Subgroups — #347×Dic₄₇ #26131104

Order 12032.

418 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688901	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688900	C₂	1	2	yes	maximal
3	M₃₂×Dic₄₇	C₂	1	2	yes	maximal
4	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#347	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093935	C₄	1	1	yes	characteristic
8	C₂×#13093934	C₄	1	1	yes	characteristic
9	C₂×#13093934	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
13	#15688901	C₄	1	4	yes	complemented
14	#15688900	C₄	1	8	yes	complemented
15	#15688898	C₄	1	4	yes	complemented
16	C₁₆×Dic₄₇	C₄	1	4	yes	complemented
17	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
18	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#347	C₄	1	4	yes	complemented
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	—	2	2	no	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	#13093964	C₂×C₄	1	4	yes	—
26	#13093964	C₂×C₄	1	4	yes	—
27	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
28	#13093935	—	2	2	no	—
29	#13093934	—	2	2	no	—
30	#13093934	C₂×C₄	1	4	yes	—
31	C₈×Dic₄₇	—	2	2	no	—
32	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
38	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
40	#97×C₄₇	C₂×C₄	1	2	yes	—
41	#97×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	—	2	1	no	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₄₇	—	2	1	no	—
45	C₂²×Dic₄₇	M₁₆	1	2	yes	—
46	C₂×C₈×C₄₇	—	2	1	no	—
47	C₂×C₈×C₄₇	—	2	1	no	—
48	C₂×C₈×C₄₇	C₄²	1	2	yes	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₄₇	—	2	1	no	—
54	C₄²×C₄₇	C₄²	1	2	yes	—
55	C₄×Dic₄₇	—	2	1	no	—
56	C₄×Dic₄₇	—	4	4	no	—
57	C₄×Dic₄₇	C₄²	1	2	yes	—
58	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
59	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
60	C₄×Dic₄₇	M₁₆	1	2	yes	—
61	C₂×C₄₇⋊C₈	—	2	1	no	—
62	C₂×C₄₇⋊C₈	—	2	1	no	—
63	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
64	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
65	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
66	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
67	C₁₆×C₄₇	C₄²	1	8	yes	complemented
68	C₄₇⋊C₁₆	C₄²	1	8	yes	complemented
69	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₄₇	—	2	4	no	—
79	C₂×Dic₄₇	—	2	2	no	—
80	C₂×Dic₄₇	—	4	2	no	—
81	C₂×Dic₄₇	#97	1	2	yes	—
82	C₂×Dic₄₇	#97	1	2	yes	—
83	C₈×C₄₇	—	2	2	no	—
84	C₈×C₄₇	C₂×C₄²	1	2	yes	—
85	C₈×C₄₇	C₂×C₄²	1	2	yes	—
86	C₄₇⋊C₈	—	2	2	no	—
87	C₄₇⋊C₈	#97	1	4	yes	—
88	C₄×#347	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
91	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
93	C₄×C₄₇	—	4	2	no	—
94	C₄×C₄₇	C₄³	1	1	yes	characteristic
95	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
96	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
98	Dic₄₇	—	2	2	no	—
99	Dic₄₇	—	4	2	no	—
100	Dic₄₇	#347	1	4	yes	direct product factor
101	C₂×#347	—	47	2	no	—
102	C₂×#347	—	47	1	no	—
103	C₂×#347	D₄₇	1	1	yes	characteristic
104	C₄×M₃₂	—	47	2	no	—
105	C₄×#97	—	47	1	no	—
106	C₂×C₄₇	—	2	1	no	—
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	C₄×#97	1	1	yes	characteristic
109	C₂×C₄₇	C₂×#347	1	2	yes	—
110	C₂×M₃₂	—	47	2	no	—
111	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
112	C₂×#97	—	47	1	no	—
113	C₂×#97	—	47	1	no	—
114	C₂×#97	—	47	1	no	—
