Subgroups — #26125328

Order 12032.

248 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
2	#20241677	C₂	1	1	yes	maximal, characteristic
3	#20241310	C₂	1	2	yes	maximal, complemented
4	#20241308	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₄	1	2	yes	—
7	#15689176	C₄	1	1	yes	characteristic
8	C₄×#10625329	C₄	1	1	yes	characteristic
9	#15688899	C₂²	1	1	yes	characteristic
10	#15688899	C₂²	1	1	yes	characteristic
11	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
12	#15688845	C₄	1	2	yes	—
13	#15688844	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₄₇×C₂²⋊C₁₆	C₄	1	2	yes	—
17	C₄₇×#346	C₂²	1	2	yes	—
18	C₂×#10625329	—	2	1	no	—
19	#13093964	—	2	4	no	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
24	#13093935	—	2	1	no	—
25	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₄₇	C₄²	1	1	yes	characteristic
45	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
46	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
47	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	#10625329	—	4	2	no	—
49	C₂×C₄₇⋊C₈	—	2	2	no	—
50	C₂×C₄₇⋊C₈	—	2	2	no	—
51	C₁₆×C₄₇	—	2	4	no	—
52	C₄₇⋊C₁₆	—	2	2	no	—
53	C₄₇⋊C₁₆	M₁₆	1	4	yes	complemented
54	C₂³×C₄₇	#97	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	4	1	no	—
59	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₄₇	—	2	1	no	—
66	C₈×C₄₇	—	2	1	no	—
67	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
68	C₈×C₄₇	#131	1	2	yes	—
69	C₄₇⋊C₈	—	4	2	no	—
70	C₄₇⋊C₈	—	4	2	no	—
71	#11414	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	2	1	no	—
73	C₂²×C₄₇	—	2	1	no	—
74	C₂²×C₄₇	—	2	1	no	—
75	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	—	2	1	no	—
80	C₄×C₄₇	#347	1	2	yes	—
81	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
82	C₄×C₄₇	#405	1	1	yes	characteristic
83	#2163	D₄₇	1	1	yes	characteristic
84	C₄×M₃₂	—	47	1	no	—
85	#1451	—	47	1	no	—
86	#1313	—	47	2	no	—
87	#1313	—	47	2	no	—
88	C₂×C₄₇	—	4	1	no	—
89	C₂×C₄₇	#1451	1	1	yes	characteristic
90	C₂×C₄₇	C₂×#347	1	1	yes	characteristic
91	C₂×C₄₇	#2110	1	1	yes	characteristic
92	C₂×M₃₂	—	47	2	no	—
93	#431	—	47	1	no	—
94	C₄×M₁₆	—	47	1	no	—
95	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
96	C₄⋊C₁₆	Dic₄₇	1	2	yes	—
97	C₂²⋊C₁₆	Dic₄₇	1	2	yes	—
98	#346	—	47	1	no	—
99	#346	—	47	1	no	—
100	#346	C₂×D₄₇	1	2	yes	—
101	C₄×C₁₆	—	47	2	no	—
102	#322	—	47	2	no	—
103	#322	—	47	2	no	—
104	C₄₇	#11414	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	94	1	no	—
106	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
109	M₃₂	—	94	4	no	—
110	C₂×C₁₆	—	47	2	no	—
111	C₂×C₁₆	—	47	2	no	—
112	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
113	C₄⋊C₈	—	94	1	no	—
114	C₂²⋊C₈	—	94	1	no	—
115	#97	—	94	1	no	—
116	C₄×C₈	—	47	2	no	—
117	C₄×C₈	—	47	2	no	—
118	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
119	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
122	C₂²×C₄	#10625324	1	1	yes	characteristic
123	C₂²×C₄	#10625324	1	1	yes	characteristic
124	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
125	M₁₆	—	188	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	94	2	no	—
129	C₂×C₈	—	94	2	no	—
130	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
134	C₄²	#10625324	1	1	yes	characteristic
135	C₄²	#10625324	1	1	yes	characteristic
136	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₁₆	—	47	4	no	—
140	C₁₆	—	94	2	no	—
141	C₂³	#13093934	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#13093934	1	1	yes	characteristic
147	C₂×C₄	#13093934	1	1	yes	characteristic
148	C₂×C₄	#13093934	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
150	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
151	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	188	2	no	—
155	C₈	—	188	2	no	—
156	C₈	M₁₆×D₄₇	1	2	yes	—
157	C₈	#13094031	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂×#13093934	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	#15688900	1	2	yes	—
167	C₄	M₁₆×Dic₄₇	1	1	yes	characteristic
168	C₄	#15689079	1	1	yes	characteristic
169	C₂	—	4	1	no	—
170	C₂	#20241678	1	1	yes	characteristic
171	C₂	C₂×#15688900	1	1	yes	characteristic
172	C₂	#20244992	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2163	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂³×C₄₇	#97
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#2163	D₄₇
hypocenter	C₄₇	#11414
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #20241310	C₂	semidirect
4. #20241308	C₂	semidirect
53. C₄₇⋊C₁₆	M₁₆	semidirect
104. C₄₇	#11414	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093964	2
#20241677	2
#20241310	2
#20241308	2
C₄₇×#2163	2
#11414	47