Subgroups — #26132651

Order 12032.

302 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	#20243545	C₂	1	1	yes	maximal, characteristic
3	#20241488	C₂	1	4	yes	maximal
4	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	—
6	C₂×#13093956	—	2	1	no	—
7	C₂×#13093956	C₂²	1	2	yes	—
8	#15688993	—	2	1	no	—
9	#15688899	C₄	1	2	yes	complemented
10	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	complemented
11	C₂×#108×C₄₇	—	2	1	no	—
12	C₂×#108×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	—	2	2	no	—
16	C₂×C₄⋊Dic₄₇	—	4	2	no	—
17	#13093964	—	2	2	no	—
18	#13093964	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	#13093956	—	2	4	no	—
21	#13093956	—	4	2	no	—
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	#108×C₄₇	—	2	4	no	—
27	#108×C₄₇	—	4	2	no	—
28	C₄⋊C₈×C₄₇	—	4	2	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₄₇	—	4	2	no	—
36	M₁₆×C₄₇	—	4	2	no	—
37	M₁₆×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	D₈	1	2	yes	—
42	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₄₇	#41	1	2	yes	—
46	C₂×C₈×C₄₇	#41	1	2	yes	—
47	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₄₇	SD₁₆	1	2	yes	—
50	C₄⋊Dic₄₇	—	4	2	no	—
51	C₄⋊Dic₄₇	—	4	2	no	—
52	C₄⋊Dic₄₇	—	8	4	no	—
53	C₄₇⋊C₁₆	—	2	4	no	—
54	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	#106	1	2	yes	—
57	C₂×C₄×C₄₇	#106	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₄₇	—	8	4	no	—
65	C₂×Dic₄₇	—	8	2	no	—
66	C₈×C₄₇	—	8	2	no	—
67	C₈×C₄₇	#107	1	2	yes	—
68	C₈×C₄₇	#107	1	2	yes	—
69	C₈×C₄₇	#117	1	2	yes	—
70	C₈×C₄₇	#125	1	2	yes	—
71	#12886	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
74	C₂²×C₄₇	#486	1	1	yes	characteristic
75	C₂²×C₄₇	#487	1	1	yes	characteristic
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	#365	1	4	yes	—
78	C₄×C₄₇	#382	1	1	yes	characteristic
79	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
80	C₄×C₄₇	#486	1	1	yes	characteristic
81	C₄×C₄₇	#487	1	1	yes	characteristic
82	Dic₄₇	—	16	2	no	—
83	C₄×M₃₂	—	47	1	no	—
84	#1848	D₄₇	1	1	yes	characteristic
85	#1844	—	47	1	no	—
86	#1383	—	47	4	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	#1834	1	1	yes	characteristic
90	C₂×C₄₇	C₂×#365	1	2	yes	—
91	C₂×M₃₂	—	47	2	no	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	C₂×D₄₇	1	2	yes	—
94	C₂×#106	—	47	2	no	—
95	C₂×#106	—	94	1	no	—
96	#423	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
98	#384	—	94	1	no	—
99	#346	—	47	2	no	—
100	C₄×C₁₆	—	47	2	no	—
101	C₄₇	#12886	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	94	2	no	—
106	C₂×C₄⋊C₄	—	188	2	no	—
107	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
108	M₃₂	—	94	2	no	—
109	M₃₂	—	94	2	no	—
110	C₂×C₁₆	—	47	4	no	—
111	#108	—	2	4	no	—
112	#108	—	4	2	no	—
113	#106	—	94	4	no	—
114	#106	—	188	2	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄×C₈	Dic₉₄	1	1	yes	characteristic
117	C₄×C₈	Dic₉₄	1	1	yes	characteristic
118	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
119	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
120	C₂²×C₄	—	188	2	no	—
121	C₂²×C₄	#10625336	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
128	C₂×C₈	#10625334	1	2	yes	—
129	C₂×C₈	#10625337	1	2	yes	—
130	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
132	C₂×C₈	#10625352	1	2	yes	—
133	C₂×C₈	#10625354	1	2	yes	—
134	C₄⋊C₄	—	376	4	no	—
135	C₄⋊C₄	—	188	2	no	—
136	C₄⋊C₄	—	188	2	no	—
137	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
138	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
139	C₄²	#10625335	1	2	yes	—
140	C₁₆	—	94	4	no	—
141	C₂³	C₂×#10625336	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	376	4	no	—
144	C₂×C₄	—	376	2	no	—
145	C₂×C₄	#13093955	1	2	yes	—
146	C₂×C₄	#13093955	1	2	yes	—
147	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#13094011	1	1	yes	characteristic
149	C₂×C₄	#13094013	1	1	yes	characteristic
150	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
151	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
152	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	#13093957	1	2	yes	—
155	C₈	#13093957	1	2	yes	—
156	C₈	#13093982	1	2	yes	—
157	C₈	#13094015	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#13093955	1	1	yes	characteristic
160	C₂²	#15689471	1	1	yes	characteristic
161	C₂²	#15689473	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	752	2	no	—
164	C₄	#15688947	1	4	yes	—
165	C₄	#15688988	1	1	yes	characteristic
166	C₄	C₂×#13093957	1	1	yes	characteristic
167	C₄	#15689469	1	1	yes	characteristic
168	C₄	#15689475	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#20243515	1	1	yes	characteristic
172	C₂	C₂×#15688947	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13093955
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂×#106
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#12886
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #15688899	C₄	semidirect
10. C₄×C₄₇⋊C₁₆	C₄	semidirect
101. C₄₇	#12886	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
#20243545	2
#20241488	2
C₄₇×#1848	2
#12886	47