Subgroups — #26125269

Order 12032.

298 conjugacy classes of subgroups.

The subgroup lattice has 136 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	1	yes	maximal, characteristic
2	#20241671	C₂	1	1	yes	maximal, characteristic
3	#20241307	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
6	#15689176	C₄	1	2	yes	—
7	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
8	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	#15688844	C₄	1	2	yes	—
10	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×M₃₂×C₄₇	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#346	C₄	1	2	yes	—
14	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄₇⋊C₁₆	—	2	4	no	—
16	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₄₇⋊C₈)	C₈	1	4	yes	—
18	C₂²⋊(C₄₇⋊C₈)	C₈	1	4	yes	—
19	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₄₇	—	2	4	no	—
24	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
34	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
35	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₄²	1	1	yes	characteristic
38	C₄²×C₄₇	C₂×C₈	1	2	yes	—
39	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	C₂×C₈	1	8	yes	—
41	C₁₆×C₄₇	—	2	4	no	—
42	C₄₇⋊C₁₆	—	2	4	no	—
43	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#97	1	2	yes	—
48	C₂×C₄×C₄₇	#97	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
51	C₈×C₄₇	#97	1	2	yes	—
52	C₈×C₄₇	#97	1	2	yes	—
53	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
54	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
55	C₄₇⋊C₈	—	2	8	no	—
56	#11401	—	47	1	no	maximal, Sylow
57	C₂²×C₄₇	—	2	2	no	—
58	C₂²×C₄₇	#322	1	2	yes	—
59	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	#322	1	2	yes	—
62	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
63	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
64	C₄×C₄₇	#504	1	4	yes	—
65	C₄×M₃₂	D₄₇	1	1	yes	characteristic
66	C₂×C₄×C₁₆	—	47	1	no	—
67	#1450	—	47	1	no	—
68	#1312	—	47	4	no	—
69	C₂×C₄₇	—	2	2	no	—
70	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
71	C₂×C₄₇	#2108	1	2	yes	—
72	C₂×M₃₂	Dic₄₇	1	2	yes	—
73	C₂²×C₁₆	—	47	2	no	—
74	#431	—	47	2	no	—
75	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
76	#346	Dic₄₇	1	2	yes	—
77	C₄×C₁₆	—	47	2	no	—
78	C₄×C₁₆	—	47	2	no	—
79	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
80	#322	—	47	2	no	—
81	C₈²	—	47	2	no	—
82	C₄₇	#11401	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
86	M₃₂	—	2	4	no	—
87	C₂×C₁₆	—	47	4	no	—
88	C₂×C₁₆	—	94	4	no	—
89	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
90	C₄⋊C₈	—	47	4	no	—
91	C₂²⋊C₈	—	47	4	no	—
92	C₄×C₈	—	47	4	no	—
93	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
94	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
97	C₂²×C₄	C₈×D₄₇	1	2	yes	—
98	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
99	C₂×C₈	—	47	8	no	—
100	C₂×C₈	#10625324	1	4	yes	—
101	C₂×C₈	#10625329	1	4	yes	—
102	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
104	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
106	C₄²	C₈×D₄₇	1	2	yes	—
107	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
108	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
109	C₁₆	—	2	4	no	—
110	C₁₆	—	94	4	no	—
111	C₂³	C₈×Dic₄₇	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
114	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
115	C₂×C₄	#13093934	1	2	yes	—
116	C₂×C₄	#13093935	1	2	yes	—
117	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
119	C₈	—	94	8	no	—
120	C₈	#13093934	1	2	yes	—
121	C₈	#13093935	1	2	yes	—
122	C₈	C₂×#10625324	1	2	yes	—
123	C₈	C₂×#10625329	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	#15688845	1	2	yes	—
126	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	#15688845	1	2	yes	—
129	C₄	C₂×#13093934	1	1	yes	characteristic
130	C₄	C₂×#13093935	1	1	yes	characteristic
131	C₄	#15689541	1	4	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#15688845	1	1	yes	characteristic
134	C₂	#20244987	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄×M₃₂×C₄₇	C₂
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#11401
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20241307	C₂	semidirect
82. C₄₇	#11401	semidirect

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
#20241671	2
#20241307	2
C₄×M₃₂×C₄₇	2
#11401	47