Subgroups — #26134900

Order 12032.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 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	#20241493	C₂	1	4	yes	maximal
4	C₄₇×#3160	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	—	2	1	no	—
10	C₄×C₄₇⋊C₁₆	—	2	1	no	—
11	C₄₇×#533	—	2	1	no	—
12	C₄₇×#500	C₄	1	2	yes	complemented
13	C₄₇×#499	—	2	1	no	—
14	C₄₇×#499	C₄	1	2	yes	complemented
15	C₄₇×#498	—	2	1	no	—
16	C₄₇×#427	—	2	1	no	—
17	C₄₇×#427	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	4	2	no	—
21	#13093964	—	2	2	no	—
22	#13093964	Q₈	1	4	yes	complemented
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	#13093956	—	2	4	no	—
25	#13093956	—	4	2	no	—
26	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₄₇	—	2	2	no	—
29	#125×C₄₇	—	2	2	no	—
30	C₂²⋊Q₈×C₄₇	—	4	2	no	—
31	C₄×Q₈×C₄₇	—	4	2	no	—
32	#117×C₄₇	—	2	2	no	—
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	#107×C₄₇	—	2	2	no	—
35	#107×C₄₇	C₂×C₄	1	4	yes	—
36	#106×C₄₇	—	2	2	no	—
37	#106×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₄₇	—	4	2	no	—
43	C₂²×C₄×C₄₇	D₈	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₄₇	—	4	2	no	—
46	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
49	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₄₇	#41	1	2	yes	—
54	C₄⋊C₄×C₄₇	—	2	4	no	—
55	C₄⋊C₄×C₄₇	—	4	4	no	—
56	C₄⋊C₄×C₄₇	—	4	2	no	—
57	C₂²⋊C₄×C₄₇	—	4	2	no	—
58	C₄²×C₄₇	—	4	2	no	—
59	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₄₇	SD₁₆	1	2	yes	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	8	4	no	—
65	C₄₇⋊C₁₆	—	2	4	no	—
66	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₄₇	—	8	4	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	4	4	no	—
70	C₂×C₄×C₄₇	#102	1	2	yes	—
71	C₂×C₄×C₄₇	#102	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₄₇	—	8	4	no	—
79	C₂×Dic₄₇	—	8	2	no	—
80	C₈×C₄₇	#103	1	2	yes	—
81	C₈×C₄₇	#103	1	2	yes	—
82	C₈×C₄₇	C₄×Q₈	1	2	yes	—
83	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
84	#13287	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
87	C₂²×C₄₇	#486	1	1	yes	characteristic
88	C₂²×C₄₇	#500	1	1	yes	characteristic
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	8	4	no	—
91	C₄×C₄₇	#360	1	4	yes	—
92	C₄×C₄₇	#391	1	1	yes	characteristic
93	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
94	C₄×C₄₇	#487	1	1	yes	characteristic
95	C₄×C₄₇	#498	1	1	yes	characteristic
96	Dic₄₇	—	16	2	no	—
97	#3160	D₄₇	1	1	yes	characteristic
98	C₄×M₃₂	—	47	1	no	—
99	#1844	—	47	1	no	—
100	#1386	—	47	4	no	—
101	C₂×C₄₇	—	2	1	no	—
102	C₂×C₄₇	—	2	1	no	—
103	C₂×C₄₇	#1990	1	1	yes	characteristic
104	C₂×C₄₇	C₂×#360	1	2	yes	—
105	#533	—	2	1	no	—
106	C₂×M₃₂	—	47	2	no	—
107	#500	Dic₄₇	1	2	yes	complemented
108	#499	—	2	1	no	—
109	#499	Dic₄₇	1	2	yes	complemented
110	#498	—	2	1	no	—
111	#427	—	2	1	no	—
112	#427	C₂×D₄₇	1	2	yes	—
113	C₂×#106	—	47	2	no	—
114	C₂×#106	—	94	1	no	—
115	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
116	#384	—	94	1	no	—
117	#346	—	94	1	no	—
118	C₄×C₁₆	—	94	1	no	—
119	C₄₇	#13287	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	2	no	—
123	#125	—	2	2	no	—
124	C₂²⋊Q₈	—	4	2	no	—
125	C₄×Q₈	—	4	2	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	—	94	2	no	—
128	C₂×C₄⋊C₄	—	188	2	no	—
129	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
130	M₃₂	—	47	4	no	—
131	M₃₂	—	94	2	no	—
132	C₂×C₁₆	—	94	2	no	—
133	#107	—	2	2	no	—
134	#107	C₂×Dic₄₇	1	4	yes	—
135	#106	—	2	2	no	—
136	#106	—	2	2	no	—
137	#106	—	94	4	no	—
138	#106	—	188	2	no	—
139	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
143	C₂×Q₈	—	4	2	no	—
144	C₂²×C₄	—	188	2	no	—
145	C₂²×C₄	#10625334	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₂×C₈	#10625336	1	2	yes	—
148	C₂×C₈	#10625337	1	2	yes	—
149	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
150	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₄₇	1	2	yes	—
152	C₂×C₈	#10625354	1	2	yes	—
153	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
154	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	376	4	no	—
159	C₄⋊C₄	—	188	2	no	—
160	C₄⋊C₄	—	188	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	#10625335	1	2	yes	—
164	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
165	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
166	C₁₆	—	94	4	no	—
167	C₂³	C₂×#10625334	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	376	4	no	—
172	C₂×C₄	—	376	2	no	—
173	C₂×C₄	#13093945	1	2	yes	—
174	C₂×C₄	#13093945	1	2	yes	—
175	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
176	C₂×C₄	#13094013	1	1	yes	characteristic
177	C₂×C₄	#13094022	1	1	yes	characteristic
178	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
179	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
180	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
181	C₈	#13093948	1	2	yes	—
182	C₈	#13093948	1	2	yes	—
183	C₈	Q₈×Dic₄₇	1	2	yes	—
184	C₈	#13094000	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#13093945	1	1	yes	characteristic
187	C₂²	#15689471	1	1	yes	characteristic
188	C₂²	#15689526	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	8	4	no	—
191	C₄	—	752	2	no	—
192	C₄	#15688933	1	4	yes	—
193	C₄	#15689030	1	1	yes	characteristic
194	C₄	C₂×#13093948	1	1	yes	characteristic
195	C₄	#15689475	1	1	yes	characteristic
196	C₄	#15689519	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#20244350	1	1	yes	characteristic
200	C₂	C₂×#15688933	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#3160	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#3160	D₄₇
hypocenter	C₄₇	#13287
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
12. C₄₇×#500	C₄	semidirect
14. C₄₇×#499	C₄	semidirect
22. #13093964	Q₈	semidirect
107. #500	Dic₄₇	semidirect
109. #499	Dic₄₇	semidirect
119. C₄₇	#13287	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
#20241493	2
C₄₇×#3160	2
#13287	47