Subgroups — #26131107

Order 12032.

402 conjugacy classes of subgroups.

The subgroup lattice has 206 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	#20243076	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×#13093934	C₄	1	1	yes	characteristic
8	C₂×#13093934	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15688969	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	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	#13093964	C₂×C₄	1	4	yes	—
25	#13093964	C₂×C₄	1	4	yes	—
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
27	#13093934	—	2	2	no	—
28	#13093934	C₂×C₄	1	4	yes	—
29	C₄×C₄₇⋊C₈	—	2	2	no	—
30	C₈×Dic₄₇	—	2	2	no	—
31	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
32	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₄₇⋊#95	—	2	2	no	—
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	C₄²	1	2	yes	—
57	C₄×Dic₄₇	C₄²	1	2	yes	—
58	C₄×Dic₄₇	M₁₆	1	2	yes	—
59	C₂×C₄₇⋊C₈	—	2	1	no	—
60	C₂×C₄₇⋊C₈	—	2	1	no	—
61	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
62	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
63	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
64	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
65	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
66	C₁₆×C₄₇	C₄²	1	8	yes	—
67	C₄₇⋊C₁₆	C₄²	1	8	yes	—
68	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	4	2	no	—
71	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
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×M₁₆	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×Dic₄₇	—	2	2	no	—
78	C₂×Dic₄₇	—	4	2	no	—
79	C₂×Dic₄₇	—	4	2	no	—
80	C₂×Dic₄₇	#97	1	4	yes	—
81	C₈×C₄₇	—	2	2	no	—
82	C₈×C₄₇	C₂×C₄²	1	2	yes	—
83	C₈×C₄₇	C₂×C₄²	1	2	yes	—
84	C₄₇⋊C₈	—	2	2	no	—
85	C₄₇⋊C₈	#97	1	2	yes	—
86	C₄₇⋊C₈	#97	1	2	yes	—
87	#12541	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
90	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
91	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₄₇	—	4	2	no	—
93	C₄×C₄₇	C₄³	1	1	yes	characteristic
94	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
95	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
96	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
97	Dic₄₇	—	2	4	no	—
98	Dic₄₇	—	4	2	no	—
99	C₂×#347	—	47	2	no	—
100	C₂×#347	—	47	1	no	—
101	C₂×#347	D₄₇	1	1	yes	characteristic
102	C₄×M₃₂	—	47	2	no	—
103	#1726	—	47	1	no	—
104	C₂×C₄₇	—	2	1	no	—
105	C₂×C₄₇	—	2	1	no	—
106	C₂×C₄₇	C₄×#97	1	1	yes	characteristic
107	C₂×C₄₇	#2110	1	2	yes	—
108	C₂×M₃₂	—	47	2	no	—
109	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
110	C₂×#97	—	47	1	no	—
111	C₂×#97	—	47	1	no	—
112	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
113	C₂×C₄×C₈	—	47	1	no	—
114	C₂×C₄×C₈	—	47	1	no	—
115	C₂×C₄×C₈	—	47	1	no	—
116	#376	—	47	1	no	—
117	#347	—	47	8	no	—
118	#347	—	47	4	no	—
119	#347	Dic₄₇	1	4	yes	complemented
120	#346	—	47	4	no	—
121	C₄×C₁₆	—	47	4	no	—
122	C₄₇	#12541	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	47	1	no	—
124	C₂²×C₈	—	47	1	no	—
125	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	C₂×C₄²	—	47	1	no	—
128	C₂×C₄²	—	47	1	no	—
129	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
130	M₃₂	—	47	4	no	—
131	M₃₂	—	47	4	no	—
132	M₃₂	C₄×D₄₇	1	4	yes	—
133	M₃₂	C₂×Dic₄₇	1	4	yes	—
134	C₂×C₁₆	—	47	4	no	—
135	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
136	#97	—	47	4	no	—
137	#97	—	94	2	no	—
138	#97	C₄×D₄₇	1	2	yes	—
139	#97	C₂×Dic₄₇	1	2	yes	—
140	C₄×C₈	—	47	2	no	—
141	C₄×C₈	—	47	2	no	—
142	C₄×C₈	—	94	2	no	—
143	C₄×C₈	—	94	2	no	—
144	#95	—	94	2	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	47	2	no	—
147	C₂²×C₄	—	94	1	no	—
148	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	47	1	no	—
152	C₂×C₈	—	47	1	no	—
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	47	2	no	—
155	C₂×C₈	—	47	2	no	—
156	C₂×C₈	—	94	1	no	—
157	C₂×C₈	—	94	1	no	—
158	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
162	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
163	C₄²	—	2	1	no	—
164	C₄²	—	47	2	no	—
165	C₄²	—	47	2	no	—
166	C₄²	—	47	2	no	—
167	C₄²	—	94	1	no	—
168	C₄²	C₄×Dic₄₇	1	2	yes	—
169	C₁₆	—	47	8	no	—
170	C₁₆	C₄×Dic₄₇	1	8	yes	—
171	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	47	4	no	—
175	C₂×C₄	—	94	2	no	—
176	C₂×C₄	—	188	2	no	—
177	C₂×C₄	—	188	2	no	—
178	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
179	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
182	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	47	2	no	—
186	C₈	—	47	2	no	—
187	C₈	—	94	2	no	—
188	C₈	C₄²×D₄₇	1	2	yes	—
189	C₈	C₂×C₄×Dic₄₇	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	#97×D₄₇	1	1	yes	characteristic
192	C₂²	M₁₆×Dic₄₇	1	1	yes	characteristic
193	C₂²	#15689075	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	94	4	no	—
196	C₄	—	188	2	no	—
197	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
198	C₄	#97×D₄₇	1	1	yes	characteristic
199	C₄	M₁₆×Dic₄₇	1	1	yes	characteristic
200	C₄	#15689075	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#97×Dic₄₇	1	1	yes	characteristic
204	C₂	#20244991	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	#12541
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
119. #347	Dic₄₇	semidirect
122. C₄₇	#12541	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
#20243076	2
C₂×C₄₇×#347	2
#12541	47