Subgroups — C₂×#11718857 #13123726

Order 1664.

472 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	2	yes	maximal, complemented
3	C₂×#88077	C₂	1	2	yes	maximal, complemented
4	#11718857	C₂	1	8	yes	maximal, direct product factor
5	C₂×#88059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#41	C₂²	1	2	yes	—
8	C₁₃×#135	C₄	1	4	yes	complemented
9	C₁₃×#135	C₂²	1	4	yes	—
10	C₂×C₁₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₃×SD₁₆	C₄	1	2	yes	complemented
12	C₂×C₁₃×D₈	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#88202	—	2	4	no	—
15	#88202	C₂²	1	8	yes	complemented
16	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×#6099	—	2	2	no	—
18	C₂×#6099	C₂²	1	2	yes	—
19	C₂²×#1055	—	2	2	no	—
20	C₂²×#1055	C₂²	1	1	yes	characteristic
21	#88077	C₂²	1	8	yes	complemented
22	#88059	C₂²	1	4	yes	complemented
23	C₈×Dic₁₃	C₂²	1	4	yes	complemented
24	C₁₃×#41	—	2	2	no	—
25	C₁₃×#41	—	2	2	no	—
26	C₁₃×#41	C₂×C₄	1	4	yes	—
27	C₁₃×#41	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₁₃	—	2	2	no	—
30	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₃	—	2	1	no	—
34	C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
35	C₁₃×SD₁₆	C₂×C₄	1	8	yes	complemented
36	C₁₃×D₈	C₂×C₄	1	4	yes	complemented
37	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₂³	1	2	yes	—
41	C₄×Dic₁₃	—	2	1	no	—
42	C₄×Dic₁₃	C₂³	1	2	yes	—
43	#6099	—	2	2	no	—
44	#6099	—	2	8	no	—
45	#6099	C₂³	1	4	yes	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	8	no	—
48	C₂×#1055	D₄	1	1	yes	characteristic
49	C₂×#1055	D₄	1	1	yes	characteristic
50	C₂×#1055	D₄	1	2	yes	—
51	C₂×#1055	C₂³	1	2	yes	—
52	C₂×#443	—	13	1	no	maximal, Sylow
53	C₂³×C₁₃	—	2	2	no	—
54	C₂³×C₁₃	#41	1	1	yes	characteristic
55	Q₈×C₁₃	—	2	2	no	—
56	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
57	D₄×C₁₃	—	2	4	no	—
58	D₄×C₁₃	—	2	4	no	—
59	D₄×C₁₃	—	2	2	no	—
60	D₄×C₁₃	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂×Dic₁₃	—	2	4	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	#1055	—	2	4	no	—
75	#1055	C₂×D₄	1	2	yes	—
76	#1055	C₂×D₄	1	2	yes	—
77	C₂×#135	D₁₃	1	1	yes	characteristic
78	C₂×#131	—	13	2	no	—
79	#443	—	13	8	no	—
80	C₂×#108	—	13	1	no	—
81	C₂×#104	—	13	2	no	—
82	C₂×C₄×C₈	—	13	1	no	—
83	C₂²×C₁₃	—	2	2	no	—
84	C₂²×C₁₃	—	2	4	no	—
85	C₂²×C₁₃	—	4	4	no	—
86	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
89	C₄×C₁₃	—	2	4	no	—
90	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
91	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
92	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
94	Dic₁₃	—	2	4	no	—
95	C₂×#41	C₂×D₁₃	1	2	yes	—
96	#135	Dic₁₃	1	4	yes	complemented
97	#135	C₂×D₁₃	1	4	yes	—
98	C₂×Q₁₆	Dic₁₃	1	1	yes	characteristic, complemented
99	C₂×SD₁₆	Dic₁₃	1	2	yes	complemented
100	C₂×D₈	Dic₁₃	1	1	yes	characteristic, complemented
101	#131	—	13	8	no	—
102	#131	—	26	4	no	—
103	C₂×M₁₆	—	13	2	no	—
104	C₂×M₁₆	—	26	2	no	—
105	C₂²×C₈	—	13	1	no	—
106	C₂²×C₈	—	26	2	no	—
107	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
108	C₂×C₄²	—	13	1	no	—
109	#108	—	13	4	no	—
110	#104	—	13	8	no	—
111	C₄×C₈	—	13	4	no	—
112	C₂×C₁₃	—	2	1	no	—
113	C₂×C₁₃	—	2	1	no	—
