Subgroups — #26122100

Order 12032.

950 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×#15688989	C₂	1	2	yes	maximal
2	C₂×#15688886	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
5	#15688989	—	2	4	no	—
6	#15688989	C₄	1	8	yes	complemented
7	C₂×C₄₇⋊#95	—	2	4	no	—
8	#15688886	—	2	8	no	—
9	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	—	2	4	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	4	yes	complemented
13	C₂³×Dic₄₇	—	2	4	no	—
14	C₂×C₄⋊Dic₄₇	—	2	8	no	—
15	C₂×C₄⋊Dic₄₇	—	2	8	no	—
16	C₂×C₄⋊Dic₄₇	—	2	4	no	—
17	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
18	C₄₇⋊#95	—	2	16	no	—
19	C₂³×C₄×C₄₇	D₄	1	2	yes	—
20	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	—	2	4	no	—
22	C₂²×C₈×C₄₇	—	2	4	no	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₄₇	D₄	1	4	yes	—
25	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	D₄	1	4	yes	—
29	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
30	C₄×C₈×C₄₇	—	2	8	no	—
31	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	—	2	8	no	—
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
41	C₂²×C₄×C₄₇	#41	1	4	yes	—
42	C₂²×Dic₄₇	—	2	8	no	—
43	C₂²×Dic₄₇	—	4	4	no	—
44	C₂²×Dic₄₇	—	4	4	no	—
45	C₂²×Dic₄₇	—	4	16	no	—
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	—	2	4	no	—
48	C₂×C₈×C₄₇	—	2	8	no	—
49	C₂×C₈×C₄₇	—	2	8	no	—
50	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
51	C₄²×C₄₇	C₄²	1	4	yes	—
52	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
54	C₄⋊Dic₄₇	—	2	16	no	—
55	C₄⋊Dic₄₇	—	4	8	no	—
56	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
57	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₄₇	#124	1	2	yes	—
59	C₂³×C₄₇	#127	1	1	yes	characteristic
60	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂³×C₄₇	#135	1	8	yes	—
62	C₂×C₄×C₄₇	—	2	16	no	—
63	C₂×C₄×C₄₇	—	2	8	no	—
64	C₂×C₄×C₄₇	#95	1	8	yes	—
65	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₄₇	#117	1	2	yes	—
70	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
71	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
72	C₂×C₄×C₄₇	#121	1	4	yes	—
73	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
74	C₂×C₄×C₄₇	#125	1	2	yes	—
75	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
76	C₂×Dic₄₇	—	4	8	no	—
77	C₂×Dic₄₇	—	4	16	no	—
78	C₂×Dic₄₇	—	8	8	no	—
79	C₂×Dic₄₇	—	8	8	no	—
80	C₈×C₄₇	—	2	8	no	—
81	#10842	—	47	1	no	maximal, Sylow
82	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
83	C₂²×C₄₇	#382	1	1	yes	characteristic
84	C₂²×C₄₇	#384	1	1	yes	characteristic
85	C₂²×C₄₇	#390	1	2	yes	—
86	C₂²×C₄₇	#391	1	2	yes	—
87	C₂²×C₄₇	#416	1	8	yes	—
88	C₂²×C₄₇	#427	1	4	yes	—
89	C₂²×C₄₇	#488	1	8	yes	—
90	C₂²×C₄₇	#495	1	4	yes	—
91	C₂²×C₄₇	#499	1	4	yes	—
92	C₄×C₄₇	—	2	8	no	—
93	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
94	C₄×C₄₇	#380	1	2	yes	—
95	C₄×C₄₇	#389	1	4	yes	—
96	Dic₄₇	—	8	8	no	—
97	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
98	C₂×#382	—	47	2	no	—
99	C₂×#340	—	47	4	no	—
100	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
101	C₂×C₄₇	#1735	1	2	yes	—
102	C₂×C₄₇	#1835	1	2	yes	—
103	C₂×C₄₇	#1846	1	2	yes	—
104	C₂×C₄₇	#1958	1	4	yes	—
105	C₂×C₄₇	#1991	1	4	yes	—
106	C₂³×C₈	C₂×D₄₇	1	2	yes	—
