Subgroups — #26225136

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253902	C₂	1	1	yes	maximal, characteristic
2	#20252318	C₂	1	2	yes	maximal
3	#20244364	C₂	1	2	yes	maximal
4	#20244139	C₂	1	4	yes	maximal
5	#20243275	C₂	1	4	yes	maximal
6	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
8	#15689680	C₂²	1	2	yes	—
9	#15689528	C₄	1	2	yes	complemented
10	#15689525	C₄	1	4	yes	complemented
11	#15689521	C₄	1	2	yes	complemented
12	#15689250	—	2	4	no	—
13	#15689247	—	2	4	no	—
14	#15689161	C₂²	1	4	yes	—
15	#15689151	C₂²	1	4	yes	—
16	#15689081	C₂²	1	4	yes	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688884	C₂²	1	8	yes	—
22	C₂×#108×C₄₇	C₂²	1	4	yes	—
23	C₄₇×#423	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
27	#13094024	C₂×C₄	1	4	yes	—
28	#13094016	C₂×C₄	1	4	yes	—
29	#13094003	—	2	4	no	—
30	C₄×Dic₉₄	—	2	4	no	—
31	#13093983	C₂³	1	4	yes	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
34	#13093958	C₂×C₄	1	8	yes	—
35	#13093956	C₂×C₄	1	8	yes	—
36	Dic₄₇⋊C₈	—	2	8	no	—
37	#13093934	—	2	4	no	—
38	#13093934	—	2	4	no	—
39	#13093934	Q₈	1	8	yes	complemented
40	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
41	C₈×Dic₄₇	—	2	4	no	—
42	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#108×C₄₇	—	2	8	no	—
46	C₄⋊C₈×C₄₇	—	2	4	no	—
47	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	#41	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₄₇	—	2	2	no	—
53	C₂×Dic₉₄	—	2	4	no	—
54	M₁₆×C₄₇	—	2	8	no	—
55	C₂×C₈×C₄₇	—	2	4	no	—
56	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₄₇	#41	1	4	yes	—
60	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²⋊Dic₄₇	—	2	4	no	—
64	C₄⋊Dic₄₇	—	2	4	no	—
65	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
66	Dic₄₇⋊C₄	—	2	8	no	—
67	C₄×Dic₄₇	—	2	4	no	—
68	C₄×Dic₄₇	—	2	2	no	—
69	C₄×Dic₄₇	—	4	4	no	—
70	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
71	C₂×C₄₇⋊C₈	—	2	2	no	—
72	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
73	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
74	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
75	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
83	C₂×Dic₄₇	—	2	8	no	—
84	C₂×Dic₄₇	—	2	4	no	—
85	C₂×Dic₄₇	—	4	4	no	—
86	Dic₉₄	—	4	8	no	—
87	C₈×C₄₇	—	4	4	no	—
88	C₈×C₄₇	C₄×Q₈	1	4	yes	—
89	C₈×C₄₇	C₄×Q₈	1	4	yes	—
90	C₄₇⋊C₈	—	2	2	no	—
91	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
92	#21734	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂²×C₄₇	#533	1	2	yes	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	#444	1	4	yes	—
98	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
99	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₄×C₄₇	#532	1	1	yes	characteristic
101	Dic₄₇	—	4	4	no	—
102	Dic₄₇	—	4	8	no	—
103	#3160	—	47	1	no	—
104	#2960	—	47	2	no	—
105	#1994	—	47	2	no	—
106	#1946	—	47	4	no	—
107	#1848	D₄₇	1	1	yes	characteristic
108	#1773	—	47	4	no	—
109	C₄×#97	—	47	1	no	—
110	C₂×C₄₇	—	2	2	no	—
111	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄₇	C₂×#444	1	2	yes	—
113	#533	—	47	2	no	—
114	#500	—	47	2	no	—
115	#499	—	47	4	no	—
116	#498	—	47	2	no	—
117	#446	—	94	4	no	—
118	C₈×Q₈	—	94	4	no	—
119	C₂×#108	C₂×D₄₇	1	4	yes	—
