Subgroups — C₂×#16840236 #21459005

Order 7808.

1920 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	C₂³×D₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#16840236	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×D₂₄₄	C₂	1	4	yes	maximal, complemented
4	C₂×#14519739	C₂	1	4	yes	maximal, complemented
5	C₂×#14519736	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₄×D₆₁	C₂	1	4	yes	maximal, complemented
7	C₂×#117×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×D₆₁	C₂²	1	2	yes	—
9	C₂²×D₂₄₄	C₄	1	16	yes	complemented
10	C₂²×D₂₄₄	C₂²	1	2	yes	—
11	C₂²×D₂₄₄	C₂²	1	2	yes	—
12	C₂²×D₂₄₄	C₂²	1	8	yes	—
13	C₂²×C₄×D₆₁	C₂²	1	8	yes	complemented
14	C₄×D₂₄₄	C₂²	1	32	yes	complemented
15	#14519739	C₂²	1	32	yes	complemented
16	#14519736	C₂²	1	8	yes	complemented
17	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	complemented
18	C₂×C₂²⋊Dic₆₁	C₂²	1	2	yes	—
19	C₂×#11758198	C₂²	1	8	yes	complemented
20	C₂²⋊C₄×D₆₁	C₂²	1	32	yes	complemented
21	C₂×C₄×Dic₆₁	C₂²	1	2	yes	complemented
22	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
23	#117×C₆₁	C₂²	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂²	1	2	yes	complemented
27	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
28	C₂³×D₆₁	—	4	8	no	—
29	C₂³×D₆₁	C₂×C₄	1	16	yes	complemented
30	C₂³×D₆₁	C₂³	1	4	yes	—
31	C₂³×D₆₁	C₂³	1	8	yes	—
32	C₂²×C₄×C₆₁	C₂×C₄	1	8	yes	complemented
33	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
37	C₂²×Dic₆₁	C₂³	1	4	yes	—
38	C₂×D₂₄₄	C₂×C₄	1	16	yes	complemented
39	C₂×D₂₄₄	C₂×C₄	1	16	yes	complemented
40	C₂×D₂₄₄	C₂×C₄	1	64	yes	complemented
41	C₂×D₂₄₄	C₂³	1	8	yes	—
42	C₂×D₂₄₄	C₂³	1	8	yes	—
43	C₂×C₄×D₆₁	—	4	16	no	—
44	C₂×C₄×D₆₁	C₂³	1	32	yes	complemented
45	C₄⋊C₄×C₆₁	C₂³	1	8	yes	complemented
46	C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
47	C₄²×C₆₁	C₂³	1	8	yes	complemented
48	C₂²⋊Dic₆₁	C₂³	1	8	yes	—
49	#11758198	C₂³	1	32	yes	complemented
50	C₄⋊Dic₆₁	C₂³	1	8	yes	complemented
51	C₄×Dic₆₁	C₂³	1	8	yes	complemented
52	C₂³×C₆₁	—	2	4	no	—
53	C₂³×C₆₁	C₂²×C₄	1	4	yes	—
54	C₂³×C₆₁	C₂⁴	1	2	yes	—
55	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₂²×D₆₁	—	2	16	no	—
57	C₂²×D₆₁	—	4	32	no	—
58	C₂²×D₆₁	—	4	16	no	—
59	C₂²×D₆₁	C₂²×C₄	1	16	yes	—
60	C₂²×D₆₁	C₂²×C₄	1	32	yes	complemented
61	C₂²×D₆₁	C₂⁴	1	8	yes	—
62	C₂×C₄×C₆₁	—	2	8	no	—
63	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₆₁	C₂²×C₄	1	16	yes	complemented
66	C₂×C₄×C₆₁	C₂⁴	1	8	yes	—
67	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
68	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
69	C₂×Dic₆₁	—	2	8	no	—
