	C₁₃×#1970	1	1	yes	characteristic
254	C₂	C₁₃×#1974	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. C₄²×C₈×C₁₃	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
47. #9844	C₁₃	direct (N × H)
75. C₄²×C₁₃	D₈	semidirect
81. #1962	C₂×C₁₃	semidirect
82. C₄²×C₈	C₂×C₁₃	semidirect
83. #1567	C₂×C₁₃	semidirect
84. #1492	C₂×C₁₃	semidirect
111. C₈×C₁₃	#104	semidirect
113. #500	C₄×C₁₃	semidirect
114. #493	C₄×C₁₃	semidirect
115. #486	C₄×C₁₃	semidirect
201. C₄²	C₁₃×D₈	semidirect
206. C₁₃	#9844	direct (N × H)
233. C₈	C₁₃×#104	semidirect

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
C₄²×C₈×C₁₃	2
C₁₃×#1567	2
C₁₃×#1492	2
#9844	13

Subgroups — C13×#9844 #15696183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9844 #15696183