120	#427	—	47	4	no	—
121	#424	—	47	4	no	—
122	#423	C₂×D₄₇	1	2	yes	—
123	#405	—	47	4	no	—
124	C₂×#97	—	47	2	no	—
125	C₂×#97	—	47	2	no	—
126	C₂×C₄×C₈	—	47	1	no	—
127	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
128	C₄³	—	47	1	no	—
129	#339	—	47	8	no	—
130	C₄₇	#21734	1	1	yes	characteristic, complemented, Sylow
131	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
132	C₂²×C₈	—	47	1	no	—
133	C₂²×C₈	—	47	1	no	—
134	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
135	C₄⋊Q₈	—	47	4	no	—
136	#125	—	47	4	no	—
137	C₂²⋊Q₈	—	94	4	no	—
138	C₄×Q₈	—	94	4	no	—
139	#117	—	47	4	no	—
140	C₂×C₄²	—	47	2	no	—
141	C₂×C₄²	—	94	2	no	—
142	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	#108	—	2	8	no	—
144	#107	—	47	8	no	—
145	#106	—	47	8	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	94	8	no	—
148	#97	—	47	8	no	—
149	#97	—	94	4	no	—
150	#97	—	94	4	no	—
151	C₄×C₈	—	47	4	no	—
152	C₄×C₈	—	94	4	no	—
153	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
154	C₄×C₈	C₄×D₄₇	1	2	yes	—
155	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
156	C₂×Q₈	—	94	4	no	—
157	C₂²×C₄	—	94	2	no	—
158	C₂²×C₄	#10625354	1	2	yes	—
159	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	47	2	no	—
163	C₂×C₈	—	47	4	no	—
164	C₂×C₈	—	47	2	no	—
165	C₂×C₈	—	94	2	no	—
166	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
167	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
168	C₂×C₈	#10625352	1	4	yes	—
169	C₂×C₈	Q₈×D₄₇	1	4	yes	—
170	C₄⋊C₄	—	47	8	no	—
171	C₄⋊C₄	—	94	8	no	—
172	C₄⋊C₄	—	94	4	no	—
173	C₂²⋊C₄	—	94	4	no	—
174	C₄²	—	47	4	no	—
175	C₄²	—	94	4	no	—
176	C₄²	—	94	2	no	—
177	C₄²	—	188	4	no	—
178	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
179	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
180	C₄²	D₄×D₄₇	1	2	yes	—
181	C₂³	C₂×#10625354	1	1	yes	characteristic
182	Q₈	—	188	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	94	8	no	—
185	C₂×C₄	—	94	4	no	—
186	C₂×C₄	—	188	4	no	—
187	C₂×C₄	#13093986	1	2	yes	—
188	C₂×C₄	#13093986	1	2	yes	—
189	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
190	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
191	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
192	C₂×C₄	C₂×#10625352	1	2	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	47	4	no	—
195	C₈	—	94	2	no	—
196	C₈	#13093989	1	4	yes	—
197	C₈	#13093989	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×#13093986	1	1	yes	characteristic
200	C₂²	#15689678	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	188	4	no	—
203	C₄	—	188	8	no	—
204	C₄	#15689244	1	4	yes	—
205	C₄	C₂×#13093989	1	2	yes	—
206	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
207	C₄	#15689671	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	#20245955	1	1	yes	characteristic
210	C₂	C₂×#15689244	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#21734
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #15689528	C₄	semidirect
10. #15689525	C₄	semidirect
11. #15689521	C₄	semidirect
39. #13093934	Q₈	semidirect
130. C₄₇	#21734	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
#20252318	2
#20244364	2
#20244139	2
#20243275	2
C₄×#13093934	2
C₄₇×#1848	2
#21734	47