Subgroups — #26103978

Order 12032.

190 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688949	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#366	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093958	—	2	1	no	—
5	C₂×C₄₇⋊C₃₂	—	2	1	no	—
6	C₂×C₄₇⋊C₃₂	—	2	1	no	—
7	C₂×C₄₇⋊C₃₂	C₄	1	2	yes	complemented
8	#15688949	—	2	1	no	—
9	#15688949	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×#107×C₄₇	—	2	1	no	—
12	C₄₇×#366	—	2	1	no	—
13	C₄₇×#366	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₄₇	—	4	1	no	—
15	#13093958	—	2	2	no	—
16	#13093958	—	4	1	no	—
17	C₄₇⋊C₃₂	—	2	2	no	—
18	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
20	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
26	#107×C₄₇	—	2	2	no	—
27	#107×C₄₇	—	4	1	no	—
28	C₂²×C₄×C₄₇	—	8	1	no	—
29	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
30	C₂²×Dic₄₇	—	8	1	no	—
31	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
37	C₄⋊C₄×C₄₇	—	4	2	no	—
38	C₄⋊C₄×C₄₇	—	8	1	no	—
39	C₄⋊Dic₄₇	—	4	2	no	—
40	C₄⋊Dic₄₇	—	8	1	no	—
41	C₁₆×C₄₇	C₄²	1	1	yes	characteristic
42	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₁₆×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₄₇	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	8	2	no	—
47	C₂×C₄×C₄₇	—	16	1	no	—
48	C₂×C₄×C₄₇	—	16	1	no	—
49	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
52	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₄₇	Q₃₂	1	1	yes	characteristic
55	C₂×Dic₄₇	—	8	2	no	—
56	C₂×Dic₄₇	—	16	1	no	—
57	C₂×Dic₄₇	—	16	1	no	—
58	C₈×C₄₇	#95	1	1	yes	characteristic
59	C₈×C₄₇	#103	1	1	yes	characteristic
60	C₈×C₄₇	#103	1	1	yes	characteristic
61	C₈×C₄₇	#106	1	1	yes	characteristic
62	#7471	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#357	1	1	yes	characteristic
64	C₂²×C₄₇	#357	1	1	yes	characteristic
65	C₂²×C₄₇	#366	1	1	yes	characteristic
66	C₂²×C₄₇	D₃₂	1	1	yes	characteristic
67	C₂²×C₄₇	SD₆₄	1	1	yes	characteristic
68	C₂²×C₄₇	SD₆₄	1	1	yes	characteristic
69	C₂²×C₄₇	Q₆₄	1	1	yes	characteristic
70	C₄×C₄₇	—	16	2	no	—
71	C₄×C₄₇	#340	1	1	yes	characteristic
72	C₄×C₄₇	#358	1	1	yes	characteristic
73	C₄×C₄₇	#358	1	1	yes	characteristic
74	C₄×C₄₇	#367	1	1	yes	characteristic
75	Dic₄₇	—	16	2	no	—
76	C₂²×C₃₂	—	47	1	no	—
77	C₂×#366	—	47	1	no	—
78	C₂×#366	D₄₇	1	1	yes	characteristic
79	C₂×C₄₇	#1380	1	1	yes	characteristic
80	C₂×C₄₇	#1415	1	1	yes	characteristic
81	C₂×C₄₇	#1415	1	1	yes	characteristic
82	C₂×C₄₇	#1416	1	1	yes	characteristic
83	C₂×C₄₇	#1416	1	1	yes	characteristic
84	C₂×C₄₇	#1423	1	1	yes	characteristic
85	C₂×C₄₇	#1424	1	1	yes	characteristic
86	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
87	C₂×#107	—	2	1	no	—
88	C₂×#107	—	94	1	no	—
89	C₂×C₃₂	—	47	2	no	—
90	C₂×C₃₂	—	94	1	no	—
91	C₂×C₃₂	—	94	1	no	—
92	#366	—	2	1	no	—
93	#366	—	47	2	no	—
94	#366	—	94	1	no	—
