Subgroups — #26115653

Order 12032.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243552	C₂	1	1	yes	maximal, characteristic
2	#20241952	C₂	1	4	yes	maximal
3	#20241671	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#3160	C₂	1	1	yes	maximal, characteristic
5	#15689176	C₂²	1	2	yes	—
6	C₂×#13093958	—	2	1	no	—
7	C₂×#13093956	—	2	1	no	—
8	#15688989	C₂²	1	2	yes	—
9	#15688860	—	2	8	no	—
10	#15688844	—	2	1	no	—
11	C₈×C₄₇⋊C₈	—	2	1	no	—
12	C₄₇×#533	C₂²	1	2	yes	—
13	C₄₇×#500	C₄	1	2	yes	complemented
14	C₄₇×#499	—	2	1	no	—
15	C₄₇×#499	—	2	1	no	—
16	C₄₇×#498	C₄	1	2	yes	complemented
17	C₄₇×#427	—	2	1	no	—
18	C₄₇×#427	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	#13093958	—	4	2	no	—
22	#13093956	—	4	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄₇⋊#95	—	2	4	no	—
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
30	#125×C₄₇	—	2	2	no	—
31	C₂²⋊Q₈×C₄₇	—	2	4	no	—
32	C₄×Q₈×C₄₇	—	2	4	no	—
33	#117×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	#107×C₄₇	—	2	2	no	—
36	#107×C₄₇	—	2	2	no	—
37	#106×C₄₇	—	2	2	no	—
38	#106×C₄₇	—	2	2	no	—
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₄₇	—	4	2	no	—
47	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
53	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
54	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
55	C₄⋊C₄×C₄₇	—	2	4	no	—
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄⋊C₄×C₄₇	—	4	4	no	—
58	C₂²⋊C₄×C₄₇	—	4	2	no	—
59	C₄²×C₄₇	—	4	2	no	—
60	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₄₇	#41	1	2	yes	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	2	4	no	—
66	C₂³×C₄₇	#127	1	1	yes	characteristic
67	Q₈×C₄₇	—	8	4	no	—
68	C₂×C₄×C₄₇	—	4	2	no	—
69	C₂×C₄×C₄₇	—	4	4	no	—
70	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₄₇	#121	1	2	yes	—
73	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×Q₁₆	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	#103	1	2	yes	—
83	C₈×C₄₇	#103	1	2	yes	—
84	C₄₇⋊C₈	—	4	4	no	—
85	#9755	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	#390	1	1	yes	characteristic
88	C₂²×C₄₇	#492	1	1	yes	characteristic
89	C₂²×C₄₇	#494	1	1	yes	characteristic
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	8	4	no	—
93	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
94	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
95	C₄×C₄₇	#487	1	1	yes	characteristic
96	C₄×C₄₇	#487	1	1	yes	characteristic
97	Dic₄₇	—	16	2	no	—
98	#3160	D₄₇	1	1	yes	characteristic
99	#1846	—	47	1	no	—
100	#1518	—	47	4	no	—
101	#1450	—	47	1	no	—
102	C₂×C₄₇	—	2	1	no	—
103	C₂×C₄₇	—	2	1	no	—
104	C₂×C₄₇	#1956	1	1	yes	characteristic
105	C₂×C₄₇	#1976	1	2	yes	—
106	#533	C₂×D₄₇	1	2	yes	—
107	#500	Dic₄₇	1	2	yes	complemented
108	#499	—	2	1	no	—
109	#499	—	2	1	no	—
110	#498	Dic₄₇	1	2	yes	complemented
111	#431	—	47	2	no	—
112	#427	—	2	1	no	—
113	#427	—	2	1	no	—
114	C₂×#107	—	94	1	no	—
115	C₂×#106	—	94	1	no	—
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	#382	—	47	2	no	—
118	#328	—	94	8	no	—
119	#322	—	94	1	no	—
120	C₈²	—	94	1	no	—
121	C₄₇	#9755	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	2	4	no	—
127	C₄×Q₈	—	2	4	no	—
128	#117	—	2	2	no	—
129	C₂×C₄⋊C₄	—	94	2	no	—
130	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
131	#107	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	—	188	2	no	—
134	#106	—	2	2	no	—
135	#106	—	2	2	no	—
136	#106	—	188	2	no	—
137	C₄⋊C₈	—	94	2	no	—
138	C₂²⋊C₈	—	47	4	no	—
139	C₄×C₈	—	94	2	no	—
140	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
144	#95	—	94	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂²×C₄	—	188	2	no	—
147	C₂²×C₄	D₄×D₄₇	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₈	—	94	4	no	—
150	C₂×C₈	#10625336	1	2	yes	—
151	C₂×C₈	#10625336	1	2	yes	—
152	C₂×C₈	#10625337	1	2	yes	—
153	C₂×C₈	#10625337	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	188	2	no	—
162	C₄⋊C₄	—	188	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	#10625352	1	2	yes	—
166	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₂³	#13094020	1	1	yes	characteristic
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	376	4	no	—
173	C₂×C₄	—	376	2	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
176	C₂×C₄	#13093995	1	2	yes	—
177	C₂×C₄	#13094011	1	1	yes	characteristic
178	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
179	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
180	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
181	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
182	C₈	—	188	4	no	—
183	C₈	#13093948	1	2	yes	—
184	C₈	#13093948	1	2	yes	—
185	C₈	#13093948	1	2	yes	—
186	C₈	#13093948	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#15689026	1	1	yes	characteristic
189	C₂²	#15689497	1	1	yes	characteristic
190	C₂²	#15689502	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	—	752	2	no	—
195	C₄	C₂×#13093948	1	1	yes	characteristic
196	C₄	C₂×#13093948	1	1	yes	characteristic
197	C₄	#15689473	1	1	yes	characteristic
198	C₄	#15689473	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#20244191	1	1	yes	characteristic
202	C₂	#20244287	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689026
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3160	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3160	D₄₇
hypocenter	C₄₇	#9755
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
13. C₄₇×#500	C₄	semidirect
16. C₄₇×#498	C₄	semidirect
107. #500	Dic₄₇	semidirect
110. #498	Dic₄₇	semidirect
121. C₄₇	#9755	semidirect

Maximal subgroups

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

subgroup	index
#20243552	2
#20241952	2
#20241671	2
C₄₇×#3160	2
#9755	47