Subgroups — C₁₃×#9820 #15696159

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#492	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1990	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1957	C₂	1	2	yes	maximal
4	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×SD₁₆	—	2	2	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
9	C₁₃×#492	—	2	2	no	—
10	C₁₃×#492	C₄	1	4	yes	complemented
11	C₂×C₁₃×#106	—	2	2	no	—
12	C₂×C₁₃×#102	—	2	4	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#384	C₂²	1	2	yes	—
16	C₁₃×#340	—	2	4	no	—
17	C₈²×C₁₃	—	2	2	no	—
18	C₂²×Q₈×C₁₃	—	2	1	no	—
19	C₂²×D₄×C₁₃	—	2	2	no	—
20	C₂²×D₄×C₁₃	—	2	1	no	—
21	C₂×C₁₃×SD₁₆	—	2	8	no	—
22	C₂×C₁₃×SD₁₆	—	4	4	no	—
23	C₂×C₁₃×SD₁₆	—	4	4	no	—
24	C₂²×C₈×C₁₃	D₄	1	4	yes	—
25	C₂²×C₈×C₁₃	D₄	1	2	yes	—
26	C₁₃×C₄⋊Q₈	—	2	1	no	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
29	C₁₃×#127	—	2	1	no	—
30	C₁₃×#127	—	2	2	no	—
31	C₁₃×#127	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×#106	—	4	4	no	—
38	C₁₃×#102	—	4	8	no	—
39	C₄×C₈×C₁₃	—	2	2	no	—
40	C₄×C₈×C₁₃	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₃	D₄	1	2	yes	—
43	#9820	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₁₃	—	4	1	no	—
45	C₂×Q₈×C₁₃	—	2	2	no	—
46	C₂×Q₈×C₁₃	—	4	1	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	4	no	—
51	C₂×D₄×C₁₃	—	4	2	no	—
52	C₂×D₄×C₁₃	—	4	1	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	4	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	4	4	no	—
57	C₂²×C₄×C₁₃	—	4	1	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₁₃×SD₁₆	—	4	16	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₁₃	D₈	1	4	yes	—
66	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
67	C₂×C₈×C₁₃	SD₁₆	1	8	yes	complemented
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₁₃×C₄⋊C₄	—	4	4	no	—
72	C₁₃×C₄⋊C₄	—	4	4	no	—
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	#41	1	2	yes	—
76	C₂×#492	C₂×C₁₃	1	1	yes	characteristic
77	#1990	C₂×C₁₃	1	2	yes	complemented
78	#1957	C₂×C₁₃	1	2	yes	—
79	#1844	C₂×C₁₃	1	1	yes	characteristic, complemented
80	C₂×C₈²	C₂×C₁₃	1	1	yes	characteristic, complemented
81	C₂³×C₁₃	—	4	2	no	—
82	C₂³×C₁₃	—	8	1	no	—
83	C₂³×C₁₃	—	8	2	no	—
84	C₂³×C₁₃	—	8	1	no	—
85	C₂³×C₁₃	—	8	2	no	—
86	C₂³×C₁₃	#127	1	1	yes	characteristic
87	Q₈×C₁₃	—	4	4	no	—
88	Q₈×C₁₃	—	8	2	no	—
89	D₄×C₁₃	—	4	4	no	—
90	D₄×C₁₃	—	4	8	no	—
91	D₄×C₁₃	—	8	4	no	—
92	D₄×C₁₃	—	8	2	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	8	1	no	—
95	C₂×C₄×C₁₃	—	8	1	no	—
96	C₂×C₄×C₁₃	—	8	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	2	yes	—
101	C₂×C₄×C₁₃	#121	1	2	yes	—
102	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
106	C₂×C₄×C₁₃	#135	1	4	yes	—
107	C₈×C₁₃	—	2	8	no	—
108	C₈×C₁₃	#102	1	8	yes	complemented
109	C₂²×SD₁₆	—	2	2	no	—
110	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
111	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
112	#492	—	2	2	no	—
113	#492	C₄×C₁₃	1	4	yes	complemented
114	C₂×#106	—	2	2	no	—
115	C₂×#102	—	2	4	no	—
116	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
118	#384	C₂²×C₁₃	1	2	yes	—
119	#340	—	2	4	no	—
120	C₈²	—	2	2	no	—
121	C₂²×C₁₃	—	8	2	no	—
122	C₂²×C₁₃	—	8	4	no	—
123	C₂²×C₁₃	—	16	1	no	—
124	C₂²×C₁₃	—	16	2	no	—
125	C₂²×C₁₃	—	16	1	no	—
126	C₂²×C₁₃	#390	1	1	yes	characteristic
127	C₂²×C₁₃	#492	1	1	yes	characteristic
128	C₂²×C₁₃	#492	1	2	yes	—
