Subgroups — C₂×#11719453 #13129583

Order 1664.

1484 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6127	C₂	1	2	yes	maximal, complemented
2	C₂²×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719453	C₂	1	16	yes	maximal, direct product factor
4	C₂×#88163	C₂	1	4	yes	maximal, complemented
5	C₂×#88162	C₂	1	4	yes	maximal, complemented
6	C₂×#88158	C₂	1	4	yes	maximal, complemented
7	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₂²⋊D₁₃	—	2	2	no	—
9	C₂×#6127	—	2	2	no	—
10	C₂×#6127	—	2	8	no	—
11	C₂×#6127	—	2	2	no	—
12	C₂×#6127	C₂²	1	4	yes	—
13	C₂×#6127	C₂²	1	4	yes	complemented
14	C₂×#6127	C₂²	1	16	yes	complemented
15	C₂²×D₅₂	C₂²	1	2	yes	complemented
16	C₂²×C₄×D₁₃	—	2	2	no	—
17	C₂²×Dic₂₆	C₂²	1	2	yes	complemented
18	C₂×C₁₃×M₁₆	C₂²	1	4	yes	complemented
19	C₂×C₁₃×M₁₆	C₂²	1	8	yes	complemented
20	C₂²×C₈×C₁₃	C₂²	1	2	yes	complemented
21	#88163	C₂²	1	32	yes	complemented
22	#88162	C₂²	1	32	yes	complemented
23	C₂×Dic₅₂	C₂²	1	4	yes	complemented
24	#88158	C₂²	1	32	yes	complemented
25	C₂×D₁₀₄	C₂²	1	4	yes	complemented
26	C₂×#6095	C₂²	1	8	yes	complemented
27	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
28	C₂³×D₁₃	—	2	2	no	—
29	C₂²×C₄×C₁₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	D₄	1	4	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂×C₂²⋊D₁₃	—	2	4	no	—
36	C₂×C₂²⋊D₁₃	—	2	4	no	—
37	C₂×C₂²⋊D₁₃	—	2	16	no	—
38	C₂²×Dic₁₃	—	2	2	no	—
39	#6127	—	2	8	no	—
40	#6127	—	2	8	no	—
41	#6127	—	2	16	no	—
42	#6127	—	2	16	no	—
43	#6127	C₂³	1	16	yes	—
44	#6127	C₂³	1	16	yes	complemented
45	C₂×D₅₂	—	2	4	no	—
46	C₂×D₅₂	—	2	2	no	—
47	C₂×D₅₂	C₂³	1	4	yes	—
48	C₂×D₅₂	C₂³	1	8	yes	complemented
49	C₂×C₄×D₁₃	—	2	4	no	—
50	C₂×C₄×D₁₃	—	2	16	no	—
51	C₂×C₄×D₁₃	—	2	4	no	—
52	C₂×Dic₂₆	—	2	4	no	—
53	C₂×Dic₂₆	—	2	2	no	—
54	C₂×Dic₂₆	C₂³	1	4	yes	—
55	C₂×Dic₂₆	C₂³	1	8	yes	complemented
56	C₁₃×M₁₆	C₂³	1	16	yes	complemented
57	C₂×C₈×C₁₃	C₂³	1	4	yes	—
58	C₂×C₈×C₁₃	C₂³	1	8	yes	complemented
59	Dic₅₂	C₂³	1	16	yes	complemented
60	D₁₀₄	C₂³	1	16	yes	complemented
61	#6095	C₂³	1	32	yes	complemented
62	C₂×#575	—	13	1	no	maximal, Sylow
63	C₂³×C₁₃	—	4	2	no	—
64	C₂³×C₁₃	C₂×D₄	1	2	yes	—
65	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₁₃	C₂×D₄	1	4	yes	—
67	C₂²×D₁₃	—	2	4	no	—
68	C₂²×D₁₃	—	2	8	no	—
69	C₂²×D₁₃	—	4	8	no	—
70	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₁₃	C₂×D₄	1	8	yes	—
73	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
75	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
78	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
79	C₂²⋊D₁₃	—	2	16	no	—
80	C₂²⋊D₁₃	—	2	16	no	—
81	C₂×Dic₁₃	—	2	4	no	—
82	C₂×Dic₁₃	—	2	8	no	—
