Subgroups — C₂×#13131215 #16807121

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	#13131215	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718259	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718495	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
8	C₂×C₄×D₅₂	C₄	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	1	yes	characteristic
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	C₂×D₁₃⋊C₁₆	—	2	8	no	—
13	C₂×D₁₃⋊C₁₆	C₂²	1	8	yes	complemented
14	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
15	C₂×#88060	C₄	1	2	yes	—
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718495	C₄	1	4	yes	—
18	#11718495	C₂²	1	4	yes	—
19	#11718259	C₂²	1	8	yes	complemented
20	C₄⋊#6092	C₂²	1	4	yes	complemented
21	C₄×#6092	C₂²	1	4	yes	complemented
22	C₂²×D₅₂	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#6103	C₈	1	2	yes	—
27	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂³	1	1	yes	characteristic
30	C₂×C₈×D₁₃	—	2	4	no	—
31	C₂×C₈×D₁₃	—	2	4	no	—
32	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₁₃	C₂³	1	4	yes	—
34	C₄×D₅₂	C₈	1	4	yes	—
35	C₄×D₅₂	C₂×C₄	1	4	yes	—
36	C₂×#6092	—	2	4	no	—
37	C₂×#6092	—	2	2	no	—
38	C₂×#6092	D₄	1	4	yes	complemented
39	C₂×#6092	C₂³	1	2	yes	—
40	C₂×#6092	C₂³	1	4	yes	—
41	C₂×#6092	C₂³	1	2	yes	—
42	D₁₃⋊C₁₆	—	2	16	no	—
43	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	#88060	C₂×C₄	1	8	yes	—
45	C₄×#1055	C₂×C₄	1	2	yes	—
46	C₄×#1055	C₂³	1	2	yes	—
47	C₂×D₄×C₁₆	—	13	1	no	maximal, Sylow
48	C₂³×D₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
53	C₂×D₅₂	C₁₆	1	8	yes	complemented
54	C₂×D₅₂	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₁₃	—	2	4	no	—
56	C₂×C₄×D₁₃	—	2	4	no	—
57	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
61	C₄²×C₁₃	C₂×C₈	1	2	yes	—
62	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
63	#6103	C₂×C₈	1	8	yes	—
64	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	C₂²×C₄	1	2	yes	—
67	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
68	C₂×#1055	C₂×D₄	1	2	yes	—
69	C₂×#1055	#41	1	2	yes	—
70	C₂×#1055	C₂⁴	1	1	yes	characteristic
71	C₈×D₁₃	—	2	8	no	—
72	C₈×D₁₃	—	2	4	no	—
73	C₈×D₁₃	—	2	4	no	—
74	#6092	—	2	4	no	—
75	#6092	—	2	2	no	—
76	#6092	C₂×D₄	1	4	yes	complemented
77	C₂³×C₁₆	—	13	2	no	—
78	C₂×C₈×D₄	—	13	1	no	—
79	D₄×C₁₆	—	13	8	no	—
80	C₂×C₄⋊C₁₆	—	13	1	no	—
81	C₂×C₂²⋊C₁₆	—	13	2	no	—
82	C₂×C₄×C₁₆	—	13	1	no	—
83	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂²×D₁₃	—	2	4	no	—
85	C₂²×D₁₃	—	2	4	no	—
86	C₂²×D₁₃	C₂×C₁₆	1	8	yes	complemented
87	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
90	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₃	#131	1	2	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×Dic₁₃	—	2	4	no	—
97	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
98	D₅₂	C₂×C₁₆	1	16	yes	complemented
99	C₄×D₁₃	—	2	4	no	—
100	C₄×D₁₃	—	2	4	no	—
101	C₄×D₁₃	—	2	8	no	—
102	C₈×C₁₃	—	2	4	no	—
103	#1055	—	2	2	no	—
104	#1055	C₄×D₄	1	2	yes	—
105	#1055	C₂²×D₄	1	1	yes	characteristic
106	#1055	C₂×#41	1	1	yes	characteristic
107	C₂³×C₈	—	13	2	no	—
108	C₂×C₄×D₄	—	13	1	no	—
109	C₂²×C₁₆	—	13	1	no	—
110	C₂²×C₁₆	—	13	1	no	—
111	C₂²×C₁₆	—	13	2	no	—
112	C₂²×C₁₆	—	13	8	no	—
113	C₂²×C₁₆	—	26	8	no	—
114	C₈×D₄	—	13	4	no	—
115	C₈×D₄	—	13	4	no	—
116	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
117	C₂×C₂²⋊C₈	—	13	2	no	—
118	C₂×C₄×C₈	—	13	1	no	—
119	C₄⋊C₁₆	—	13	4	no	—
120	C₂²⋊C₁₆	—	13	8	no	—
121	C₄×C₁₆	—	13	4	no	—
122	C₂²×C₁₃	C₈×D₄	1	2	yes	—
123	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
124	C₂²×C₁₃	#504	1	2	yes	—
125	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
126	C₂×D₁₃	—	2	4	no	—
127	C₂×D₁₃	—	2	4	no	—
128	C₂×D₁₃	—	2	16	no	—
129	C₂×D₁₃	C₂²×C₁₆	1	8	yes	complemented
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	C₈×D₄	1	2	yes	—
132	C₄×C₁₃	C₂²×C₁₆	1	4	yes	complemented
133	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