129	C₂²×C₁₃	#493	1	1	yes	characteristic
130	C₂²×C₁₃	#495	1	2	yes	—
131	C₄×C₁₃	—	8	2	no	—
132	C₄×C₁₃	—	16	2	no	—
133	C₄×C₁₃	C₂×#102	1	2	yes	—
134	C₄×C₁₃	C₄×SD₁₆	1	4	yes	—
135	C₄×C₁₃	#465	1	4	yes	—
136	C₄×C₁₃	#486	1	2	yes	—
137	C₂²×Q₈	—	2	1	no	—
138	C₂²×D₄	—	2	1	no	—
139	C₂²×D₄	—	2	2	no	—
140	C₂×SD₁₆	—	2	8	no	—
141	C₂×SD₁₆	—	4	4	no	—
142	C₂×SD₁₆	—	4	4	no	—
143	C₂²×C₈	D₄×C₁₃	1	4	yes	—
144	C₂²×C₈	D₄×C₁₃	1	2	yes	—
145	C₄⋊Q₈	—	2	1	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
148	#127	—	2	1	no	—
149	#127	—	2	2	no	—
150	#127	C₂×C₄×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
156	#106	—	4	4	no	—
157	#102	—	4	8	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
161	C₄×C₈	D₄×C₁₃	1	2	yes	—
162	C₂×C₁₃	—	16	2	no	—
163	C₂×C₁₃	#1705	1	1	yes	characteristic
164	C₂×C₁₃	#1706	1	2	yes	—
165	C₂×C₁₃	#1708	1	1	yes	characteristic
166	C₂×C₁₃	#1957	1	1	yes	characteristic
167	C₂×C₁₃	#1968	1	2	yes	—
168	C₂⁴	—	4	1	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	4	1	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
185	SD₁₆	—	4	16	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
189	C₂×C₈	C₁₃×D₈	1	4	yes	—
190	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
191	C₂×C₈	C₁₃×SD₁₆	1	8	yes	complemented
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
198	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₄²	C₁₃×#41	1	2	yes	—
200	C₁₃	#9820	1	1	yes	characteristic, direct product factor, Sylow
201	C₂³	—	4	2	no	—
202	C₂³	—	8	1	no	—
203	C₂³	—	8	2	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	2	no	—
206	C₂³	C₁₃×#127	1	1	yes	characteristic
207	Q₈	—	4	4	no	—
208	Q₈	—	8	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	8	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
221	C₂×C₄	C₁₃×#121	1	2	yes	—
222	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
223	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
224	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
225	C₂×C₄	C₂×C₁₃×SD₁₆	1	4	yes	—
226	C₂×C₄	C₁₃×#135	1	4	yes	—
227	C₈	—	2	8	no	—
228	C₈	C₁₃×#102	1	8	yes	complemented
229	C₂²	—	8	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	—	16	1	no	—
232	C₂²	—	16	2	no	—
233	C₂²	—	16	1	no	—
234	C₂²	C₁₃×#390	1	1	yes	characteristic
235	C₂²	C₁₃×#492	1	1	yes	characteristic
236	C₂²	C₁₃×#492	1	2	yes	—
237	C₂²	C₁₃×#493	1	1	yes	characteristic
238	C₂²	C₁₃×#495	1	2	yes	—
239	C₄	—	8	2	no	—
240	C₄	—	16	2	no	—
241	C₄	C₂×C₁₃×#102	1	2	yes	—
242	C₄	C₄×C₁₃×SD₁₆	1	4	yes	—
243	C₄	C₁₃×#465	1	4	yes	—
244	C₄	C₁₃×#486	1	2	yes	—
245	C₂	—	16	2	no	—
246	C₂	C₁₃×#1705	1	1	yes	characteristic
247	C₂	C₁₃×#1706	1	2	yes	—
248	C₂	C₁₃×#1708	1	1	yes	characteristic
249	C₂	C₁₃×#1957	1	1	yes	characteristic
250	C₂	C₁₃×#1968	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1990	C₂	semidirect
4. C₁₃×#1844	C₂	semidirect
5. C₂×C₈²×C₁₃	C₂	semidirect
10. C₁₃×#492	C₄	semidirect
43. #9820	C₁₃	direct (N × H)
67. C₂×C₈×C₁₃	SD₁₆	semidirect
77. #1990	C₂×C₁₃	semidirect
79. #1844	C₂×C₁₃	semidirect
80. C₂×C₈²	C₂×C₁₃	semidirect
108. C₈×C₁₃	#102	semidirect
113. #492	C₄×C₁₃	semidirect
191. C₂×C₈	C₁₃×SD₁₆	semidirect
200. C₁₃	#9820	direct (N × H)
228. C₈	C₁₃×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#492	2
C₁₃×#1990	2
C₁₃×#1957	2
C₁₃×#1844	2
C₂×C₈²×C₁₃	2
#9820	13

Subgroups — C13×#9820 #15696159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9820 #15696159