95	#366	Dic₄₇	1	2	yes	complemented
96	C₄₇	#7471	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	4	1	no	—
99	C₂×C₄⋊C₄	—	188	1	no	—
100	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
103	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
106	#107	—	2	2	no	—
107	#107	—	4	1	no	—
108	#107	—	94	2	no	—
109	#107	—	188	1	no	—
110	C₃₂	—	94	2	no	—
111	C₂²×C₄	—	8	1	no	—
112	C₂²×C₄	—	376	1	no	—
113	C₂²×C₄	#10625334	1	1	yes	characteristic
114	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
115	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
116	C₂×C₈	#10625335	1	1	yes	characteristic
117	C₂×C₈	#10625336	1	1	yes	characteristic
118	C₂×C₈	#10625337	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	376	1	no	—
123	C₄⋊C₄	—	188	2	no	—
124	C₁₆	C₄×Dic₄₇	1	1	yes	characteristic
125	C₁₆	Dic₄₇⋊C₄	1	1	yes	characteristic
126	C₁₆	C₄⋊Dic₄₇	1	1	yes	characteristic
127	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
128	C₂³	#13093965	1	1	yes	characteristic
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	376	2	no	—
133	C₂×C₄	—	752	1	no	—
134	C₂×C₄	—	752	1	no	—
135	C₂×C₄	#13093944	1	1	yes	characteristic
136	C₂×C₄	#13093945	1	1	yes	characteristic
137	C₂×C₄	#13093957	1	1	yes	characteristic
138	C₂×C₄	#13093967	1	1	yes	characteristic
139	C₂×C₄	#13093968	1	1	yes	characteristic
140	C₂×C₄	#13093970	1	1	yes	characteristic
141	C₈	C₄₇⋊#95	1	1	yes	characteristic
142	C₈	#13093947	1	1	yes	characteristic
143	C₈	#13093948	1	1	yes	characteristic
144	C₈	#13093955	1	1	yes	characteristic
145	C₂²	#15688923	1	1	yes	characteristic
146	C₂²	#15688924	1	1	yes	characteristic
147	C₂²	#15688948	1	1	yes	characteristic
148	C₂²	#15688958	1	1	yes	characteristic
149	C₂²	#15688960	1	1	yes	characteristic
150	C₂²	#15688961	1	1	yes	characteristic
151	C₂²	#15688963	1	1	yes	characteristic
152	C₄	—	16	2	no	—
153	C₄	—	752	2	no	—
154	C₄	#15688885	1	1	yes	characteristic
155	C₄	#15688926	1	1	yes	characteristic
156	C₄	#15688927	1	1	yes	characteristic
157	C₄	#15688950	1	1	yes	characteristic
158	C₂	#20241479	1	1	yes	characteristic
159	C₂	#20241565	1	1	yes	characteristic
160	C₂	#20241566	1	1	yes	characteristic
161	C₂	#20241568	1	1	yes	characteristic
162	C₂	#20241569	1	1	yes	characteristic
163	C₂	#20241587	1	1	yes	characteristic
164	C₂	#20241589	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093965
commutator	C₁₆×C₄₇	C₄²
Frattini	C₂²×C₁₆	C₂×D₄₇
Fitting	C₂×C₄₇×#366	C₂
socle, prime radical	C₂³×C₄₇	D₁₆
cosocle	C₂²×C₁₆×C₄₇	C₂²
hypercenter	C₂×#366	D₄₇
hypocenter	C₄₇	#7471
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
7. C₂×C₄₇⋊C₃₂	C₄	semidirect
9. #15688949	C₄	semidirect
13. C₄₇×#366	C₄	semidirect
95. #366	Dic₄₇	semidirect
96. C₄₇	#7471	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄₇⋊C₃₂	2
C₂×#15688949	2
C₂×C₄₇×#366	2
#7471	47