114	C₂×C₁₃	—	4	4	no	—
115	C₂×C₁₃	#443	1	2	yes	complemented
116	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
117	#41	—	2	2	no	—
118	#41	—	2	2	no	—
119	#41	C₂×Dic₁₃	1	4	yes	—
120	#41	C₂²×D₁₃	1	4	yes	—
121	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	26	1	no	—
126	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
127	Q₁₆	C₂×Dic₁₃	1	4	yes	complemented
128	SD₁₆	C₂×Dic₁₃	1	8	yes	complemented
129	D₈	C₂×Dic₁₃	1	4	yes	complemented
130	M₁₆	—	13	4	no	—
131	M₁₆	—	26	8	no	—
132	M₁₆	—	26	2	no	—
133	C₂×C₈	—	13	2	no	—
134	C₂×C₈	—	13	1	no	—
135	C₂×C₈	—	13	1	no	—
136	C₂×C₈	—	13	2	no	—
137	C₂×C₈	—	26	8	no	—
138	C₂×C₈	—	26	2	no	—
139	C₂×C₈	C₂×Dic₁₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₃	1	2	yes	—
142	C₂×C₈	C₂²×D₁₃	1	2	yes	—
143	C₄²	—	13	2	no	—
144	C₄²	—	26	1	no	—
145	C₁₃	C₂×#443	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	2	2	no	—
147	C₂³	#6129	1	1	yes	characteristic
148	Q₈	—	2	2	no	—
149	Q₈	C₂²×Dic₁₃	1	4	yes	—
150	D₄	—	2	4	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	2	no	—
153	D₄	C₂²×Dic₁₃	1	4	yes	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	26	4	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
160	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
161	C₂×C₄	#6129	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
163	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
165	C₈	—	13	2	no	—
166	C₈	—	13	2	no	—
167	C₈	—	26	4	no	—
168	C₈	C₂²×Dic₁₃	1	2	yes	—
169	C₈	C₂²×Dic₁₃	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	D₄×Dic₁₃	1	1	yes	characteristic
174	C₂²	D₄×Dic₁₃	1	1	yes	characteristic
175	C₂²	C₂×#6129	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	26	4	no	—
178	C₄	D₄×Dic₁₃	1	1	yes	characteristic
179	C₄	D₄×Dic₁₃	1	1	yes	characteristic
180	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
181	C₄	C₂³×Dic₁₃	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	4	4	no	—
185	C₂	#11718857	1	2	yes	direct product factor
186	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₄×D₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×#135	D₁₃
hypocenter	C₁₃	C₂×#443
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂×#88077	C₂	semidirect
4. #11718857	C₂	direct (N × H)
5. C₂×#88059	C₂	semidirect
6. C₂×C₈×Dic₁₃	C₂	semidirect
8. C₁₃×#135	C₄	semidirect
10. C₂×C₁₃×Q₁₆	C₄	semidirect
11. C₂×C₁₃×SD₁₆	C₄	semidirect
12. C₂×C₁₃×D₈	C₄	semidirect
15. #88202	C₂²	semidirect
21. #88077	C₂²	semidirect
22. #88059	C₂²	semidirect
23. C₈×Dic₁₃	C₂²	semidirect
34. C₁₃×Q₁₆	C₂×C₄	semidirect
35. C₁₃×SD₁₆	C₂×C₄	semidirect
36. C₁₃×D₈	C₂×C₄	semidirect
96. #135	Dic₁₃	semidirect
98. C₂×Q₁₆	Dic₁₃	semidirect
99. C₂×SD₁₆	Dic₁₃	semidirect
100. C₂×D₈	Dic₁₃	semidirect
115. C₂×C₁₃	#443	semidirect
127. Q₁₆	C₂×Dic₁₃	semidirect
128. SD₁₆	C₂×Dic₁₃	semidirect
129. D₈	C₂×Dic₁₃	semidirect
145. C₁₃	C₂×#443	semidirect
185. C₂	#11718857	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×C₁₃×#135	2
C₂×#88202	2
C₂×#88077	2
#11718857	2
C₂×#88059	2
C₂×C₈×Dic₁₃	2
C₂×#443	13

Subgroups — C2×#11718857 #13123726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11718857 #13123726