Subgroups — C₂²×#14519737 #21458987

Order 7808.

2528 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14519737	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#11758196	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11758198	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₆₁	C₄	1	16	yes	complemented
11	C₂²×C₂²⋊D₆₁	C₂²	1	12	yes	—
12	C₂³×Dic₆₁	—	2	8	no	—
13	C₂³×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₂³×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂³×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂³×Dic₆₁	C₂²	1	12	yes	complemented
17	C₂²×C₄×D₆₁	—	2	8	no	—
18	C₂²×C₄×D₆₁	C₂²	1	12	yes	complemented
19	#14519737	C₂²	1	64	yes	direct product factor
20	C₂×#11758196	C₂²	1	12	yes	complemented
21	C₂×#11758198	C₂²	1	12	yes	complemented
22	C₂×C₄×Dic₆₁	C₂²	1	12	yes	complemented
23	C₂⁵×C₆₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₆₁	C₂²	1	12	yes	complemented
27	C₂⁴×C₆₁	—	2	4	no	—
28	C₂⁴×C₆₁	—	2	4	no	—
29	C₂⁴×C₆₁	C₂×C₄	1	8	yes	complemented
30	C₂⁴×C₆₁	C₂³	1	6	yes	—
31	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
32	C₂³×D₆₁	—	2	4	no	—
33	C₂³×D₆₁	—	2	4	no	—
34	C₂³×D₆₁	C₂×C₄	1	8	yes	complemented
35	C₂³×D₆₁	C₂³	1	6	yes	—
36	C₂²×C₄×C₆₁	—	2	4	no	—
37	C₂²×C₄×C₆₁	—	2	4	no	—
38	C₂²×C₄×C₆₁	C₂³	1	6	yes	—
39	C₂²×C₄×C₆₁	C₂³	1	6	yes	—
40	C₂×C₂²⋊D₆₁	C₂×C₄	1	96	yes	complemented
41	C₂×C₂²⋊D₆₁	C₂³	1	16	yes	—
42	C₂²×Dic₆₁	—	2	4	no	—
43	C₂²×Dic₆₁	—	2	48	no	—
44	C₂²×Dic₆₁	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₆₁	D₄	1	8	yes	complemented
46	C₂²×Dic₆₁	C₂³	1	6	yes	—
47	C₂²×Dic₆₁	C₂³	1	6	yes	—
48	C₂²×Dic₆₁	C₂³	1	6	yes	—
49	C₂²×Dic₆₁	C₂³	1	16	yes	complemented
50	C₂×C₄×D₆₁	—	2	48	no	—
51	C₂×C₄×D₆₁	C₂³	1	16	yes	complemented
52	C₂²⋊C₄×C₆₁	C₂³	1	16	yes	complemented
53	#11758198	C₂³	1	16	yes	complemented
54	#11758196	C₂³	1	16	yes	complemented
55	C₄×Dic₆₁	C₂³	1	16	yes	complemented
56	C₂³×C₆₁	—	2	32	no	—
57	C₂³×C₆₁	—	2	12	no	—
58	C₂³×C₆₁	—	2	12	no	—
59	C₂³×C₆₁	C₂²×C₄	1	4	yes	—
60	C₂³×C₆₁	C₂²×C₄	1	24	yes	complemented
61	C₂³×C₆₁	C₂×D₄	1	4	yes	—
62	C₂³×C₆₁	#41	1	4	yes	—
63	C₂³×C₆₁	C₂⁴	1	4	yes	—
64	C₂³×C₆₁	C₂⁴	1	3	yes	—
65	C₂²×D₆₁	—	2	32	no	—
66	C₂²×D₆₁	—	2	12	no	—
67	C₂²×D₆₁	—	2	12	no	—
68	C₂²×D₆₁	C₂²×C₄	1	24	yes	complemented
69	C₂²×D₆₁	C₂⁴	1	4	yes	—
70	C₂×C₄×C₆₁	—	2	12	no	—
71	C₂×C₄×C₆₁	—	2	12	no	—
72	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
73	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
74	C₂²⋊D₆₁	C₂²×C₄	1	64	yes	complemented
75	C₂×Dic₆₁	—	2	12	no	—
76	C₂×Dic₆₁	—	2	32	no	—
77	C₂×Dic₆₁	C₂²×C₄	1	24	yes	complemented
78	C₂×Dic₆₁	C₂×D₄	1	24	yes	complemented
79	C₂×Dic₆₁	C₂⁴	1	4	yes	—
80	C₂×Dic₆₁	C₂⁴	1	4	yes	—
81	C₂×Dic₆₁	C₂⁴	1	4	yes	—
82	C₄×D₆₁	—	2	32	no	—
83	C₂²×C₆₁	—	2	48	no	—
84	C₂²×C₆₁	—	2	4	no	—
