Subgroups — C₂×#20245667 #26371760

Order 12032.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689525	C₂	1	2	yes	maximal
2	#20245667	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688951	C₂	1	2	yes	maximal
4	C₂×#15688949	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	—	2	2	no	—
7	#15689525	—	2	4	no	—
8	#15689525	C₂²	1	8	yes	—
9	C₂×#13093958	C₂²	1	4	yes	—
10	C₂×#13093956	—	2	2	no	—
11	#15688951	C₂²	1	8	yes	—
12	#15688949	C₂²	1	8	yes	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₄₇	C₂²	1	4	yes	—
17	#13094016	—	2	8	no	—
18	#13094016	—	4	4	no	—
19	C₂×C₄⋊Dic₄₇	—	2	4	no	—
20	C₂×Dic₄₇⋊C₄	—	4	4	no	—
21	#13093958	—	2	4	no	—
22	#13093958	C₂³	1	8	yes	—
23	#13093956	—	2	8	no	—
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	Q₈	1	4	yes	—
28	C₂×C₁₆×C₄₇	Q₈	1	4	yes	—
29	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
30	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
31	C₄×C₈×C₄₇	D₄	1	2	yes	—
32	C₄×C₈×C₄₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₇	D₈	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₄₇	—	4	4	no	—
36	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
42	C₄²×C₄₇	D₈	1	2	yes	—
43	C₄²×C₄₇	C₂×D₄	1	2	yes	—
44	C₄⋊Dic₄₇	—	2	8	no	—
45	C₄⋊Dic₄₇	—	4	4	no	—
46	Dic₄₇⋊C₄	—	4	16	no	—
47	C₁₆×C₄₇	C₂×Q₈	1	4	yes	—
48	C₁₆×C₄₇	C₂×Q₈	1	4	yes	—
49	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₄₇	—	4	8	no	—
57	C₂×Dic₄₇	—	8	8	no	—
58	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
59	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
60	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
61	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
62	C₂×#2253	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#500	1	2	yes	—
64	C₂²×C₄₇	#508	1	4	yes	—
65	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₄₇	—	2	4	no	—
67	C₄×C₄₇	#500	1	2	yes	—
68	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
70	Dic₄₇	—	8	8	no	—
71	C₂×#499	—	47	2	no	—
72	#2253	—	47	8	no	—
73	C₂×#367	—	47	2	no	—
74	C₂×#366	—	47	2	no	—
75	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
76	C₂×C₄₇	#2253	1	4	yes	complemented
77	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
78	C₂×C₄₇	C₂×#508	1	2	yes	—
79	C₂×#125	—	94	2	no	—
80	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
82	#499	—	47	8	no	—
83	#499	—	94	4	no	—
84	C₂×#107	—	47	4	no	—
85	C₂×#106	—	94	2	no	—
86	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
87	#367	—	47	8	no	—
88	#366	—	47	8	no	—
89	C₄×C₁₆	C₂×D₄₇	1	4	yes	—
90	C₄₇	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
93	#125	—	94	8	no	—
94	#125	—	188	4	no	—
95	C₂×C₄⋊C₄	—	94	4	no	—
96	C₂×C₄⋊C₄	—	188	4	no	—
97	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
98	C₂×C₁₆	Dic₉₄	1	4	yes	—
99	C₂×C₁₆	Dic₉₄	1	4	yes	—
100	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
101	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
102	#107	—	47	8	no	—
103	#107	—	94	4	no	—
104	#106	—	94	8	no	—
105	C₄×C₈	D₁₈₈	1	2	yes	—
106	C₄×C₈	C₂²×D₄₇	1	2	yes	—
107	C₂²×C₄	—	188	4	no	—
108	C₂²×C₄	D₃₇₆	1	2	yes	—
109	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
110	C₂×C₈	Dic₁₈₈	1	4	yes	—
111	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
112	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
113	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
114	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
115	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
116	C₄⋊C₄	—	94	8	no	—
117	C₄⋊C₄	—	188	16	no	—
118	C₄⋊C₄	—	188	4	no	—
119	C₄²	D₃₇₆	1	2	yes	—
120	C₄²	C₂×D₁₈₈	1	2	yes	—
121	C₁₆	C₂×Dic₉₄	1	4	yes	—
122	C₁₆	C₂×Dic₉₄	1	4	yes	—
123	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	376	8	no	—
126	C₂×C₄	—	188	8	no	—
127	C₂×C₄	#13094024	1	2	yes	—
128	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
129	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
130	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
131	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
132	C₈	#13094024	1	2	yes	—
133	C₈	C₂×Dic₁₈₈	1	4	yes	—
134	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
135	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
136	C₂²	#15689528	1	2	yes	—
137	C₂²	#15689555	1	4	yes	—
138	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	376	8	no	—
141	C₄	#15689528	1	2	yes	—
142	C₄	C₂×#13094024	1	1	yes	characteristic
143	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
144	C₂	#20245667	1	4	yes	direct product factor
145	C₂	C₂×#15689528	1	1	yes	characteristic
146	C₂	C₂×#15689555	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	C₂×#2253
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20245667	C₂	direct (N × H)
76. C₂×C₄₇	#2253	semidirect
90. C₄₇	C₂×#2253	semidirect
144. C₂	#20245667	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×#15689525	2
#20245667	2
C₂×#15688951	2
C₂×#15688949	2
C₂×C₄×C₁₆×C₄₇	2
C₂×#2253	47

Subgroups — C2×#20245667 #26371760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245667 #26371760