Subgroups — C₁₃×#9706 #15696045

Order 3328.

850 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×#2867	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#331	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#127	—	2	1	no	—
6	C₂×C₁₃×#127	—	2	2	no	—
7	C₂×C₁₃×#127	—	2	1	no	—
8	C₂×C₁₃×#127	C₄	1	4	yes	—
9	C₂×C₁₃×#127	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	—	2	1	no	—
13	C₄³×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#336	—	2	1	no	—
15	C₁₃×#336	—	2	2	no	—
16	C₁₃×#331	—	2	8	no	—
17	C₂²×D₄×C₁₃	—	2	4	no	—
18	C₂²×D₄×C₁₃	—	2	4	no	—
19	C₂²×D₄×C₁₃	—	2	2	no	—
20	C₂²×D₄×C₁₃	—	2	2	no	—
21	C₂²×D₄×C₁₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₁₃×#127	—	2	4	no	—
25	C₁₃×#127	—	2	4	no	—
26	C₁₃×#127	—	4	8	no	—
27	C₁₃×#127	—	4	4	no	—
28	C₁₃×#127	—	4	8	no	—
29	C₁₃×#127	—	4	2	no	—
30	C₁₃×#127	—	4	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊C₈	—	2	4	no	—
37	C₁₃×C₄⋊C₈	—	2	2	no	—
38	C₁₃×C₄⋊C₈	D₄	1	4	yes	complemented
39	C₁₃×#97	—	4	2	no	—
40	#9706	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₁₃	—	2	4	no	—
42	C₂×D₄×C₁₃	—	2	8	no	—
43	C₂×D₄×C₁₃	—	4	8	no	—
44	C₂×D₄×C₁₃	—	4	8	no	—
45	C₂×D₄×C₁₃	—	4	8	no	—
46	C₂×D₄×C₁₃	—	4	8	no	—
47	C₂×D₄×C₁₃	—	4	8	no	—
48	C₂×D₄×C₁₃	—	4	8	no	—
49	C₂×D₄×C₁₃	—	4	8	no	—
50	C₂×D₄×C₁₃	—	4	4	no	—
51	C₂×D₄×C₁₃	—	4	2	no	—
52	C₂×D₄×C₁₃	—	4	2	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	4	8	no	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	4	2	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	—	4	4	no	—
67	C₄²×C₁₃	—	2	4	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	4	no	—
70	C₄²×C₁₃	D₈	1	2	yes	—
71	C₄²×C₁₃	SD₁₆	1	2	yes	—
72	C₄²×C₁₃	C₂×D₄	1	2	yes	—
73	C₄²×C₁₃	#41	1	2	yes	—
74	#2867	C₂×C₁₃	1	1	yes	characteristic
75	#1831	C₂×C₁₃	1	1	yes	characteristic, complemented
76	#1767	C₂×C₁₃	1	1	yes	characteristic, complemented
77	C₂×#331	C₂×C₁₃	1	4	yes	complemented
78	C₂³×C₁₃	—	4	4	no	—
79	C₂³×C₁₃	—	4	8	no	—
80	C₂³×C₁₃	—	4	4	no	—
81	C₂³×C₁₃	—	4	8	no	—
82	C₂³×C₁₃	—	4	4	no	—
83	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	D₄×C₁₃	—	8	16	no	—
85	D₄×C₁₃	—	8	16	no	—
86	D₄×C₁₃	—	8	8	no	—
87	D₄×C₁₃	—	8	4	no	—
88	D₄×C₁₃	—	8	4	no	—
89	D₄×C₁₃	—	8	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	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	#100	1	4	yes	—
97	C₂×C₄×C₁₃	#102	1	4	yes	—
98	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₃	#121	1	2	yes	—
100	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
106	C₈×C₁₃	—	4	4	no	—
107	C₈×C₁₃	—	8	2	no	—
108	C₂×#127	—	2	1	no	—
109	C₂×#127	—	2	2	no	—
110	C₂×#127	—	2	1	no	—
111	C₂×#127	C₄×C₁₃	1	4	yes	—
112	C₂×#127	C₂²×C₁₃	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
115	C₂×#97	—	2	1	no	—
116	C₄³	C₂²×C₁₃	1	1	yes	characteristic
117	#336	—	2	2	no	—
118	#336	—	2	1	no	—
119	#331	—	2	8	no	—
120	C₂²×C₁₃	—	8	8	no	—
121	C₂²×C₁₃	—	8	8	no	—
122	C₂²×C₁₃	—	8	4	no	—
123	C₂²×C₁₃	—	8	4	no	—
124	C₂²×C₁₃	—	8	4	no	—
125	C₂²×C₁₃	#390	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×#100	1	2	yes	—
127	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
128	C₂²×C₁₃	#418	1	2	yes	—
129	C₂²×C₁₃	#418	1	1	yes	characteristic
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	—	2	4	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	#459	1	2	yes	—
134	C₄×C₁₃	#460	1	2	yes	—