85	C₂²×C₆₁	—	2	4	no	—
86	C₂²×C₆₁	C₄×D₄	1	16	yes	complemented
87	C₂²×C₆₁	C₂³×C₄	1	6	yes	—
88	C₂²×C₆₁	C₂³×C₄	1	8	yes	complemented
89	C₂²×C₆₁	C₂²×D₄	1	6	yes	—
90	C₂²×C₆₁	C₂×#41	1	6	yes	—
91	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
92	C₂×D₆₁	—	2	48	no	—
93	C₂×D₆₁	—	2	4	no	—
94	C₂×D₆₁	—	2	4	no	—
95	C₂×D₆₁	C₂³×C₄	1	8	yes	complemented
96	C₄×C₆₁	—	2	4	no	—
97	C₄×C₆₁	—	2	4	no	—
98	Dic₆₁	—	2	4	no	—
99	Dic₆₁	C₂³×C₄	1	8	yes	complemented
100	Dic₆₁	C₂²×D₄	1	8	yes	complemented
101	C₂²×C₄×D₄	—	61	1	no	maximal, Sylow
102	C₂×C₆₁	—	2	8	no	—
103	C₂×C₆₁	C₂×C₄×D₄	1	12	yes	complemented
104	C₂×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
105	C₂×C₆₁	C₂³×D₄	1	1	yes	characteristic
106	C₂×C₆₁	C₂²×#41	1	1	yes	characteristic
107	D₆₁	—	2	8	no	—
108	C₂³×D₄	—	61	1	no	—
109	C₂⁴×C₄	—	61	1	no	—
110	C₂⁴×C₄	—	61	1	no	—
111	C₂×C₄×D₄	—	61	24	no	—
112	C₂²×C₄⋊C₄	—	61	1	no	—
113	C₂²×C₂²⋊C₄	—	61	1	no	—
114	C₂²×C₂²⋊C₄	D₆₁	1	1	yes	characteristic, complemented
115	C₂²×C₄²	—	61	1	no	—
116	C₆₁	C₂²×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
117	C₂⁵	—	61	1	no	—
118	C₂⁵	C₂×D₆₁	1	1	yes	characteristic
119	C₂²×D₄	—	61	16	no	—
120	C₂²×D₄	—	61	12	no	—
121	C₂³×C₄	—	61	1	no	—
122	C₂³×C₄	—	61	1	no	—
123	C₂³×C₄	—	61	12	no	—
124	C₂³×C₄	—	61	1	no	—
125	C₂³×C₄	—	61	12	no	—
126	C₂³×C₄	—	122	8	no	—
127	C₂³×C₄	—	122	8	no	—
128	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
129	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
130	C₄×D₄	—	61	64	no	—
131	C₂×C₄⋊C₄	—	61	12	no	—
132	C₂×C₂²⋊C₄	—	61	12	no	—
133	C₂×C₂²⋊C₄	C₂×D₆₁	1	12	yes	complemented
134	C₂×C₄²	—	61	12	no	—
135	C₂⁴	—	2	4	no	—
136	C₂⁴	—	2	4	no	—
137	C₂⁴	—	61	8	no	—
138	C₂⁴	—	61	6	no	—
139	C₂⁴	—	122	4	no	—
140	C₂⁴	—	122	4	no	—
141	C₂⁴	C₄×D₆₁	1	8	yes	complemented
142	C₂⁴	C₂²×D₆₁	1	6	yes	—
143	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
144	C₂×D₄	—	61	96	no	—
145	C₂×D₄	—	61	16	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	61	8	no	—
149	C₂²×C₄	—	61	8	no	—
150	C₂²×C₄	—	61	6	no	—
151	C₂²×C₄	—	61	6	no	—
152	C₂²×C₄	—	61	6	no	—
153	C₂²×C₄	—	61	16	no	—
154	C₂²×C₄	—	61	16	no	—
155	C₂²×C₄	—	122	48	no	—
156	C₂²×C₄	—	122	48	no	—
157	C₂²×C₄	—	122	4	no	—
158	C₂²×C₄	C₂²×D₆₁	1	6	yes	—
159	C₂²×C₄	C₂²×D₆₁	1	6	yes	—
160	C₄⋊C₄	—	61	16	no	—
161	C₂²⋊C₄	—	61	16	no	—
162	C₂²⋊C₄	C₂²×D₆₁	1	16	yes	complemented
163	C₄²	—	61	16	no	—
164	C₂³	—	2	12	no	—
165	C₂³	—	2	12	no	—
166	C₂³	—	2	32	no	—
167	C₂³	—	61	24	no	—
168	C₂³	—	61	4	no	—
169	C₂³	—	122	12	no	—
170	C₂³	—	122	12	no	—
171	C₂³	—	122	32	no	—
172	C₂³	C₂×C₄×D₆₁	1	4	yes	—
173	C₂³	C₂×C₄×D₆₁	1	24	yes	complemented
174	C₂³	D₄×D₆₁	1	4	yes	—
175	C₂³	#11758224	1	4	yes	—
176	C₂³	C₂³×D₆₁	1	4	yes	—
177	C₂³	C₂³×D₆₁	1	3	yes	—
178	D₄	—	61	64	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	61	24	no	—
