Subgroups — C₂×#13131170 #16806994

Order 3328.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	#13131170	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718260	C₂	1	4	yes	maximal, complemented
4	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂×#88106	—	2	8	no	—
9	C₂×#88106	C₂²	1	8	yes	—
10	C₂²×#6092	C₂²	1	4	yes	—
11	C₂×C₈×Dic₁₃	C₄	1	4	yes	—
12	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
13	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
14	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
15	#11718260	C₂²	1	16	yes	complemented
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×Dic₁₃	C₈	1	2	yes	—
18	C₂²×C₈×C₁₃	—	2	4	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₈	1	4	yes	—
23	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	D₄	1	4	yes	—
29	C₂²×#1055	Q₈	1	4	yes	—
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	C₂²×#1055	C₂³	1	2	yes	—
32	#88106	—	2	16	no	—
33	C₂×#6092	—	2	8	no	—
34	C₂×#6092	C₂³	1	8	yes	—
35	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
36	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
37	C₈×Dic₁₃	C₂³	1	4	yes	—
38	C₂×#2150	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	2	8	no	—
47	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
48	C₂²×Dic₁₃	C₂×C₈	1	8	yes	—
49	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₁₃	—	2	8	no	—
51	C₂×C₈×C₁₃	—	2	4	no	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
53	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
54	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
55	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
56	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
57	C₂×#1055	C₄⋊C₄	1	8	yes	—
58	C₂×#1055	C₄⋊C₄	1	4	yes	—
59	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
60	C₂×#1055	C₂²×C₄	1	2	yes	—
61	C₂×#1055	C₂×D₄	1	2	yes	—
62	C₂×#1055	C₂×D₄	1	4	yes	—
63	C₂×#1055	C₂×Q₈	1	2	yes	—
64	C₂×#1055	C₂×Q₈	1	4	yes	—
65	C₂×#1055	C₂⁴	1	1	yes	characteristic
66	#6092	—	2	8	no	—
67	C₂²×M₃₂	—	13	2	no	—
68	C₂²×C₄×C₈	—	13	1	no	—
69	#2150	—	13	8	no	—
70	C₂×C₄⋊C₁₆	—	13	4	no	—
71	C₂³×C₁₃	—	2	4	no	—
72	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
73	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
74	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
76	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃	—	2	4	no	—
85	C₂×Dic₁₃	—	2	16	no	—
86	C₂×Dic₁₃	M₃₂	1	8	yes	complemented
87	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
88	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
89	C₈×C₁₃	—	2	4	no	—
90	#1055	C₂×C₄⋊C₄	1	2	yes	—
91	#1055	C₂×C₄⋊C₄	1	4	yes	—
92	#1055	C₂²×D₄	1	1	yes	characteristic
93	#1055	C₂²×Q₈	1	1	yes	characteristic
94	C₂³×C₈	—	13	1	no	—
95	C₂³×C₈	#261	1	1	yes	characteristic
96	C₂²×C₄²	—	13	1	no	—
97	C₂×M₃₂	—	13	8	no	—
98	C₂×M₃₂	—	26	8	no	—
99	C₂²×C₁₆	—	13	4	no	—
100	C₂×C₄×C₈	—	13	2	no	—
101	C₂×C₄×C₈	—	13	4	no	—
102	C₂×C₄×C₈	—	13	4	no	—
103	C₂×C₄×C₈	—	13	2	no	—
104	C₄⋊C₁₆	—	13	16	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	8	no	—
107	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
108	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
109	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
110	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
112	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
