Subgroups — C₂×#20241306 #26125264

Order 12032.

350 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₇⋊C₁₆	C₂	1	2	yes	maximal
2	#20241306	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	—
5	C₄×C₄₇⋊C₁₆	C₄	1	4	yes	—
6	C₄×C₄₇⋊C₁₆	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₈²×C₄₇	C₄	1	2	yes	—
11	C₈²×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄₇⋊C₁₆	—	2	8	no	—
13	C₂×C₄₇⋊C₁₆	C₂×C₄	1	8	yes	—
14	C₂²×C₈×C₄₇	C₈	1	2	yes	—
15	C₂²×C₈×C₄₇	C₈	1	2	yes	—
16	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₄₇	C₈	1	2	yes	—
19	C₄×C₈×C₄₇	C₈	1	2	yes	—
20	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	—	2	4	no	—
29	C₂×C₈×C₄₇	C₁₆	1	8	yes	complemented
30	C₂×C₈×C₄₇	C₄²	1	2	yes	—
31	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₄₇	M₁₆	1	8	yes	—
34	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
35	C₄²×C₄₇	C₄²	1	1	yes	characteristic
36	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₄₇⋊C₁₆	—	2	8	no	—
40	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₄₇	#97	1	4	yes	—
45	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	—
46	C₂×C₄×C₄₇	M₃₂	1	4	yes	—
47	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₄₇	—	2	4	no	—
51	C₈×C₄₇	#97	1	4	yes	—
52	C₈×C₄₇	C₂×C₁₆	1	8	yes	complemented
53	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
54	C₂×#1312	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	#322	1	2	yes	—
56	C₂²×C₄₇	C₄×C₁₆	1	2	yes	—
57	C₂²×C₄₇	#346	1	2	yes	—
58	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
59	C₄×C₄₇	#322	1	2	yes	—
60	C₄×C₄₇	C₄×C₁₆	1	2	yes	—
61	C₄×C₄₇	#346	1	2	yes	—
62	C₄×C₄₇	C₂×#97	1	2	yes	—
63	C₄×C₄₇	C₂²×C₁₆	1	2	yes	—
64	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
65	C₂×C₄×C₁₆	—	47	2	no	—
66	C₂×C₈²	D₄₇	1	1	yes	characteristic
67	#1312	—	47	4	no	—
68	C₂×C₄₇	#1312	1	4	yes	complemented
69	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
70	C₂×C₄₇	C₂×C₄×C₁₆	1	1	yes	characteristic
71	C₂×C₄₇	C₂×#346	1	1	yes	characteristic
72	C₂²×C₁₆	—	47	4	no	—
73	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
74	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
76	C₄×C₁₆	—	47	4	no	—
77	C₄×C₁₆	—	47	4	no	—
78	C₈²	Dic₄₇	1	2	yes	—
79	C₈²	C₂×D₄₇	1	2	yes	—
80	C₄₇	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₄₇⋊C₈	1	2	yes	—
82	C₂²×C₈	C₄₇⋊C₈	1	2	yes	—
83	C₂²×C₈	C₄×D₄₇	1	2	yes	—
84	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
85	C₂×C₁₆	—	47	8	no	—
86	C₂×C₁₆	—	94	8	no	—
87	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
88	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
89	C₄×C₈	C₄×D₄₇	1	2	yes	—
90	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
91	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
92	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
93	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
96	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₄₇⋊C₁₆	1	8	yes	complemented
99	C₂×C₈	#10625324	1	8	yes	—
100	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
101	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
102	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
103	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
104	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
105	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
106	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
107	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
108	C₁₆	—	94	8	no	—
109	C₂³	C₄×C₄₇⋊C₈	1	1	yes	characteristic
110	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
111	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
112	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
113	C₂×C₄	#13093934	1	4	yes	—
114	C₂×C₄	C₂×C₄₇⋊C₁₆	1	4	yes	—
115	C₂×C₄	#13093964	1	4	yes	—
116	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	#13093934	1	4	yes	—
121	C₈	C₂×C₄₇⋊C₁₆	1	8	yes	complemented
122	C₈	C₂×#10625324	1	4	yes	—
123	C₂²	#15688844	1	2	yes	—
124	C₂²	C₄×C₄₇⋊C₁₆	1	2	yes	—
125	C₂²	#15688899	1	2	yes	—
126	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
127	C₄	#15688844	1	2	yes	—
128	C₄	C₄×C₄₇⋊C₁₆	1	2	yes	—
129	C₄	#15688899	1	2	yes	—
130	C₄	C₂×#13093934	1	2	yes	—
131	C₄	C₂²×C₄₇⋊C₁₆	1	2	yes	—
132	C₄	C₂×#13093964	1	2	yes	—
133	C₂	#20241306	1	4	yes	direct product factor
134	C₂	C₂×#15688844	1	1	yes	characteristic
135	C₂	C₂×C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
136	C₂	C₂×#15688899	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₄×C₁₆
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	C₂×#1312
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #20241306	C₂	direct (N × H)
29. C₂×C₈×C₄₇	C₁₆	semidirect
52. C₈×C₄₇	C₂×C₁₆	semidirect
68. C₂×C₄₇	#1312	semidirect
80. C₄₇	C₂×#1312	semidirect
98. C₂×C₈	C₄₇⋊C₁₆	semidirect
121. C₈	C₂×C₄₇⋊C₁₆	semidirect
133. C₂	#20241306	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×C₄×C₄₇⋊C₁₆	2
#20241306	2
C₂×C₈²×C₄₇	2
C₂×#1312	47

Subgroups — C2×#20241306 #26125264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241306 #26125264