Subgroups — #346×Dic₄₇ #26131098

Order 12032.

510 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688898	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₄₇	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
13	#15688899	C₄	1	4	yes	complemented
14	#15688898	C₄	1	8	yes	complemented
15	C₁₆×Dic₄₇	C₄	1	8	yes	complemented
16	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#346	C₄	1	4	yes	complemented
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₄₇	C₈	1	4	yes	—
22	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
27	C₄×C₄₇⋊C₈	C₈	1	4	yes	—
28	C₈×Dic₄₇	C₈	1	4	yes	—
29	C₈×Dic₄₇	C₈	1	4	yes	—
30	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
31	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
43	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₄₇	C₄²	1	2	yes	—
45	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₄₇	C₄²	1	2	yes	—
52	C₄²×C₄₇	C₂×C₈	1	2	yes	—
53	C₄×Dic₄₇	—	2	8	no	—
54	C₄×Dic₄₇	C₄²	1	2	yes	—
55	C₄×Dic₄₇	C₄²	1	2	yes	—
56	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
57	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
58	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
59	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
60	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
61	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
62	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
63	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
64	C₁₆×C₄₇	C₄²	1	8	yes	complemented
65	C₄₇⋊C₁₆	C₄²	1	8	yes	complemented
66	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×Dic₄₇	—	2	4	no	—
76	C₂×Dic₄₇	C₄×C₈	1	4	yes	—
77	C₂×Dic₄₇	C₄×C₈	1	4	yes	—
78	C₂×Dic₄₇	M₃₂	1	8	yes	—
79	C₈×C₄₇	C₄×C₈	1	4	yes	—
80	C₈×C₄₇	C₂×C₄²	1	2	yes	—
81	C₈×C₄₇	C₂×C₄²	1	2	yes	—
82	C₄₇⋊C₈	C₄×C₈	1	4	yes	—
83	C₄₇⋊C₈	C₄×C₈	1	4	yes	—
84	C₄×#346	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×M₃₂	1	4	yes	—
89	C₄×C₄₇	—	2	4	no	—
90	C₄×C₄₇	C₄³	1	1	yes	characteristic
91	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
94	Dic₄₇	—	2	4	no	—
95	Dic₄₇	#346	1	8	yes	direct product factor
96	C₂×#346	—	47	2	no	—
97	C₂×#346	—	47	1	no	—
98	C₂×#346	D₄₇	1	1	yes	characteristic
99	C₂×C₄×C₁₆	—	47	2	no	—
100	C₄²×C₈	—	47	1	no	—
101	C₂×C₄₇	C₄²×C₈	1	1	yes	characteristic
102	C₂×C₄₇	C₂×#346	1	2	yes	—
103	C₂×C₄₇	C₄×M₃₂	1	2	yes	—
104	C₂×C₄₇	C₄×M₃₂	1	2	yes	—
105	C₂²×C₁₆	—	47	2	no	—
106	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
107	C₂×C₄×C₈	—	47	1	no	—
108	C₂×C₄×C₈	—	47	1	no	—
109	C₂×C₄×C₈	—	47	1	no	—
110	C₂×C₄×C₈	—	47	1	no	—
111	C₂×C₄×C₈	—	47	1	no	—
112	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
113	C₄³	—	47	1	no	—
114	#346	—	47	8	no	—
115	#346	—	47	4	no	—
116	#346	Dic₄₇	1	4	yes	direct product factor
117	C₄×C₁₆	—	47	8	no	—
118	C₄₇	C₄×#346	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	47	1	no	—
120	C₂²×C₈	—	47	1	no	—
121	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
123	C₂×C₄²	—	47	1	no	—
124	C₂×C₄²	—	47	1	no	—
125	C₂×C₄²	—	47	4	no	—
126	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
127	C₂×C₁₆	—	47	4	no	—
128	C₂×C₁₆	—	47	4	no	—
129	C₂×C₁₆	—	47	4	no	—
130	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
131	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
132	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
133	C₄×C₈	—	47	4	no	—
134	C₄×C₈	—	47	4	no	—
135	C₄×C₈	—	47	4	no	—
136	C₄×C₈	—	47	4	no	—
137	C₄×C₈	—	47	4	no	—
138	C₄×C₈	C₄×D₄₇	1	2	yes	—
139	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
140	C₂²×C₄	—	47	2	no	—
141	C₂²×C₄	—	47	2	no	—
142	C₂²×C₄	C₈×D₄₇	1	2	yes	—
143	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
144	C₂×C₈	—	47	2	no	—
145	C₂×C₈	—	47	2	no	—
146	C₂×C₈	—	47	2	no	—
147	C₂×C₈	—	47	2	no	—
148	C₂×C₈	—	47	2	no	—
149	C₂×C₈	—	47	2	no	—
150	C₂×C₈	C₈×D₄₇	1	2	yes	—
151	C₂×C₈	C₈×D₄₇	1	2	yes	—
152	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
155	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
156	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
157	C₄²	—	47	2	no	—
158	C₄²	—	47	2	no	—
159	C₄²	—	47	2	no	—
160	C₄²	—	47	2	no	—
161	C₄²	—	94	8	no	—
162	C₄²	C₈×D₄₇	1	2	yes	—
163	C₄²	C₄×Dic₄₇	1	2	yes	—
164	C₁₆	—	47	8	no	—
165	C₁₆	C₄×Dic₄₇	1	8	yes	complemented
166	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	47	8	no	—
169	C₂×C₄	—	47	4	no	—
170	C₂×C₄	—	47	4	no	—
171	C₂×C₄	—	94	4	no	—
172	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
173	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
174	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
176	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
179	C₈	—	47	4	no	—
180	C₈	—	47	4	no	—
181	C₈	C₈×Dic₄₇	1	4	yes	—
182	C₈	C₄²×D₄₇	1	2	yes	—
183	C₈	C₂×C₄×Dic₄₇	1	2	yes	—
184	C₂²	C₄×C₈×D₄₇	1	1	yes	characteristic
185	C₂²	C₄×C₈×D₄₇	1	1	yes	characteristic
186	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
187	C₂²	M₃₂×D₄₇	1	4	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	47	8	no	—
190	C₄	—	94	4	no	—
191	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
192	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
193	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
194	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
195	C₂	C₄×C₈×Dic₄₇	1	1	yes	characteristic
196	C₂	#346×D₄₇	1	2	yes	—
197	C₂	M₃₂×Dic₄₇	1	2	yes	—
198	C₂	#20244981	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂×C₄₇	C₄²×C₈
Fitting	C₂×C₄₇×#346	C₂
socle, prime radical	C₂³×C₄₇	C₂²×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	C₄×#346
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #15688899	C₄	semidirect
14. #15688898	C₄	semidirect
15. C₁₆×Dic₄₇	C₄	semidirect
18. C₄₇×#346	C₄	semidirect
64. C₁₆×C₄₇	C₄²	semidirect
65. C₄₇⋊C₁₆	C₄²	semidirect
95. Dic₄₇	#346	direct (N × H)
116. #346	Dic₄₇	direct (N × H)
118. C₄₇	C₄×#346	semidirect
165. 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×#15688899	2
C₂×#15688898	2
C₂×C₁₆×Dic₄₇	2
C₄×C₈×Dic₄₇	2
C₂×C₄₇×#346	2
C₄×#346	47

Subgroups — #346×Dic47 #26131098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #346×Dic₄₇ #26131098