Subgroups — C₂×#13127977 #16769431

Order 3328.

2126 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719457	C₂	1	2	yes	maximal, complemented
2	#13127977	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718569	C₂	1	6	yes	maximal, complemented
4	C₂×#11718565	C₂	1	3	yes	maximal, complemented
5	C₂²×D₂₀₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88256	—	2	2	no	—
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	#11719457	—	2	8	no	—
10	#11719457	C₂²	1	16	yes	complemented
11	C₂×#88162	—	2	6	no	—
12	C₂×#88158	C₂²	1	6	yes	complemented
13	C₂²×D₁₀₄	C₂²	1	6	yes	complemented
14	C₁₃×#504	C₂²	1	8	yes	complemented
15	C₂×C₁₃×M₃₂	C₂²	1	3	yes	complemented
16	C₂²×C₁₃×C₁₆	C₂²	1	3	yes	complemented
17	#11718569	C₂²	1	48	yes	complemented
18	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
19	#11718565	C₂²	1	24	yes	complemented
20	C₂×#88040	C₂²	1	6	yes	complemented
21	C₂×D₂₀₈	C₂²	1	3	yes	complemented
22	C₂×D₂₀₈	C₂²	1	6	yes	complemented
23	C₂×D₂₀₈	C₂²	1	24	yes	complemented
24	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
25	#88256	—	2	16	no	—
26	#88256	—	4	8	no	—
27	C₂×#6131	—	4	2	no	—
28	C₂×D₄×D₁₃	—	4	6	no	—
29	C₂×#6127	—	2	6	no	—
30	C₂²×D₅₂	—	2	6	no	—
31	C₁₃×#131	D₄	1	4	yes	—
32	C₁₃×#131	C₂³	1	4	yes	—
33	C₂×C₁₃×M₁₆	D₄	1	3	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
35	#88162	—	2	48	no	—
36	C₂×Dic₅₂	C₂³	1	2	yes	—
37	#88158	—	2	12	no	—
38	#88158	C₂³	1	24	yes	complemented
39	C₂×D₁₀₄	—	2	6	no	—
40	C₂×D₁₀₄	—	2	12	no	—
41	C₂×D₁₀₄	C₂³	1	6	yes	—
42	C₂×D₁₀₄	C₂³	1	24	yes	complemented
43	C₂×#6095	—	2	6	no	—
44	C₁₃×M₃₂	C₂³	1	12	yes	complemented
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	3	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	12	yes	complemented
48	Dic₁₀₄	C₂³	1	4	yes	complemented
49	#88040	C₂³	1	24	yes	complemented
50	D₂₀₈	C₂³	1	12	yes	complemented
51	D₂₀₈	C₂³	1	24	yes	complemented
52	C₂×#3415	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	—	4	6	no	—
54	C₁₃×#41	D₈	1	4	yes	—
55	C₁₃×#41	C₂×D₄	1	4	yes	—
56	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
57	C₂×D₄×C₁₃	D₈	1	3	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
59	C₂×C₂²⋊D₁₃	—	4	6	no	—
60	#6131	—	4	16	no	—
61	D₄×D₁₃	—	4	48	no	—
62	#6127	—	2	24	no	—
63	#6127	—	4	12	no	—
64	C₂×D₅₂	—	2	6	no	—
65	C₂×D₅₂	—	2	24	no	—
66	C₂×D₅₂	—	4	6	no	—
67	C₂×D₅₂	—	4	12	no	—
68	C₂×C₄×D₁₃	—	4	6	no	—
69	C₂×Dic₂₆	—	2	2	no	—
70	C₁₃×M₁₆	C₂×D₄	1	12	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
75	Dic₅₂	—	2	2	no	—
76	Dic₅₂	C₂⁴	1	4	yes	—
77	D₁₀₄	—	2	6	no	—
78	D₁₀₄	—	2	24	no	—
79	D₁₀₄	C₂⁴	1	12	yes	—
80	#6095	—	2	24	no	—
81	C₁₃×C₁₆	C₂⁴	1	2	yes	—
82	C₁₃×C₁₆	C₂⁴	1	6	yes	—
83	C₂×#576	—	13	2	no	—
84	#3415	—	13	16	no	—
85	C₂×#509	—	13	6	no	—
86	C₂×#508	—	13	3	no	—
87	C₂²×D₁₆	—	13	3	no	—
88	C₂×#504	D₁₃	1	1	yes	characteristic, complemented