135	Dic₁₃	—	2	4	no	—
136	C₂²×D₄	—	13	1	no	—
137	C₂³×C₄	—	13	2	no	—
138	C₂²×C₈	—	13	1	no	—
139	C₂²×C₈	—	13	1	no	—
140	C₂²×C₈	—	13	4	no	—
141	C₂²×C₈	—	13	4	no	—
142	C₂²×C₈	—	26	4	no	—
143	C₂²×C₈	—	26	4	no	—
144	C₂²×C₈	C₂×#261	1	2	yes	—
145	C₄×D₄	—	13	4	no	—
146	C₄×D₄	—	13	4	no	—
147	C₂×C₄⋊C₄	—	13	1	no	—
148	C₂×C₂²⋊C₄	—	13	2	no	—
149	C₂×C₄²	C₂×#261	1	1	yes	characteristic
150	C₂×C₁₆	—	13	4	no	—
151	C₂×C₁₆	—	13	2	no	—
152	C₂×C₁₆	—	13	2	no	—
153	C₂×C₁₆	—	13	4	no	—
154	C₂×C₁₆	—	26	16	no	—
155	C₂×C₁₆	—	26	2	no	—
156	C₂×C₁₆	—	26	4	no	—
157	C₄⋊C₈	C₂×#261	1	4	yes	—
158	C₂²⋊C₈	—	13	8	no	—
159	C₄×C₈	—	13	2	no	—
160	C₄×C₈	—	13	2	no	—
161	C₂×C₁₃	D₄×C₁₆	1	4	yes	complemented
162	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
163	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
164	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
165	D₁₃	—	2	8	no	—
166	C₂⁴	—	13	2	no	—
167	C₂×D₄	—	13	8	no	—
168	C₂×D₄	—	13	4	no	—
169	C₂²×C₄	—	13	2	no	—
170	C₂²×C₄	—	13	4	no	—
171	C₂²×C₄	—	13	4	no	—
172	C₂²×C₄	—	26	4	no	—
173	C₂²×C₄	—	26	4	no	—
174	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
175	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	13	1	no	—
181	C₂×C₈	—	13	1	no	—
182	C₂×C₈	—	13	2	no	—
183	C₂×C₈	—	26	4	no	—
184	C₂×C₈	—	26	4	no	—
185	C₂×C₈	—	26	8	no	—
186	C₂×C₈	—	26	2	no	—
187	C₂×C₈	C₂²×#261	1	4	yes	—
188	C₄⋊C₄	—	13	4	no	—
189	C₂²⋊C₄	—	13	8	no	—
190	C₄²	D₁₃⋊C₈	1	2	yes	—
191	C₄²	C₂²×#261	1	2	yes	—
192	C₁₆	—	13	4	no	—
193	C₁₆	—	26	4	no	—
194	C₁₆	—	26	2	no	—
195	C₁₃	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	13	8	no	—
197	C₂³	—	13	4	no	—
198	C₂³	—	26	4	no	—
199	C₂³	—	26	4	no	—
200	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
201	D₄	—	13	16	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	13	4	no	—
204	C₂×C₄	—	26	4	no	—
205	C₂×C₄	—	26	8	no	—
206	C₂×C₄	—	26	4	no	—
207	C₂×C₄	—	26	4	no	—
208	C₂×C₄	D₁₃⋊C₁₆	1	4	yes	complemented
209	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
210	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
211	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
212	C₂×C₄	D₄×#261	1	2	yes	—
213	C₂×C₄	#88240	1	2	yes	—
214	C₂×C₄	C₂³×#261	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	13	1	no	—
217	C₈	—	13	1	no	—
218	C₈	—	13	2	no	—
219	C₈	—	26	2	no	—
220	C₂²	—	13	8	no	—
221	C₂²	—	26	4	no	—
222	C₂²	—	26	4	no	—
223	C₂²	—	26	16	no	—
224	C₂²	#11719073	1	2	yes	—
225	C₂²	C₂×D₁₃⋊C₁₆	1	2	yes	—
226	C₂²	#11719096	1	2	yes	—
227	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	26	4	no	—
230	C₄	#11719073	1	2	yes	—
231	C₄	C₂×D₁₃⋊C₁₆	1	4	yes	complemented
232	C₄	C₂×D₄×#261	1	1	yes	characteristic
233	C₄	C₂×#88240	1	1	yes	characteristic
234	C₂	—	26	8	no	—
235	C₂	#13131215	1	4	yes	direct product factor
236	C₂	C₂×#11719073	1	1	yes	characteristic
237	C₂	C₂²×D₁₃⋊C₁₆	1	1	yes	characteristic
238	C₂	C₂×#11719096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×D₅₂	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₃⋊C₁₆	C₂	semidirect
2. #13131215	C₂	direct (N × H)
3. C₂×C₄⋊#6092	C₂	semidirect
4. C₂×#11718259	C₂	semidirect
5. C₂×C₄×#6092	C₂	semidirect
13. C₂×D₁₃⋊C₁₆	C₂²	semidirect
19. #11718259	C₂²	semidirect
20. C₄⋊#6092	C₂²	semidirect
21. C₄×#6092	C₂²	semidirect
38. C₂×#6092	D₄	semidirect
53. C₂×D₅₂	C₁₆	semidirect
76. #6092	C₂×D₄	semidirect
86. C₂²×D₁₃	C₂×C₁₆	semidirect
89. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
98. D₅₂	C₂×C₁₆	semidirect
129. C₂×D₁₃	C₂²×C₁₆	semidirect
132. C₄×C₁₃	C₂²×C₁₆	semidirect
161. C₂×C₁₃	D₄×C₁₆	semidirect
195. C₁₃	C₂×D₄×C₁₆	semidirect
208. C₂×C₄	D₁₃⋊C₁₆	semidirect
231. C₄	C₂×D₁₃⋊C₁₆	semidirect
235. C₂	#13131215	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₃⋊C₁₆	2
#13131215	2
C₂×C₄⋊#6092	2
C₂×#11718259	2
C₂×C₄×#6092	2
C₂×#11718495	2
C₂×D₄×C₁₆	13

Subgroups — C2×#13131215 #16807121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131215 #16807121