Subgroups — #26132446

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688949	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	#15688949	—	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	D₁₆	1	2	yes	—
57	C₂³×C₄₇	Q₃₂	1	2	yes	—
58	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	#107	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#121	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#135	1	2	yes	—
68	C₂×Dic₄₇	—	8	4	no	—
69	C₂×Dic₄₇	—	8	4	no	—
70	C₂×Dic₄₇	—	8	4	no	—
71	C₈×C₄₇	—	2	2	no	—
72	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₈×C₄₇	#117	1	1	yes	characteristic
74	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
75	#12834	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	—	2	2	no	—
78	C₂²×C₄₇	—	2	4	no	—
79	C₂²×C₄₇	—	2	4	no	—
80	C₂²×C₄₇	#366	1	4	yes	complemented
81	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
82	C₂²×C₄₇	#466	1	2	yes	—
83	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
84	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
85	C₂²×C₄₇	#508	1	2	yes	—
86	C₄×C₄₇	—	2	2	no	—
87	C₄×C₄₇	#380	1	1	yes	characteristic
88	C₄×C₄₇	#427	1	1	yes	characteristic
89	C₄×C₄₇	#467	1	2	yes	—
90	Dic₄₇	—	16	4	no	—
91	C₂³×C₁₆	D₄₇	1	1	yes	characteristic
92	C₂×#366	—	47	2	no	—
93	#1808	—	47	2	no	—
94	#1380	—	47	2	no	—
95	C₂×C₄₇	—	2	4	no	—
96	C₂×C₄₇	#1808	1	1	yes	characteristic
97	C₂×C₄₇	C₂×#366	1	1	yes	characteristic
98	C₂×C₄₇	#2158	1	1	yes	characteristic
99	C₂×C₄₇	#2215	1	2	yes	—
100	C₂×C₄₇	#2216	1	2	yes	—
101	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
102	C₂²×C₁₆	—	2	4	no	—
103	C₂²×C₁₆	Dic₄₇	1	4	yes	complemented
104	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
105	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
106	C₂×#107	—	47	4	no	—
107	#380	—	94	2	no	—
108	#366	—	94	4	no	—
109	#340	—	94	2	no	—
110	C₄₇	#12834	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	Dic₉₄	1	2	yes	—
115	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
117	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	94	4	no	—
119	C₂×C₂²⋊C₄	—	188	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	8	no	—
122	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
123	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	—
125	#107	—	94	4	no	—
126	#107	—	94	4	no	—
127	#95	—	188	2	no	—
128	C₂⁴	D₃₇₆	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	188	4	no	—
132	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
133	C₂²×C₄	Dic₁₈₈	1	2	yes	—
134	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
135	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
141	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
142	C₂×C₈	#10625350	1	2	yes	—
143	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
145	C₄⋊C₄	—	188	4	no	—
146	C₄⋊C₄	—	188	4	no	—
147	C₂²⋊C₄	—	376	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊Dic₄₇	1	4	yes	complemented
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	#13093958	1	2	yes	—
153	C₂³	D₇₅₂	1	2	yes	—
154	C₂³	Dic₃₇₆	1	2	yes	—
155	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	376	4	no	—
161	C₂×C₄	—	376	4	no	—
162	C₂×C₄	—	376	4	no	—
163	C₂×C₄	#13093958	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
165	C₂×C₄	#13093998	1	2	yes	—
166	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
167	C₂×C₄	#13094046	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
170	C₈	#13093983	1	1	yes	characteristic
171	C₈	#13094003	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	#15688949	1	4	yes	complemented
177	C₂²	C₂×#13093958	1	1	yes	characteristic
178	C₂²	#15689359	1	2	yes	—
179	C₂²	C₂×D₇₅₂	1	1	yes	characteristic
180	C₂²	C₂×Dic₃₇₆	1	1	yes	characteristic
181	C₂²	#15689555	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	752	4	no	—
184	C₄	#15688983	1	1	yes	characteristic
185	C₄	#15689161	1	1	yes	characteristic
186	C₄	#15689364	1	2	yes	—
187	C₂	—	2	4	no	—
188	C₂	#20243417	1	1	yes	characteristic
189	C₂	C₂×#15688949	1	1	yes	characteristic
190	C₂	#20245172	1	1	yes	characteristic
191	C₂	#20245474	1	2	yes	—
192	C₂	#20245479	1	2	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	#12834
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×#15688949	C₂	semidirect
3. #20241480	C₂	semidirect
11. C₂²×C₁₆×C₄₇	C₄	semidirect
52. C₁₆×C₄₇	C₂²⋊C₄	semidirect
80. C₂²×C₄₇	#366	semidirect
103. C₂²×C₁₆	Dic₄₇	semidirect
110. C₄₇	#12834	semidirect
149. C₁₆	C₂²⋊Dic₄₇	semidirect
176. C₂²	#15688949	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688949	2
#20243417	2
#20241480	2
C₂³×C₁₆×C₄₇	2
#12834	47