113	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
115	Dic₁₃	—	2	4	no	—
116	Dic₁₃	C₂×M₃₂	1	8	yes	complemented
117	C₂³×C₄	—	13	2	no	—
118	C₂³×C₄	C₂×#261	1	1	yes	characteristic
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₂²×C₈	—	13	4	no	—
122	C₂²×C₈	—	13	2	no	—
123	C₂²×C₈	—	13	2	no	—
124	C₂²×C₈	—	13	1	no	—
125	C₂²×C₈	—	13	1	no	—
126	C₂²×C₈	C₂×#261	1	4	yes	—
127	C₂²×C₈	C₂×#261	1	2	yes	—
128	C₂×C₄²	—	13	1	no	—
129	C₂×C₄²	—	13	4	no	—
130	C₂×C₄²	—	13	1	no	—
131	C₂×C₄²	—	13	4	no	—
132	C₂×C₄²	—	13	2	no	—
133	M₃₂	—	26	16	no	—
134	C₂×C₁₆	—	13	8	no	—
135	C₂×C₁₆	—	26	8	no	—
136	C₄×C₈	—	13	4	no	—
137	C₄×C₈	—	13	4	no	—
138	C₄×C₈	—	13	8	no	—
139	C₂×C₁₃	—	2	4	no	—
140	C₂×C₁₃	#2150	1	4	yes	complemented
141	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
143	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
144	C₂²×C₄	—	13	2	no	—
145	C₂²×C₄	—	13	8	no	—
146	C₂²×C₄	—	13	2	no	—
147	C₂²×C₄	—	26	8	no	—
148	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
149	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
150	C₂²×C₄	C₄⋊#261	1	4	yes	—
151	C₂²×C₄	#6122	1	4	yes	—
152	C₂²×C₄	C₂²×#261	1	2	yes	—
153	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	13	4	no	—
158	C₂×C₈	—	13	8	no	—
159	C₂×C₈	—	13	4	no	—
160	C₂×C₈	—	13	4	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	1	no	—
163	C₂×C₈	—	13	1	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	C₂²×#261	1	4	yes	—
166	C₄²	—	13	4	no	—
167	C₄²	—	13	4	no	—
168	C₄²	—	13	4	no	—
169	C₄²	—	13	4	no	—
170	C₁₆	—	26	8	no	—
171	C₁₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
172	C₂³	—	2	4	no	—
173	C₂³	#88112	1	4	yes	—
174	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
175	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
176	C₂×C₄	—	13	8	no	—
177	C₂×C₄	—	13	4	no	—
178	C₂×C₄	—	13	4	no	—
179	C₂×C₄	—	26	16	no	—
180	C₂×C₄	—	26	4	no	—
181	C₂×C₄	#88112	1	8	yes	—
182	C₂×C₄	#88112	1	4	yes	—
183	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
184	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
186	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
187	C₂×C₄	C₂×#6122	1	4	yes	—
188	C₂×C₄	C₂×#6122	1	2	yes	—
189	C₂×C₄	C₂³×#261	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	13	1	no	—
192	C₈	—	13	2	no	—
193	C₈	—	13	4	no	—
194	C₈	—	13	1	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	D₁₃⋊M₃₂	1	4	yes	—
198	C₂²	C₂×#88112	1	4	yes	—
199	C₂²	C₂×#88112	1	2	yes	—
200	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
201	C₄	—	13	8	no	—
202	C₄	—	26	4	no	—
203	C₄	C₂×#88112	1	4	yes	—
204	C₄	C₂×#88112	1	2	yes	—
205	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
206	C₄	C₂²×#6122	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#13131170	1	4	yes	direct product factor
209	C₂	C₂²×#88112	1	1	yes	characteristic
210	C₂	C₂×D₁₃⋊M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	C₂×#2150
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #13131170	C₂	direct (N × H)
3. C₂×#11718260	C₂	semidirect
15. #11718260	C₂²	semidirect
86. C₂×Dic₁₃	M₃₂	semidirect
116. Dic₁₃	C₂×M₃₂	semidirect
140. C₂×C₁₃	#2150	semidirect
171. C₁₃	C₂×#2150	semidirect
208. C₂	#13131170	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×#88106	2
#13131170	2
C₂×#11718260	2
C₂²×C₈×Dic₁₃	2
C₂×#2150	13

Subgroups — C2×#13131170 #16806994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131170 #16806994