Subgroups — C₁₃×#11657 #15697996

Order 3328.

430 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1331	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₁₃	—	2	1	no	—
7	C₄×C₁₃×D₈	—	2	1	no	—
8	C₄×C₁₃×D₈	—	2	1	no	—
9	C₄×C₁₃×D₈	C₄	1	2	yes	—
10	C₄×C₁₃×D₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
12	C₁₃×#423	—	2	1	no	—
13	C₂×C₁₃×#102	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊C₁₆	—	2	1	no	—
16	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
18	C₁₃×#335	—	2	1	no	—
19	C₁₃×#335	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	—	4	1	no	—
22	C₂×C₁₃×D₈	—	2	4	no	—
23	C₂×C₁₃×D₈	—	2	1	no	—
24	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	—	4	1	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×D₄×C₁₃	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	4	1	no	—
31	C₄×D₄×C₁₃	—	4	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₁₃×#107	—	2	1	no	—
37	C₁₃×#107	C₂×C₄	1	2	yes	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	4	1	no	—
40	C₁₃×#102	—	2	2	no	—
41	C₁₃×#102	—	2	2	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	#11657	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂⁴×C₁₃	—	4	1	no	—
48	C₂×D₄×C₁₃	—	2	4	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	4	2	no	—
51	C₂×D₄×C₁₃	—	4	1	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	—	8	2	no	—
56	C₂²×C₄×C₁₃	—	8	2	no	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
60	C₁₃×D₈	—	2	4	no	—
61	C₁₃×D₈	—	2	2	no	—
62	C₁₃×D₈	M₁₆	1	4	yes	complemented
63	C₁₃×M₁₆	—	8	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	4	1	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	4	1	no	—
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₁₃×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₃	D₈	1	1	yes	characteristic
77	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×C₁₆	—	4	2	no	—
80	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
81	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
82	#1567	C₂×C₁₃	1	1	yes	characteristic, complemented
83	#1331	C₂×C₁₃	1	4	yes	complemented
84	C₂³×C₁₃	—	4	4	no	—
85	C₂³×C₁₃	—	4	2	no	—
86	C₂³×C₁₃	—	8	2	no	—
87	C₂³×C₁₃	—	8	2	no	—
88	C₂³×C₁₃	#102	1	1	yes	characteristic
89	D₄×C₁₃	—	2	4	no	—
90	D₄×C₁₃	—	4	4	no	—
91	D₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	4	1	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	8	2	no	—
97	C₂×C₄×C₁₃	—	8	4	no	—
98	C₂×C₄×C₁₃	—	8	2	no	—
99	C₂×C₄×C₁₃	—	8	2	no	—
100	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
103	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
104	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₈×C₁₃	—	2	1	no	—
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	8	2	no	—
112	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
113	C₂²×D₈	C₄×C₁₃	1	1	yes	characteristic
114	C₂×C₄×D₄	—	2	1	no	—
115	C₄×D₈	—	2	1	no	—
116	C₄×D₈	—	2	1	no	—
117	C₄×D₈	C₄×C₁₃	1	2	yes	—
118	C₄×D₈	C₂²×C₁₃	1	2	yes	—
119	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
120	#423	—	2	1	no	—
121	C₂×#102	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₄⋊C₁₆	—	2	1	no	—
124	C₂²⋊C₁₆	—	2	1	no	—
125	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
126	#335	—	2	1	no	—
127	#335	C₂²×C₁₃	1	2	yes	complemented
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	4	4	no	—
131	C₂²×C₁₃	—	8	8	no	—
132	C₂²×C₁₃	—	8	2	no	—
133	C₂²×C₁₃	—	8	2	no	—
134	C₂²×C₁₃	#357	1	2	yes	—
135	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	8	2	no	—
140	C₄×C₁₃	#357	1	2	yes	—
141	C₄×C₁₃	#407	1	1	yes	characteristic
142	C₄×C₁₃	#421	1	1	yes	characteristic
143	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
144	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
145	C₂²×D₄	—	2	1	no	—
146	C₂³×C₄	—	4	1	no	—
147	C₂×D₈	—	2	4	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
150	C₂×M₁₆	—	4	1	no	—
151	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
152	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
153	C₄×D₄	—	2	2	no	—
154	C₄×D₄	—	2	2	no	—
155	C₄×D₄	—	4	1	no	—
156	C₄×D₄	—	4	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₂²⋊C₄	—	4	1	no	—
159	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
160	C₂×C₁₆	—	2	2	no	—
161	#107	—	2	1	no	—
162	#107	C₂×C₄×C₁₃	1	2	yes	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	4	1	no	—
165	#102	—	2	2	no	—
166	#102	—	2	2	no	—
167	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
169	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
171	C₂×C₁₃	—	2	2	no	—
172	C₂×C₁₃	—	8	4	no	—
173	C₂×C₁₃	#1486	1	1	yes	characteristic
174	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
175	C₂×C₁₃	#2143	1	1	yes	characteristic
176	C₂⁴	—	4	1	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	8	2	no	—
185	C₂²×C₄	—	8	2	no	—
186	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
188	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
189	D₈	—	2	4	no	—
190	D₈	—	2	2	no	—
191	D₈	C₁₃×M₁₆	1	4	yes	complemented
192	M₁₆	—	8	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
205	C₄²	C₁₃×D₈	1	1	yes	characteristic
206	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₁₆	—	4	2	no	—
209	C₁₃	#11657	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³	—	4	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	—	8	2	no	—
214	C₂³	C₁₃×#102	1	1	yes	characteristic
215	D₄	—	2	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
227	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
228	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
229	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
230	C₂×C₄	C₁₃×SD₃₂	1	2	yes	—
231	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
233	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	8	2	no	—
238	C₈	C₂×C₁₃×M₁₆	1	2	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	4	no	—
242	C₂²	—	8	8	no	—
243	C₂²	—	8	2	no	—
244	C₂²	—	8	2	no	—
245	C₂²	C₁₃×#357	1	2	yes	—
246	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	8	2	no	—
251	C₄	C₁₃×#357	1	2	yes	—
252	C₄	C₁₃×#407	1	1	yes	characteristic
253	C₄	C₁₃×#421	1	1	yes	characteristic
254	C₄	C₂×C₁₃×D₁₆	1	1	yes	characteristic
255	C₄	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	4	no	—
258	C₂	C₁₃×#1486	1	1	yes	characteristic
259	C₂	C₂×C₁₃×#357	1	1	yes	characteristic
260	C₂	C₁₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₄×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂²×C₁₃×D₈	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2162	C₂	semidirect
3. C₁₃×#1567	C₂	semidirect
4. C₁₃×#1331	C₂	semidirect
17. C₄×C₁₃×C₁₆	C₂²	semidirect
19. C₁₃×#335	C₂²	semidirect
46. #11657	C₁₃	direct (N × H)
62. C₁₃×D₈	M₁₆	semidirect
81. #2162	C₂×C₁₃	semidirect
82. #1567	C₂×C₁₃	semidirect
83. #1331	C₂×C₁₃	semidirect
125. C₄×C₁₆	C₂²×C₁₃	semidirect
127. #335	C₂²×C₁₃	semidirect
191. D₈	C₁₃×M₁₆	semidirect
209. C₁₃	#11657	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×C₄×C₁₃×D₈	2
C₁₃×#2162	2
C₁₃×#1567	2
C₁₃×#1331	2
#11657	13

Subgroups — C13×#11657 #15697996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#11657 #15697996