Subgroups — #26132705

Order 12032.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688947	C₂	1	4	yes	maximal
2	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
3	#20243545	C₂	1	1	yes	maximal, characteristic
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	#15688947	—	2	8	no	—
10	#15688899	C₄	1	2	yes	complemented
11	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	complemented
12	C₂×#108×C₄₇	—	2	1	no	—
13	C₂×#108×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×C₄⋊Dic₄₇	—	4	2	no	—
18	#13093964	D₄	1	4	yes	—
19	#13093964	Q₈	1	4	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
21	#13093956	—	2	4	no	—
22	#13093956	—	4	2	no	—
23	C₂×M₁₆×C₄₇	—	2	2	no	—
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	#108×C₄₇	—	2	4	no	—
28	#108×C₄₇	—	4	2	no	—
29	C₄⋊C₈×C₄₇	—	4	2	no	—
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₄₇	—	4	2	no	—
37	M₁₆×C₄₇	—	4	2	no	—
38	M₁₆×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
43	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
44	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₄₇	SD₁₆	1	2	yes	—
51	C₄⋊Dic₄₇	—	4	2	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	C₄⋊Dic₄₇	—	8	4	no	—
54	C₄₇⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
55	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	#106	1	2	yes	—
58	C₂×C₄×C₄₇	#106	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×Dic₄₇	—	8	4	no	—
66	C₂×Dic₄₇	—	8	2	no	—
67	C₈×C₄₇	—	8	2	no	—
68	C₈×C₄₇	#106	1	2	yes	—
69	C₈×C₄₇	#106	1	2	yes	—
70	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
72	#12903	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
75	C₂²×C₄₇	#492	1	1	yes	characteristic
76	C₂²×C₄₇	#498	1	1	yes	characteristic
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	#365	1	4	yes	complemented
79	C₄×C₄₇	#384	1	1	yes	characteristic
80	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
81	C₄×C₄₇	#492	1	1	yes	characteristic
82	C₄×C₄₇	#498	1	1	yes	characteristic
83	Dic₄₇	—	16	2	no	—
84	C₂×#365	—	47	4	no	—
85	C₄×M₃₂	—	47	1	no	—
86	#1848	D₄₇	1	1	yes	characteristic
87	#1844	—	47	1	no	—
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	—	2	1	no	—
90	C₂×C₄₇	#1844	1	1	yes	characteristic
91	C₂×C₄₇	C₂×#365	1	2	yes	—
92	C₂×M₃₂	—	47	2	no	—
93	C₂×#108	—	2	1	no	—
94	C₂×#108	C₂×D₄₇	1	2	yes	—
95	C₂×#106	—	47	2	no	—
96	C₂×#106	—	94	1	no	—
97	#423	—	2	1	no	—
98	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
99	#384	—	94	1	no	—
100	#365	—	94	8	no	—
101	#346	—	47	2	no	—
102	C₄×C₁₆	—	47	2	no	—
103	C₄₇	#12903	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	—	2	2	no	—
105	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	94	2	no	—
108	C₂×C₄⋊C₄	—	188	2	no	—
109	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
110	M₃₂	—	47	4	no	—
111	M₃₂	—	47	4	no	—
112	C₂×C₁₆	—	47	4	no	—
113	#108	—	2	4	no	—
114	#108	—	4	2	no	—
115	#106	—	94	4	no	—
116	#106	—	188	2	no	—
117	C₄⋊C₈	—	4	2	no	—
118	C₄×C₈	Dic₉₄	1	1	yes	characteristic
119	C₄×C₈	Dic₉₄	1	1	yes	characteristic
120	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
121	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
122	C₂²×C₄	—	188	2	no	—
123	C₂²×C₄	#10625336	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	M₁₆	—	4	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
129	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
130	C₂×C₈	#10625335	1	2	yes	—
131	C₂×C₈	#10625336	1	2	yes	—
132	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
134	C₂×C₈	D₄×D₄₇	1	2	yes	—
135	C₂×C₈	Q₈×D₄₇	1	2	yes	—
136	C₄⋊C₄	—	376	4	no	—
137	C₄⋊C₄	—	188	2	no	—
138	C₄⋊C₄	—	188	2	no	—
139	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
140	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
141	C₄²	#10625335	1	2	yes	—
142	C₁₆	—	47	8	no	—
143	C₂³	C₂×#10625336	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	376	4	no	—
146	C₂×C₄	—	376	2	no	—
147	C₂×C₄	#13093955	1	2	yes	—
148	C₂×C₄	#13093955	1	2	yes	—
149	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
150	C₂×C₄	#13094020	1	1	yes	characteristic
151	C₂×C₄	#13094022	1	1	yes	characteristic
152	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
153	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
154	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
155	C₈	—	8	2	no	—
156	C₈	#13093955	1	2	yes	—
157	C₈	#13093955	1	2	yes	—
158	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
159	C₈	#13094023	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#13093955	1	1	yes	characteristic
162	C₂²	#15689497	1	1	yes	characteristic
163	C₂²	#15689519	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	752	2	no	—
166	C₄	#15688947	1	4	yes	complemented
167	C₄	#15688992	1	1	yes	characteristic
168	C₄	C₂×#13093955	1	1	yes	characteristic
169	C₄	#15689495	1	1	yes	characteristic
170	C₄	#15689519	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#20243544	1	1	yes	characteristic
174	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	#12903
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
10. #15688899	C₄	semidirect
11. C₄×C₄₇⋊C₁₆	C₄	semidirect
54. C₄₇⋊C₁₆	C₄⋊C₄	semidirect
78. C₄×C₄₇	#365	semidirect
103. C₄₇	#12903	semidirect
166. C₄	#15688947	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688947	2
C₄×#13093964	2
#20243545	2
C₄₇×#1848	2
#12903	47