83	D₅₂	—	2	4	no	—
84	D₅₂	—	2	8	no	—
85	D₅₂	C₂⁴	1	8	yes	—
86	C₄×D₁₃	—	2	16	no	—
87	C₄×D₁₃	—	2	16	no	—
88	Dic₂₆	—	2	4	no	—
89	Dic₂₆	—	2	8	no	—
90	Dic₂₆	C₂⁴	1	8	yes	—
91	C₈×C₁₃	C₂⁴	1	8	yes	—
92	C₂²×#41	—	13	2	no	—
93	#575	—	13	16	no	—
94	C₂×#137	—	13	4	no	—
95	C₂×AGL(1,ℤ/8)	—	13	4	no	—
96	C₂×#135	—	13	4	no	—
97	C₂²×M₁₆	D₁₃	1	1	yes	characteristic, complemented
98	C₂²×C₁₃	—	2	4	no	—
99	C₂²×C₁₃	—	2	2	no	—
100	C₂²×C₁₃	—	4	4	no	—
101	C₂²×C₁₃	C₂²×D₄	1	4	yes	—
102	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
103	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₁₃	—	2	8	no	—
105	C₂×D₁₃	—	4	16	no	—
106	C₂×D₁₃	—	4	8	no	—
107	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₃	C₂²×D₄	1	4	yes	—
109	C₄×C₁₃	C₂²×D₄	1	2	yes	—
110	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
111	Dic₁₃	—	2	8	no	—
112	C₂×#41	—	13	4	no	—
113	C₂×#41	—	13	4	no	—
114	C₂×#41	—	13	16	no	—
115	C₂×#41	—	26	2	no	—
116	C₂×#41	—	26	8	no	—
117	C₂×#41	—	26	2	no	—
118	C₂²×Q₈	—	13	2	no	—
119	C₂²×D₄	—	13	2	no	—
120	C₂²×D₄	—	26	2	no	—
121	C₂³×C₄	—	26	2	no	—
122	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
123	#137	—	13	32	no	—
124	AGL(1,ℤ/8)	—	13	32	no	—
125	#135	—	13	32	no	—
126	C₂×Q₁₆	—	13	4	no	—
127	C₂×SD₁₆	—	13	8	no	—
128	C₂×D₈	—	13	4	no	—
129	C₂×M₁₆	C₂×D₁₃	1	4	yes	complemented
130	C₂×M₁₆	C₂×D₁₃	1	8	yes	complemented
131	C₂²×C₈	C₂×D₁₃	1	2	yes	complemented
132	C₂×C₁₃	—	2	4	no	—
133	C₂×C₁₃	—	2	2	no	—
134	C₂×C₁₃	#575	1	2	yes	complemented
135	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
136	D₁₃	—	4	8	no	—
137	C₂⁴	—	26	2	no	—
138	C₂⁴	D₅₂	1	1	yes	characteristic
139	#41	—	13	16	no	—
140	#41	—	13	16	no	—
141	#41	—	26	16	no	—
142	#41	—	26	8	no	—
143	#41	—	26	8	no	—
144	#41	—	26	16	no	—
145	C₂×Q₈	—	13	4	no	—
146	C₂×Q₈	—	13	8	no	—
147	C₂×Q₈	—	26	2	no	—
148	C₂×Q₈	—	26	4	no	—
149	C₂×D₄	—	13	4	no	—
150	C₂×D₄	—	13	8	no	—
151	C₂×D₄	—	26	4	no	—
152	C₂×D₄	—	26	4	no	—
153	C₂×D₄	—	26	16	no	—
154	C₂×D₄	—	26	2	no	—
155	C₂×D₄	—	26	4	no	—
156	C₂²×C₄	—	26	2	no	—
157	C₂²×C₄	—	26	4	no	—
158	C₂²×C₄	—	26	16	no	—
159	C₂²×C₄	—	26	4	no	—
160	C₂²×C₄	D₅₂	1	1	yes	characteristic
161	C₂²×C₄	D₅₂	1	4	yes	—
162	C₂²×C₄	D₅₂	1	2	yes	—
163	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
165	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
166	Q₁₆	—	13	16	no	—
167	SD₁₆	—	13	32	no	—
168	D₈	—	13	16	no	—
169	M₁₆	C₂²×D₁₃	1	16	yes	complemented
170	C₂×C₈	C₂²×D₁₃	1	4	yes	—
171	C₂×C₈	C₂²×D₁₃	1	8	yes	complemented
172	C₁₃	C₂×#575	1	1	yes	characteristic, complemented, Sylow
173	C₂³	—	4	2	no	—
174	C₂³	—	26	8	no	—
175	C₂³	—	26	4	no	—
176	C₂³	—	52	8	no	—
177	C₂³	C₂×D₅₂	1	2	yes	—