135	C₄×C₁₃	#490	1	2	yes	—
136	C₄×C₁₃	#496	1	2	yes	—
137	C₂²×D₄	—	2	4	no	—
138	C₂²×D₄	—	2	4	no	—
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	—	2	2	no	—
141	C₂²×D₄	C₂×C₄×C₁₃	1	4	yes	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	#127	—	2	4	no	—
145	#127	—	2	4	no	—
146	#127	—	4	8	no	—
147	#127	—	4	4	no	—
148	#127	—	4	8	no	—
149	#127	—	4	2	no	—
150	#127	—	4	2	no	—
151	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
152	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
153	C₂×C₄²	D₄×C₁₃	1	2	yes	—
154	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
156	C₄⋊C₈	—	2	4	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	D₄×C₁₃	1	4	yes	complemented
159	#97	—	4	2	no	—
160	C₂×C₁₃	—	16	4	no	—
161	C₂×C₁₃	#1545	1	2	yes	—
162	C₂×C₁₃	#1555	1	2	yes	—
163	C₂×C₁₃	#1961	1	1	yes	characteristic
164	C₂×C₁₃	#1965	1	1	yes	characteristic
165	C₂×C₁₃	#1982	1	1	yes	characteristic
166	C₂⁴	—	2	4	no	—
167	C₂×D₄	—	2	8	no	—
168	C₂×D₄	—	4	8	no	—
169	C₂×D₄	—	4	8	no	—
170	C₂×D₄	—	4	8	no	—
171	C₂×D₄	—	4	8	no	—
172	C₂×D₄	—	4	8	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	4	8	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	8	no	—
182	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
183	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	4	no	—
195	C₄²	C₁₃×D₈	1	2	yes	—
196	C₄²	C₁₃×SD₁₆	1	2	yes	—
197	C₄²	C₂×D₄×C₁₃	1	2	yes	—
198	C₄²	C₁₃×#41	1	2	yes	—
199	C₁₃	#9706	1	1	yes	characteristic, direct product factor, Sylow
200	C₂³	—	4	4	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	4	8	no	—
205	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
206	D₄	—	8	8	no	—
207	D₄	—	8	16	no	—
208	D₄	—	8	16	no	—
209	D₄	—	8	4	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	C₁₃×#100	1	4	yes	—
219	C₂×C₄	C₁₃×#102	1	4	yes	—
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	C₁₃×#127	1	1	yes	characteristic
224	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
225	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
226	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
227	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
228	C₈	—	4	4	no	—
229	C₈	—	8	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	—	8	8	no	—
232	C₂²	—	8	8	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	C₁₃×#390	1	1	yes	characteristic
236	C₂²	C₂×C₁₃×#100	1	2	yes	—
237	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
238	C₂²	C₁₃×#418	1	1	yes	characteristic
239	C₂²	C₁₃×#418	1	2	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	C₁₃×#459	1	2	yes	—
244	C₄	C₁₃×#460	1	2	yes	—
245	C₄	C₁₃×#490	1	2	yes	—
246	C₄	C₁₃×#496	1	2	yes	—
247	C₂	—	16	4	no	—
248	C₂	C₁₃×#1545	1	2	yes	—
249	C₂	C₁₃×#1555	1	2	yes	—
250	C₂	C₁₃×#1961	1	1	yes	characteristic
251	C₂	C₁₃×#1965	1	1	yes	characteristic
252	C₂	C₁₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1831	C₂	semidirect
3. C₁₃×#1767	C₂	semidirect
4. C₂×C₁₃×#331	C₂	semidirect
38. C₁₃×C₄⋊C₈	D₄	semidirect
40. #9706	C₁₃	direct (N × H)
75. #1831	C₂×C₁₃	semidirect
76. #1767	C₂×C₁₃	semidirect
77. C₂×#331	C₂×C₁₃	semidirect
158. C₄⋊C₈	D₄×C₁₃	semidirect
199. C₁₃	#9706	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×#2867	2
C₁₃×#1831	2
C₁₃×#1767	2
C₂×C₁₃×#331	2
#9706	13

Subgroups — C13×#9706 #15696045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9706 #15696045