70	C₂×Dic₆₁	C₂⁴	1	8	yes	—
71	D₂₄₄	C₂²×C₄	1	32	yes	complemented
72	D₂₄₄	C₂²×C₄	1	32	yes	complemented
73	C₄×D₆₁	—	4	32	no	—
74	C₂²×C₆₁	—	2	8	no	—
75	C₂²×C₆₁	—	2	4	no	—
76	C₂²×C₆₁	C₂³×C₄	1	4	yes	—
77	C₂²×C₆₁	C₂³×C₄	1	2	yes	—
78	C₂²×C₆₁	H(5,𝔽₂)	1	4	yes	—
79	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
80	C₂×D₆₁	—	2	32	no	—
81	C₂×D₆₁	—	2	16	no	—
82	C₂×D₆₁	—	4	32	no	—
83	C₂×D₆₁	C₂³×C₄	1	16	yes	—
84	C₄×C₆₁	—	2	8	no	—
85	C₄×C₆₁	C₂³×C₄	1	4	yes	—
86	C₄×C₆₁	C₂³×C₄	1	4	yes	—
87	Dic₆₁	—	2	8	no	—
88	C₂×#518	—	61	1	no	maximal, Sylow
89	C₂×C₆₁	—	2	4	no	—
90	C₂×C₆₁	#518	1	4	yes	complemented
91	C₂×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
92	C₂×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
93	D₆₁	—	2	16	no	—
94	C₂³×D₄	—	61	1	no	—
95	#518	—	61	16	no	—
96	C₂×C₄×D₄	—	61	4	no	—
97	C₂×C₄×D₄	—	61	4	no	—
98	C₂×#117	—	61	1	no	—
99	C₂×#117	D₆₁	1	1	yes	characteristic, complemented
100	C₂²×C₂²⋊C₄	—	61	4	no	—
101	C₆₁	C₂×#518	1	1	yes	characteristic, complemented, Sylow
102	C₂⁵	—	61	2	no	—
103	C₂²×D₄	—	61	16	no	—
104	C₂²×D₄	—	61	2	no	—
105	C₂²×D₄	—	61	2	no	—
106	C₂²×D₄	—	61	8	no	—
107	C₂³×C₄	—	61	8	no	—
108	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
109	C₄×D₄	—	61	32	no	—
110	C₄×D₄	—	61	32	no	—
111	#117	—	61	8	no	—
112	#117	C₂×D₆₁	1	8	yes	complemented
113	C₂×C₄⋊C₄	—	61	2	no	—
114	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	complemented
115	C₂×C₂²⋊C₄	—	61	2	no	—
116	C₂×C₂²⋊C₄	—	61	8	no	—
117	C₂×C₂²⋊C₄	—	61	32	no	—
118	C₂×C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
119	C₂×C₄²	—	61	2	no	—
120	C₂×C₄²	C₂×D₆₁	1	2	yes	complemented
121	C₂⁴	—	61	4	no	—
122	C₂⁴	—	61	16	no	—
123	C₂⁴	—	61	8	no	—
124	C₂⁴	—	244	8	no	—
125	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
126	C₂×D₄	—	61	16	no	—
127	C₂×D₄	—	61	16	no	—
128	C₂×D₄	—	61	64	no	—
129	C₂×D₄	—	61	8	no	—
130	C₂×D₄	—	61	8	no	—
131	C₂²×C₄	—	61	4	no	—
132	C₂²×C₄	—	61	32	no	—
133	C₂²×C₄	—	244	16	no	—
134	C₂²×C₄	C₄×D₆₁	1	8	yes	complemented
135	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
136	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
139	C₄⋊C₄	—	61	8	no	—
140	C₄⋊C₄	C₂²×D₆₁	1	8	yes	complemented
141	C₂²⋊C₄	—	61	8	no	—
142	C₂²⋊C₄	—	61	32	no	—
143	C₂²⋊C₄	C₂²×D₆₁	1	8	yes	—
144	C₄²	—	61	8	no	—
145	C₄²	C₂²×D₆₁	1	8	yes	complemented
146	C₂³	—	2	4	no	—
147	C₂³	—	61	16	no	—
148	C₂³	—	61	8	no	—
149	C₂³	—	61	32	no	—
150	C₂³	—	122	16	no	—
151	C₂³	—	244	32	no	—
152	C₂³	—	244	16	no	—
153	C₂³	C₂×C₄×D₆₁	1	4	yes	—
154	C₂³	C₂³×D₆₁	1	2	yes	—
155	C₂³	C₂³×D₆₁	1	1	yes	characteristic
156	D₄	—	61	32	no	—