182	C₂×C₄	—	61	24	no	—
183	C₂×C₄	—	61	4	no	—
184	C₂×C₄	—	61	4	no	—
185	C₂×C₄	—	61	4	no	—
186	C₂×C₄	—	122	32	no	—
187	C₂×C₄	—	122	32	no	—
188	C₂×C₄	—	122	12	no	—
189	C₂×C₄	C₂³×D₆₁	1	4	yes	—
190	C₂×C₄	C₂³×D₆₁	1	4	yes	—
191	C₂²	—	2	48	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	61	8	no	—
195	C₂²	—	122	48	no	—
196	C₂²	—	122	4	no	—
197	C₂²	—	122	4	no	—
198	C₂²	#14519737	1	16	yes	direct product factor
199	C₂²	C₂²×C₄×D₆₁	1	6	yes	—
200	C₂²	C₂²×C₄×D₆₁	1	8	yes	complemented
201	C₂²	C₂×D₄×D₆₁	1	6	yes	—
202	C₂²	C₂×#11758224	1	6	yes	—
203	C₂²	C₂⁴×D₆₁	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	61	8	no	—
207	C₄	—	61	8	no	—
208	C₄	—	122	4	no	—
209	C₂	—	2	8	no	—
210	C₂	—	122	8	no	—
211	C₂	C₂×#14519737	1	12	yes	direct product factor
212	C₂	C₂³×C₄×D₆₁	1	1	yes	characteristic
213	C₂	C₂²×D₄×D₆₁	1	1	yes	characteristic
214	C₂	C₂²×#11758224	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₆₁
commutator	C₂×C₆₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₆₁
Fitting	C₂²×C₂²⋊C₄×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂³
hypercenter	C₂²×C₂²⋊C₄	D₆₁
hypocenter	C₆₁	C₂²×C₄×D₄
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₆₁	C₂	semidirect
3. C₂³×C₄×D₆₁	C₂	semidirect
4. C₂×#14519737	C₂	direct (N × H)
5. C₂²×#11758196	C₂	semidirect
6. C₂²×#11758198	C₂	semidirect
7. C₂²×C₄×Dic₆₁	C₂	semidirect
8. C₂²×C₂²⋊C₄×C₆₁	C₂	semidirect
10. C₂²×C₂²⋊D₆₁	C₄	semidirect
16. C₂³×Dic₆₁	C₂²	semidirect
18. C₂²×C₄×D₆₁	C₂²	semidirect
19. #14519737	C₂²	direct (N × H)
20. C₂×#11758196	C₂²	semidirect
21. C₂×#11758198	C₂²	semidirect
22. C₂×C₄×Dic₆₁	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₆₁	C₂²	semidirect
29. C₂⁴×C₆₁	C₂×C₄	semidirect
34. C₂³×D₆₁	C₂×C₄	semidirect
40. C₂×C₂²⋊D₆₁	C₂×C₄	semidirect
44. C₂²×Dic₆₁	C₂×C₄	semidirect
45. C₂²×Dic₆₁	D₄	semidirect
49. C₂²×Dic₆₁	C₂³	semidirect
51. C₂×C₄×D₆₁	C₂³	semidirect
52. C₂²⋊C₄×C₆₁	C₂³	semidirect
53. #11758198	C₂³	semidirect
54. #11758196	C₂³	semidirect
55. C₄×Dic₆₁	C₂³	semidirect
60. C₂³×C₆₁	C₂²×C₄	semidirect
68. C₂²×D₆₁	C₂²×C₄	semidirect
74. C₂²⋊D₆₁	C₂²×C₄	semidirect
77. C₂×Dic₆₁	C₂²×C₄	semidirect
78. C₂×Dic₆₁	C₂×D₄	semidirect
86. C₂²×C₆₁	C₄×D₄	semidirect
88. C₂²×C₆₁	C₂³×C₄	semidirect
95. C₂×D₆₁	C₂³×C₄	semidirect
99. Dic₆₁	C₂³×C₄	semidirect
100. Dic₆₁	C₂²×D₄	semidirect
103. C₂×C₆₁	C₂×C₄×D₄	semidirect
114. C₂²×C₂²⋊C₄	D₆₁	semidirect
116. C₆₁	C₂²×C₄×D₄	semidirect
133. C₂×C₂²⋊C₄	C₂×D₆₁	semidirect
141. C₂⁴	C₄×D₆₁	semidirect
162. C₂²⋊C₄	C₂²×D₆₁	semidirect
173. C₂³	C₂×C₄×D₆₁	semidirect
198. C₂²	#14519737	direct (N × H)
200. C₂²	C₂²×C₄×D₆₁	semidirect
211. C₂	C₂×#14519737	direct (N × H)

Subgroups — C22×#14519737 #21458987

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14519737 #21458987