89	C₂³×C₁₃	C₂×D₈	1	3	yes	—
90	C₂²×D₁₃	—	4	6	no	—
91	C₂²×D₁₃	—	4	24	no	—
92	C₂²×D₁₃	—	16	12	no	—
93	Q₈×C₁₃	C₂×D₈	1	4	yes	—
94	D₄×C₁₃	C₂×D₈	1	12	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
96	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
99	C₂²⋊D₁₃	—	4	24	no	—
100	C₂×Dic₁₃	—	4	2	no	—
101	D₅₂	—	2	12	no	—
102	D₅₂	—	4	6	no	—
103	D₅₂	—	4	24	no	—
104	C₄×D₁₃	—	4	24	no	—
105	Dic₂₆	—	2	4	no	—
106	Dic₂₆	—	4	2	no	—
107	C₈×C₁₃	C₂²×D₄	1	6	yes	—
108	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
110	C₂×H(5,𝔽₂)	—	26	2	no	—
111	#576	—	13	16	no	—
112	#576	—	26	8	no	—
113	C₂×AGL(1,ℤ/8)	—	26	6	no	—
114	C₂×#135	—	13	6	no	—
115	C₂²×D₈	—	13	6	no	—
116	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
117	#509	—	13	48	no	—
118	#508	—	13	24	no	—
119	C₂×Q₃₂	—	13	1	no	—
120	C₂×SD₃₂	—	13	6	no	—
121	C₂×D₁₆	—	13	3	no	—
122	C₂×D₁₆	—	13	6	no	—
123	C₂×D₁₆	—	13	24	no	—
124	#504	C₂×D₁₃	1	8	yes	complemented
125	C₂×M₃₂	C₂×D₁₃	1	3	yes	complemented
126	C₂²×C₁₆	C₂×D₁₃	1	3	yes	complemented
127	C₂²×C₁₃	—	2	6	no	—
128	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
129	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
130	C₂×D₁₃	—	4	12	no	—
131	C₂×D₁₃	—	8	12	no	—
132	C₂×D₁₃	—	16	24	no	—
133	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₁₃	C₂²×D₈	1	6	yes	—
135	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
136	Dic₁₃	—	4	4	no	—
137	H(5,𝔽₂)	—	26	16	no	—
138	H(5,𝔽₂)	—	52	8	no	—
139	C₂×#41	—	26	6	no	—
140	C₂×#41	—	52	2	no	—
141	C₂×#41	D₅₂	1	1	yes	characteristic
142	C₂²×D₄	—	26	6	no	—
143	C₂²×D₄	—	52	6	no	—
144	AGL(1,ℤ/8)	—	26	48	no	—
145	#135	—	13	24	no	—
146	#135	—	26	12	no	—
147	C₂×Q₁₆	—	13	2	no	—
148	C₂×SD₁₆	—	26	6	no	—
149	C₂×D₈	—	13	6	no	—
150	C₂×D₈	—	13	24	no	—
151	C₂×D₈	—	26	6	no	—
152	C₂×D₈	—	26	12	no	—
153	#131	D₅₂	1	4	yes	—
154	#131	C₂²×D₁₃	1	4	yes	—
155	C₂×M₁₆	D₅₂	1	3	yes	—
156	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
157	Q₃₂	—	13	4	no	—
158	SD₃₂	—	13	24	no	—
159	D₁₆	—	13	12	no	—
160	D₁₆	—	13	24	no	—
161	M₃₂	C₂²×D₁₃	1	12	yes	complemented
162	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
163	C₂×C₁₆	C₂²×D₁₃	1	3	yes	—
164	C₂×C₁₆	C₂²×D₁₃	1	12	yes	complemented
165	C₂×C₁₃	—	2	6	no	—
166	C₂×C₁₃	#3415	1	2	yes	complemented
167	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
168	D₁₃	—	8	12	no	—
169	C₂⁴	—	52	6	no	—
170	#41	—	26	24	no	—
171	#41	—	52	16	no	—
172	#41	—	52	12	no	—
173	#41	D₁₀₄	1	4	yes	—
174	#41	C₂×D₅₂	1	4	yes	—
175	C₂×Q₈	—	26	2	no	—
176	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
177	C₂×D₄	—	26	6	no	—
178	C₂×D₄	—	26	24	no	—
179	C₂×D₄	—	52	6	no	—
180	C₂×D₄	—	52	48	no	—
181	C₂×D₄	—	52	6	no	—
182	C₂×D₄	—	52	12	no	—
183	C₂×D₄	D₁₀₄	1	3	yes	—
184	C₂²×C₄	—	52	6	no	—
185	C₂²×C₄	C₂×D₅₂	1	3	yes	—
186	Q₁₆	—	13	4	no	—
187	Q₁₆	—	26	2	no	—
188	SD₁₆	—	26	24	no	—
189	D₈	—	13	12	no	—
190	D₈	—	26	6	no	—