107	C₂²×C₄⋊C₄	—	47	4	no	—
108	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
109	C₂×C₄×C₈	—	2	4	no	—
110	C₂×C₄×C₈	Dic₄₇	1	4	yes	complemented
111	#382	—	47	8	no	—
112	#382	—	94	4	no	—
113	C₂×#95	—	94	4	no	—
114	#340	—	94	8	no	—
115	C₄₇	#10842	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	—	94	4	no	—
117	C₂³×C₄	D₁₈₈	1	2	yes	—
118	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	Dic₉₄	1	4	yes	—
122	C₂²×C₈	D₁₈₈	1	4	yes	—
123	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
124	C₂×C₄⋊C₄	—	47	8	no	—
125	C₂×C₄⋊C₄	—	94	8	no	—
126	C₂×C₄⋊C₄	—	94	8	no	—
127	C₂×C₄⋊C₄	—	94	4	no	—
128	C₂×C₄²	Dic₉₄	1	2	yes	—
129	C₂×C₄²	C₄×D₄₇	1	4	yes	—
130	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
132	C₄×C₈	—	2	8	no	—
133	#95	—	94	16	no	—
134	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	94	8	no	—
137	C₂²×C₄	—	188	4	no	—
138	C₂²×C₄	—	188	4	no	—
139	C₂²×C₄	—	188	16	no	—
140	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
141	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
142	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
145	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
146	C₂²×C₄	#10625350	1	4	yes	—
147	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	—
154	C₄⋊C₄	—	94	16	no	—
155	C₄⋊C₄	—	188	8	no	—
156	C₄²	C₄×Dic₄₇	1	4	yes	—
157	C₄²	Dic₄₇⋊C₄	1	8	yes	—
158	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
159	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
160	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
161	C₂³	#13094012	1	2	yes	—
162	C₂³	#13094021	1	1	yes	characteristic
163	C₂³	#13094024	1	1	yes	characteristic
164	C₂³	#13094046	1	8	yes	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	376	8	no	—
168	C₂×C₄	—	376	8	no	—
169	C₂×C₄	—	188	8	no	—
170	C₂×C₄	—	188	16	no	—
171	C₂×C₄	C₄₇⋊#95	1	8	yes	—
172	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
176	C₂×C₄	#13093983	1	2	yes	—
177	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
178	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
179	C₂×C₄	#13093998	1	4	yes	—
180	C₂×C₄	#13094003	1	4	yes	—
181	C₂×C₄	#13094016	1	2	yes	—
182	C₂×C₄	#13094024	1	2	yes	—
183	C₈	—	2	8	no	—
184	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
185	C₂²	#15688989	1	1	yes	characteristic
186	C₂²	#15688993	1	1	yes	characteristic
187	C₂²	#15689025	1	2	yes	—
188	C₂²	#15689029	1	2	yes	—
189	C₂²	#15689120	1	8	yes	—
190	C₂²	#15689161	1	4	yes	—
191	C₂²	#15689478	1	8	yes	—
192	C₂²	#15689507	1	4	yes	—
193	C₂²	#15689525	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	376	8	no	—
196	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
197	C₄	#15688983	1	2	yes	—
198	C₄	#15689021	1	4	yes	—
199	C₂	#20241618	1	1	yes	characteristic
200	C₂	#20243105	1	2	yes	—
201	C₂	#20243519	1	2	yes	—
202	C₂	#20243552	1	2	yes	—
203	C₂	#20244199	1	4	yes	—
204	C₂	#20244353	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#10842
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #15688989	C₄	semidirect
12. C₂×C₄×C₈×C₄₇	C₄	semidirect
110. C₂×C₄×C₈	Dic₄₇	semidirect
115. C₄₇	#10842	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688989	2
C₂×#15688886	2
C₂²×C₄×C₈×C₄₇	2
#10842	47