178	C₂³	C₂×D₅₂	1	1	yes	characteristic
179	C₂³	C₂×D₅₂	1	4	yes	—
180	Q₈	—	13	8	no	—
181	Q₈	—	26	8	no	—
182	Q₈	—	26	4	no	—
183	D₄	—	13	8	no	—
184	D₄	—	26	16	no	—
185	D₄	—	26	16	no	—
186	D₄	—	26	8	no	—
187	D₄	—	26	4	no	—
188	C₂×C₄	—	26	8	no	—
189	C₂×C₄	—	26	4	no	—
190	C₂×C₄	—	26	16	no	—
191	C₂×C₄	—	26	16	no	—
192	C₂×C₄	C₂×D₅₂	1	2	yes	—
193	C₂×C₄	C₂×D₅₂	1	8	yes	—
194	C₂×C₄	C₂×D₅₂	1	4	yes	—
195	C₂×C₄	C₂×D₅₂	1	2	yes	—
196	C₂×C₄	C₂×D₅₂	1	4	yes	—
197	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
198	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
199	C₂×C₄	C₂³×D₁₃	1	2	yes	—
200	C₂×C₄	C₂³×D₁₃	1	4	yes	—
201	C₈	C₂³×D₁₃	1	8	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	4	no	—
205	C₂²	—	26	8	no	—
206	C₂²	—	52	16	no	—
207	C₂²	—	52	8	no	—
208	C₂²	C₂²×D₅₂	1	4	yes	—
209	C₂²	C₂²×D₅₂	1	2	yes	—
210	C₂²	C₂²×D₅₂	1	1	yes	characteristic
211	C₄	—	26	8	no	—
212	C₄	C₂²×D₅₂	1	2	yes	—
213	C₄	C₂²×D₅₂	1	4	yes	—
214	C₄	C₂²×D₅₂	1	1	yes	characteristic
215	C₄	C₂⁴×D₁₃	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	—	2	2	no	—
218	C₂	—	52	8	no	—
219	C₂	#11719453	1	2	yes	direct product factor
220	C₂	C₂³×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₂
commutator, cosocle	C₄×C₁₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	C₂²×M₁₆	D₁₃
hypocenter	C₁₃	C₂×#575
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#6127	C₂	semidirect
2. C₂²×C₁₃×M₁₆	C₂	semidirect
3. #11719453	C₂	direct (N × H)
4. C₂×#88163	C₂	semidirect
5. C₂×#88162	C₂	semidirect
6. C₂×#88158	C₂	semidirect
13. C₂×#6127	C₂²	semidirect
14. C₂×#6127	C₂²	semidirect
15. C₂²×D₅₂	C₂²	semidirect
17. C₂²×Dic₂₆	C₂²	semidirect
18. C₂×C₁₃×M₁₆	C₂²	semidirect
19. C₂×C₁₃×M₁₆	C₂²	semidirect
20. C₂²×C₈×C₁₃	C₂²	semidirect
21. #88163	C₂²	semidirect
22. #88162	C₂²	semidirect
23. C₂×Dic₅₂	C₂²	semidirect
24. #88158	C₂²	semidirect
25. C₂×D₁₀₄	C₂²	semidirect
26. C₂×#6095	C₂²	semidirect
44. #6127	C₂³	semidirect
48. C₂×D₅₂	C₂³	semidirect
55. C₂×Dic₂₆	C₂³	semidirect
56. C₁₃×M₁₆	C₂³	semidirect
58. C₂×C₈×C₁₃	C₂³	semidirect
59. Dic₅₂	C₂³	semidirect
60. D₁₀₄	C₂³	semidirect
61. #6095	C₂³	semidirect
97. C₂²×M₁₆	D₁₃	semidirect
129. C₂×M₁₆	C₂×D₁₃	semidirect
130. C₂×M₁₆	C₂×D₁₃	semidirect
131. C₂²×C₈	C₂×D₁₃	semidirect
134. C₂×C₁₃	#575	semidirect
169. M₁₆	C₂²×D₁₃	semidirect
171. C₂×C₈	C₂²×D₁₃	semidirect
172. C₁₃	C₂×#575	semidirect
219. C₂	#11719453	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×#6127	2
C₂²×C₁₃×M₁₆	2
#11719453	2
C₂×#88163	2
C₂×#88162	2
C₂×#88158	2
C₂×#575	13

Subgroups — C2×#11719453 #13129583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719453 #13129583