157	D₄	—	61	32	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	61	8	no	—
160	C₂×C₄	—	122	8	no	—
161	C₂×C₄	—	244	32	no	—
162	C₂×C₄	C₂×C₄×D₆₁	1	4	yes	—
163	C₂×C₄	C₂×C₄×D₆₁	1	4	yes	—
164	C₂×C₄	C₂×C₄×D₆₁	1	16	yes	complemented
165	C₂×C₄	C₂³×D₆₁	1	8	yes	—
166	C₂×C₄	C₂³×D₆₁	1	2	yes	—
167	C₂×C₄	C₂³×D₆₁	1	2	yes	—
168	C₂²	—	2	8	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	61	16	no	—
171	C₂²	—	122	32	no	—
172	C₂²	—	122	16	no	—
173	C₂²	—	244	32	no	—
174	C₂²	C₂²×C₄×D₆₁	1	4	yes	—
175	C₂²	C₂²×C₄×D₆₁	1	2	yes	—
176	C₂²	#14519823	1	4	yes	—
177	C₂²	C₂⁴×D₆₁	1	1	yes	characteristic
178	C₄	—	2	8	no	—
179	C₄	—	122	8	no	—
180	C₄	C₂²×C₄×D₆₁	1	4	yes	—
181	C₄	C₂²×C₄×D₆₁	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	—	122	16	no	—
184	C₂	#16840236	1	4	yes	direct product factor
185	C₂	C₂³×C₄×D₆₁	1	1	yes	characteristic
186	C₂	C₂×#14519823	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₆₁
commutator	C₂×C₆₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₆₁
Fitting	C₂×#117×C₆₁	C₂
socle	C₂³×C₆₁	C₂⁴
hypercenter	C₂×#117	D₆₁
hypocenter	C₆₁	C₂×#518
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #16840236	C₂	direct (N × H)
3. C₂×C₄×D₂₄₄	C₂	semidirect
4. C₂×#14519739	C₂	semidirect
5. C₂×#14519736	C₂	semidirect
6. C₂×C₂²⋊C₄×D₆₁	C₂	semidirect
7. C₂×#117×C₆₁	C₂	semidirect
9. C₂²×D₂₄₄	C₄	semidirect
13. C₂²×C₄×D₆₁	C₂²	semidirect
14. C₄×D₂₄₄	C₂²	semidirect
15. #14519739	C₂²	semidirect
16. #14519736	C₂²	semidirect
17. C₂×C₄⋊Dic₆₁	C₂²	semidirect
19. C₂×#11758198	C₂²	semidirect
20. C₂²⋊C₄×D₆₁	C₂²	semidirect
21. C₂×C₄×Dic₆₁	C₂²	semidirect
23. #117×C₆₁	C₂²	semidirect
24. C₂×C₄⋊C₄×C₆₁	C₂²	semidirect
26. C₂×C₄²×C₆₁	C₂²	semidirect
29. C₂³×D₆₁	C₂×C₄	semidirect
32. C₂²×C₄×C₆₁	C₂×C₄	semidirect
38. C₂×D₂₄₄	C₂×C₄	semidirect
39. C₂×D₂₄₄	C₂×C₄	semidirect
40. C₂×D₂₄₄	C₂×C₄	semidirect
44. C₂×C₄×D₆₁	C₂³	semidirect
45. C₄⋊C₄×C₆₁	C₂³	semidirect
47. C₄²×C₆₁	C₂³	semidirect
49. #11758198	C₂³	semidirect
50. C₄⋊Dic₆₁	C₂³	semidirect
51. C₄×Dic₆₁	C₂³	semidirect
60. C₂²×D₆₁	C₂²×C₄	semidirect
65. C₂×C₄×C₆₁	C₂²×C₄	semidirect
71. D₂₄₄	C₂²×C₄	semidirect
72. D₂₄₄	C₂²×C₄	semidirect
90. C₂×C₆₁	#518	semidirect
99. C₂×#117	D₆₁	semidirect
101. C₆₁	C₂×#518	semidirect
112. #117	C₂×D₆₁	semidirect
114. C₂×C₄⋊C₄	C₂×D₆₁	semidirect
120. C₂×C₄²	C₂×D₆₁	semidirect
134. C₂²×C₄	C₄×D₆₁	semidirect
140. C₄⋊C₄	C₂²×D₆₁	semidirect
145. C₄²	C₂²×D₆₁	semidirect
164. C₂×C₄	C₂×C₄×D₆₁	semidirect
184. C₂	#16840236	direct (N × H)

Subgroups — C2×#16840236 #21459005

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840236 #21459005