115	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
116	C₂×C₄×C₈	—	47	1	no	—
117	C₂×C₄×C₈	—	47	1	no	—
118	C₄³	—	47	1	no	—
119	#347	—	47	8	no	—
120	#347	—	47	4	no	—
121	#347	Dic₄₇	1	4	yes	direct product factor
122	#346	—	47	4	no	—
123	C₄×C₁₆	—	47	4	no	—
124	C₄₇	C₄×#347	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	47	1	no	—
126	C₂²×C₈	—	47	1	no	—
127	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
129	C₂×C₄²	—	47	1	no	—
130	C₂×C₄²	—	47	1	no	—
131	C₂×C₄²	—	94	2	no	—
132	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
133	M₃₂	—	47	4	no	—
134	M₃₂	—	47	4	no	—
135	M₃₂	C₄×D₄₇	1	4	yes	—
136	M₃₂	C₂×Dic₄₇	1	4	yes	—
137	C₂×C₁₆	—	47	4	no	—
138	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
139	#97	—	47	4	no	—
140	#97	—	94	2	no	—
141	#97	—	94	2	no	—
142	#97	C₄×D₄₇	1	2	yes	—
143	#97	C₂×Dic₄₇	1	2	yes	—
144	C₄×C₈	—	47	2	no	—
145	C₄×C₈	—	47	2	no	—
146	C₄×C₈	—	94	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	47	2	no	—
149	C₂²×C₄	—	94	1	no	—
150	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	47	2	no	—
155	C₂×C₈	—	47	2	no	—
156	C₂×C₈	—	47	2	no	—
157	C₂×C₈	—	94	1	no	—
158	C₂×C₈	—	94	1	no	—
159	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
162	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
163	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
164	C₄²	—	2	1	no	—
165	C₄²	—	47	1	no	—
166	C₄²	—	47	1	no	—
167	C₄²	—	47	2	no	—
168	C₄²	—	47	2	no	—
169	C₄²	—	94	1	no	—
170	C₄²	—	188	4	no	—
171	C₄²	C₄×Dic₄₇	1	2	yes	—
172	C₁₆	—	47	8	no	—
173	C₁₆	C₄×Dic₄₇	1	8	yes	complemented
174	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	47	2	no	—
178	C₂×C₄	—	47	2	no	—
179	C₂×C₄	—	94	4	no	—
180	C₂×C₄	—	94	2	no	—
181	C₂×C₄	—	188	2	no	—
182	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
183	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
187	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	47	4	no	—
190	C₈	—	94	2	no	—
191	C₈	C₄²×D₄₇	1	2	yes	—
192	C₈	C₂×C₄×Dic₄₇	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	#97×D₄₇	1	1	yes	characteristic
195	C₂²	M₁₆×Dic₄₇	1	1	yes	characteristic
196	C₂²	#15689075	1	1	yes	characteristic
197	C₄	—	4	2	no	—
198	C₄	—	47	4	no	—
199	C₄	—	94	2	no	—
200	C₄	—	188	2	no	—
201	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
202	C₄	#97×D₄₇	1	1	yes	characteristic
203	C₄	M₁₆×Dic₄₇	1	1	yes	characteristic
204	C₄	#15689075	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#97×Dic₄₇	1	1	yes	characteristic
208	C₂	#347×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#347	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂³×C₄₇	C₂×M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#347	D₄₇
hypocenter	C₄₇	C₄×#347
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
13. #15688901	C₄	semidirect
14. #15688900	C₄	semidirect
15. #15688898	C₄	semidirect
16. C₁₆×Dic₄₇	C₄	semidirect
19. C₄₇×#347	C₄	semidirect
67. C₁₆×C₄₇	C₄²	semidirect
68. C₄₇⋊C₁₆	C₄²	semidirect
100. Dic₄₇	#347	direct (N × H)
121. #347	Dic₄₇	direct (N × H)
124. C₄₇	C₄×#347	semidirect
173. C₁₆	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688901	2
C₂×#15688900	2
M₃₂×Dic₄₇	2
#97×Dic₄₇	2
C₂×C₄₇×#347	2
C₄×#347	47

Subgroups — #347×Dic47 #26131104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #347×Dic₄₇ #26131104