191	D₈	—	26	24	no	—
192	M₁₆	C₂×D₅₂	1	12	yes	—
193	C₂×C₈	C₂×D₅₂	1	3	yes	—
194	C₂×C₈	C₂×D₅₂	1	6	yes	—
195	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂³×D₁₃	1	6	yes	—
197	C₁₆	C₂³×D₁₃	1	2	yes	—
198	C₁₆	C₂³×D₁₃	1	6	yes	—
199	C₁₃	C₂×#3415	1	1	yes	characteristic, complemented, Sylow
200	C₂³	—	52	6	no	—
201	C₂³	—	52	24	no	—
202	C₂³	—	208	12	no	—
203	C₂³	C₂×D₁₀₄	1	3	yes	—
204	Q₈	—	26	4	no	—
205	Q₈	—	52	2	no	—
206	Q₈	C₂×D₁₀₄	1	4	yes	—
207	D₄	—	26	12	no	—
208	D₄	—	52	24	no	—
209	D₄	—	52	24	no	—
210	D₄	—	52	6	no	—
211	D₄	C₂×D₁₀₄	1	12	yes	—
212	C₂×C₄	—	52	2	no	—
213	C₂×C₄	—	52	24	no	—
214	C₂×C₄	C₂×D₁₀₄	1	6	yes	—
215	C₂×C₄	C₂×D₁₀₄	1	3	yes	—
216	C₂×C₄	C₂²×D₅₂	1	6	yes	—
217	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
218	C₈	C₂²×D₅₂	1	1	yes	characteristic
219	C₈	C₂²×D₅₂	1	6	yes	—
220	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
221	C₂²	—	2	6	no	—
222	C₂²	—	52	12	no	—
223	C₂²	—	104	12	no	—
224	C₂²	—	208	24	no	—
225	C₂²	C₂²×D₁₀₄	1	6	yes	—
226	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
227	C₄	—	52	4	no	—
228	C₄	C₂²×D₁₀₄	1	6	yes	—
229	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
230	C₄	C₂³×D₅₂	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	104	12	no	—
233	C₂	#13127977	1	2	yes	direct product factor
234	C₂	C₂³×D₁₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₈×C₁₃	C₂⁵
Frattini	C₈	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#504	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	C₂×#3415
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#11719457	C₂	semidirect
2. #13127977	C₂	direct (N × H)
3. C₂×#11718569	C₂	semidirect
4. C₂×#11718565	C₂	semidirect
5. C₂²×D₂₀₈	C₂	semidirect
6. C₂×C₁₃×#504	C₂	semidirect
10. #11719457	C₂²	semidirect
12. C₂×#88158	C₂²	semidirect
13. C₂²×D₁₀₄	C₂²	semidirect
14. C₁₃×#504	C₂²	semidirect
15. C₂×C₁₃×M₃₂	C₂²	semidirect
16. C₂²×C₁₃×C₁₆	C₂²	semidirect
17. #11718569	C₂²	semidirect
18. C₂×Dic₁₀₄	C₂²	semidirect
19. #11718565	C₂²	semidirect
20. C₂×#88040	C₂²	semidirect
21. C₂×D₂₀₈	C₂²	semidirect
22. C₂×D₂₀₈	C₂²	semidirect
23. C₂×D₂₀₈	C₂²	semidirect
38. #88158	C₂³	semidirect
42. C₂×D₁₀₄	C₂³	semidirect
44. C₁₃×M₃₂	C₂³	semidirect
47. C₂×C₁₃×C₁₆	C₂³	semidirect
48. Dic₁₀₄	C₂³	semidirect
49. #88040	C₂³	semidirect
50. D₂₀₈	C₂³	semidirect
51. D₂₀₈	C₂³	semidirect
88. C₂×#504	D₁₃	semidirect
124. #504	C₂×D₁₃	semidirect
125. C₂×M₃₂	C₂×D₁₃	semidirect
126. C₂²×C₁₆	C₂×D₁₃	semidirect
161. M₃₂	C₂²×D₁₃	semidirect
164. C₂×C₁₆	C₂²×D₁₃	semidirect
166. C₂×C₁₃	#3415	semidirect
199. C₁₃	C₂×#3415	semidirect
233. C₂	#13127977	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×#11719457	2
#13127977	2
C₂×#11718569	2
C₂×#11718565	2
C₂²×D₂₀₈	2
C₂×C₁₃×#504	2
C₂×#3415	13

Subgroups — C2×#13127977 #16769431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127977 #16769431