Subgroups — #26132450

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688951	C₂	1	2	yes	maximal, complemented
2	#20243417	C₂	1	2	yes	maximal
3	#20241480	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093958	C₂²	1	4	yes	—
6	#15688983	—	2	2	no	—
7	#15688951	—	2	4	no	—
8	#15688886	—	2	2	no	—
9	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₄₇	—	2	4	no	—
11	C₂²×C₁₆×C₄₇	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	—	2	4	no	—
15	C₂×C₂²⋊Dic₄₇	—	4	2	no	—
16	#13093958	—	2	4	no	—
17	#13093958	—	2	4	no	—
18	C₄₇⋊#95	—	4	2	no	—
19	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	—	2	2	no	—
21	C₂²×C₈×C₄₇	—	2	2	no	—
22	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	—	2	2	no	—
27	C₂×C₁₆×C₄₇	—	2	8	no	—
28	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
31	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	4	4	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₄₇	#41	1	2	yes	—
48	C₂²⋊Dic₄₇	—	8	4	no	—
49	C₄⋊Dic₄₇	—	4	4	no	—
50	C₄⋊Dic₄₇	—	4	4	no	—
51	C₁₆×C₄₇	—	2	2	no	—
52	C₁₆×C₄₇	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₄₇	—	2	2	no	—
54	C₂³×C₄₇	—	2	2	no	—
55	C₂³×C₄₇	#107	1	2	yes	—
56	C₂³×C₄₇	SD₃₂	1	4	yes	—
57	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	4	no	—
62	C₂×C₄×C₄₇	#107	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#121	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#135	1	2	yes	—
67	C₂×Dic₄₇	—	8	4	no	—
68	C₂×Dic₄₇	—	8	4	no	—
69	C₂×Dic₄₇	—	8	4	no	—
70	C₈×C₄₇	—	2	2	no	—
71	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₄₇	#117	1	1	yes	characteristic
73	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
74	#12835	—	47	1	no	maximal, Sylow
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	—	2	8	no	—
78	C₂²×C₄₇	#367	1	4	yes	complemented
79	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
80	C₂²×C₄₇	#466	1	2	yes	—
81	C₂²×C₄₇	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₄₇	#508	1	2	yes	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	#380	1	1	yes	characteristic
85	C₄×C₄₇	#427	1	1	yes	characteristic
86	C₄×C₄₇	#467	1	2	yes	—
87	Dic₄₇	—	16	4	no	—
88	C₂³×C₁₆	D₄₇	1	1	yes	characteristic
89	C₂×#367	—	47	2	no	—
90	#1808	—	47	2	no	—
91	#1380	—	47	2	no	—
92	C₂×C₄₇	—	2	4	no	—
93	C₂×C₄₇	#1808	1	1	yes	characteristic
94	C₂×C₄₇	C₂×#367	1	1	yes	characteristic
95	C₂×C₄₇	#2158	1	1	yes	characteristic
96	C₂×C₄₇	#2217	1	4	yes	—
97	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
98	C₂²×C₁₆	—	2	4	no	—
99	C₂²×C₁₆	Dic₄₇	1	4	yes	complemented
100	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
102	C₂×#107	—	47	4	no	—
103	#380	—	94	2	no	—
104	#367	—	94	4	no	—
105	#340	—	94	2	no	—
106	C₄₇	#12835	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	Dic₉₄	1	2	yes	—
111	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
113	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	94	4	no	—
115	C₂×C₂²⋊C₄	—	188	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	—
121	#107	—	94	4	no	—
122	#107	—	94	4	no	—
123	#95	—	188	2	no	—
124	C₂⁴	D₃₇₆	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	188	4	no	—
128	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
129	C₂²×C₄	Dic₁₈₈	1	2	yes	—
130	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
131	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
137	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
138	C₂×C₈	#10625350	1	2	yes	—
139	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
141	C₄⋊C₄	—	188	4	no	—
142	C₄⋊C₄	—	188	4	no	—
143	C₂²⋊C₄	—	376	4	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₂²⋊Dic₄₇	1	4	yes	complemented
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	#13093958	1	2	yes	—
149	C₂³	#13093969	1	4	yes	—
150	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	376	4	no	—
156	C₂×C₄	—	376	4	no	—
157	C₂×C₄	—	376	4	no	—
158	C₂×C₄	#13093958	1	2	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
160	C₂×C₄	#13093998	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
162	C₂×C₄	#13094046	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
165	C₈	#13093983	1	1	yes	characteristic
166	C₈	#13094003	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	#15688951	1	4	yes	complemented
171	C₂²	C₂×#13093958	1	1	yes	characteristic
172	C₂²	#15689359	1	2	yes	—
173	C₂²	C₂×#13093969	1	2	yes	—
174	C₂²	#15689555	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	752	4	no	—
177	C₄	#15688983	1	1	yes	characteristic
178	C₄	#15689161	1	1	yes	characteristic
179	C₄	#15689364	1	2	yes	—
180	C₂	—	2	4	no	—
181	C₂	#20243417	1	1	yes	characteristic
182	C₂	C₂×#15688951	1	1	yes	characteristic
183	C₂	#20245172	1	1	yes	characteristic
184	C₂	#20245484	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂³×C₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂³×C₁₆	D₄₇
hypocenter	C₄₇	#12835
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15688951	C₂	semidirect
3. #20241480	C₂	semidirect
11. C₂²×C₁₆×C₄₇	C₄	semidirect
52. C₁₆×C₄₇	C₂²⋊C₄	semidirect
78. C₂²×C₄₇	#367	semidirect
99. C₂²×C₁₆	Dic₄₇	semidirect
106. C₄₇	#12835	semidirect
145. C₁₆	C₂²⋊Dic₄₇	semidirect
170. C₂²	#15688951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688951	2
#20243417	2
#20241480	2
C₂³×C₁₆×C₄₇	2
#12835	47