Subgroups — C₂×#16839525 #21445533

Order 7808.

512 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₆₁	C₂	1	2	yes	maximal
2	#16839525	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₆₁	—	2	2	no	—
5	C₂×C₄⋊Dic₆₁	—	2	2	no	—
6	C₂×C₄⋊Dic₆₁	—	2	4	no	—
7	C₂×C₄⋊Dic₆₁	C₄	1	8	yes	complemented
8	C₂×C₄⋊Dic₆₁	C₂²	1	4	yes	—
9	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₆₁	—	2	4	no	—
11	C₂²×C₈×C₆₁	C₄	1	4	yes	complemented
12	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂⁴×C₆₁	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₆₁	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	D₄	1	4	yes	—
18	C₂²×C₄×C₆₁	Q₈	1	2	yes	—
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₆₁	—	2	8	no	—
21	C₂²×Dic₆₁	—	2	4	no	—
22	C₂²×Dic₆₁	—	4	4	no	—
23	C₂²×Dic₆₁	—	4	4	no	—
24	C₂×C₈×C₆₁	—	2	8	no	—
25	C₂×C₈×C₆₁	—	2	4	no	—
26	C₂×C₈×C₆₁	C₂×C₄	1	4	yes	complemented
27	C₄⋊Dic₆₁	—	2	8	no	—
28	C₄⋊Dic₆₁	—	2	4	no	—
29	C₄⋊Dic₆₁	C₂×C₄	1	8	yes	complemented
30	C₂³×C₆₁	C₂²⋊C₄	1	4	yes	—
31	C₂³×C₆₁	C₄⋊C₄	1	2	yes	—
32	C₂³×C₆₁	D₈	1	2	yes	—
33	C₂³×C₆₁	SD₁₆	1	4	yes	—
34	C₂³×C₆₁	Q₁₆	1	2	yes	—
35	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₄²	1	4	yes	—
37	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
39	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₆₁	C₄⋊C₄	1	8	yes	—
41	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×Dic₆₁	—	2	8	no	—
46	C₂×Dic₆₁	—	4	16	no	—
47	C₂×Dic₆₁	—	4	4	no	—
48	C₂×Dic₆₁	—	4	4	no	—
49	C₈×C₆₁	—	2	4	no	—
50	C₂²×C₆₁	#95	1	4	yes	—
51	C₂²×C₆₁	#102	1	8	yes	—
52	C₂²×C₆₁	#103	1	8	yes	—
53	C₂²×C₆₁	#106	1	4	yes	—
54	C₂²×C₆₁	#107	1	4	yes	—
55	C₂²×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₆₁	C₂×SD₁₆	1	2	yes	—
59	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₆₁	#95	1	4	yes	—
61	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
64	Dic₆₁	—	4	8	no	—
65	C₂×#340	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	#340	1	8	yes	complemented
67	C₂×C₆₁	C₂×#95	1	1	yes	characteristic
68	C₂×C₆₁	C₂×#102	1	2	yes	—
69	C₂×C₆₁	C₂×#103	1	2	yes	—
70	C₂×C₆₁	C₂×#106	1	1	yes	characteristic
71	C₂×C₆₁	C₂×#107	1	1	yes	characteristic
72	C₂³×C₈	D₆₁	1	1	yes	characteristic
73	C₂²×C₄⋊C₄	—	61	2	no	—
74	#340	—	61	4	no	—
75	C₆₁	C₂×#340	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	—	122	2	no	—
77	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	Dic₆₁	1	4	yes	complemented
80	C₂²×C₈	C₂×D₆₁	1	2	yes	—
81	C₂×C₄⋊C₄	—	61	8	no	—
82	C₂×C₄⋊C₄	—	61	4	no	—
83	C₂×C₄⋊C₄	—	122	2	no	—
84	C₂×C₄⋊C₄	—	122	4	no	—
85	C₂⁴	D₂₄₄	1	1	yes	characteristic
86	C₂²×C₄	—	122	8	no	—
87	C₂²×C₄	—	122	4	no	—
88	C₂²×C₄	—	244	4	no	—
89	C₂²×C₄	—	244	4	no	—
90	C₂²×C₄	Dic₁₂₂	1	2	yes	—
91	C₂²×C₄	C₄×D₆₁	1	4	yes	—
92	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₆₁	1	4	yes	—
95	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×Dic₆₁	1	4	yes	complemented
99	C₄⋊C₄	—	61	8	no	—
100	C₄⋊C₄	—	122	8	no	—
101	C₄⋊C₄	—	122	4	no	—
102	C₂³	#11758190	1	4	yes	—
103	C₂³	D₄₈₈	1	2	yes	—
104	C₂³	Dic₂₄₄	1	2	yes	—
105	C₂³	C₄⋊Dic₆₁	1	2	yes	—
106	C₂³	#11758198	1	4	yes	—
107	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
108	C₂×C₄	—	122	8	no	—
109	C₂×C₄	—	244	16	no	—
110	C₂×C₄	—	244	4	no	—
111	C₂×C₄	—	244	4	no	—
112	C₂×C₄	C₄×Dic₆₁	1	4	yes	—
113	C₂×C₄	#11758196	1	8	yes	—
114	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
115	C₂×C₄	#11758198	1	4	yes	—
116	C₂×C₄	C₂²⋊Dic₆₁	1	4	yes	—
117	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
119	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₂²	#14519684	1	4	yes	—
123	C₂²	#14519699	1	8	yes	—
124	C₂²	#14519702	1	8	yes	—
125	C₂²	#14519709	1	4	yes	—
126	C₂²	#14519711	1	4	yes	—
127	C₂²	C₂×#11758198	1	2	yes	—
128	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
129	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
130	C₂²	C₂×#11758190	1	2	yes	—
131	C₂²	C₂×Dic₂₄₄	1	1	yes	characteristic
132	C₄	—	244	8	no	—
133	C₄	#14519684	1	4	yes	—
134	C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
135	C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
136	C₄	C₂×#11758196	1	2	yes	—
137	C₂	#16839525	1	8	yes	direct product factor
138	C₂	C₂×#14519684	1	1	yes	characteristic
139	C₂	C₂×#14519699	1	2	yes	—
140	C₂	C₂×#14519702	1	2	yes	—
141	C₂	C₂×#14519709	1	1	yes	characteristic
142	C₂	C₂×#14519711	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₂₄₄
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂³×C₈×C₆₁	C₂
socle	C₂⁴×C₆₁	D₄
hypercenter	C₂³×C₈	D₆₁
hypocenter	C₆₁	C₂×#340
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #16839525	C₂	direct (N × H)
7. C₂×C₄⋊Dic₆₁	C₄	semidirect
11. C₂²×C₈×C₆₁	C₄	semidirect
26. C₂×C₈×C₆₁	C₂×C₄	semidirect
29. C₄⋊Dic₆₁	C₂×C₄	semidirect
66. C₂×C₆₁	#340	semidirect
75. C₆₁	C₂×#340	semidirect
79. C₂²×C₈	Dic₆₁	semidirect
98. C₂×C₈	C₂×Dic₆₁	semidirect
137. C₂	#16839525	direct (N × H)

Subgroups — C2×#16839525 #21445533

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16839525 #21445533