Subgroups — #26225142

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	#20252323	C₂	1	2	yes	maximal
2	#20244366	C₂	1	2	yes	maximal
3	#20244142	C₂	1	4	yes	maximal
4	#20243560	C₂	1	1	yes	maximal, characteristic
5	#20243274	C₂	1	4	yes	maximal
6	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#3160	C₂	1	1	yes	maximal, characteristic
8	#15689254	—	2	4	no	—
9	Q₈×C₄₇⋊C₈	—	2	4	no	—
10	C₂×#13093960	C₂²	1	4	yes	—
11	#15689150	C₂²	1	4	yes	—
12	#15689147	C₂²	1	2	yes	—
13	#15689079	C₂²	1	4	yes	—
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×#13093934	C₂²	1	2	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
18	#15688882	C₂²	1	8	yes	—
19	C₄₇×#533	C₂²	1	2	yes	—
20	C₄₇×#500	C₄	1	2	yes	complemented
21	C₄₇×#499	C₄	1	4	yes	complemented
22	C₄₇×#498	C₄	1	2	yes	complemented
23	C₄₇×#427	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#10625329	C₂³	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
30	#13093960	—	2	8	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
33	#13093935	—	2	4	no	—
34	#13093934	—	2	4	no	—
35	#13093934	Q₈	1	8	yes	complemented
36	C₄×C₄₇⋊C₈	—	2	4	no	—
37	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
38	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
40	#125×C₄₇	C₂×C₄	1	4	yes	—
41	C₂²⋊Q₈×C₄₇	—	2	4	no	—
42	C₄×Q₈×C₄₇	—	2	4	no	—
43	#117×C₄₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#107×C₄₇	C₂×C₄	1	8	yes	—
46	#106×C₄₇	C₂×C₄	1	8	yes	—
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	C₂×Q₈×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	#41	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	2	no	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₄₇	#41	1	4	yes	—
58	C₄⋊C₄×C₄₇	—	2	8	no	—
59	C₄⋊C₄×C₄₇	—	2	4	no	—
60	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
61	C₂²⋊C₄×C₄₇	—	2	4	no	—
62	C₄²×C₄₇	—	2	4	no	—
63	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	C₄×Dic₄₇	—	2	2	no	—
67	C₄×Dic₄₇	—	4	4	no	—
68	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
69	#10625329	—	2	8	no	—
70	C₂×C₄₇⋊C₈	—	2	2	no	—
71	C₂×C₄₇⋊C₈	—	2	4	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	Q₈×C₄₇	—	4	8	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	8	no	—
79	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
85	C₂×Dic₄₇	—	2	4	no	—
86	C₂×Dic₄₇	—	4	4	no	—
87	C₈×C₄₇	C₄×Q₈	1	4	yes	—
88	C₈×C₄₇	C₄×Q₈	1	4	yes	—
89	C₄₇⋊C₈	—	2	2	no	—
90	C₄₇⋊C₈	—	4	4	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	—	4	8	no	—
98	C₄×C₄₇	#444	1	4	yes	—
99	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
100	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₄₇	#532	1	1	yes	characteristic
102	Dic₄₇	—	4	4	no	—
103	#3160	D₄₇	1	1	yes	characteristic
104	#2960	—	47	2	no	—
105	#1994	—	47	2	no	—
106	#1946	—	47	4	no	—
107	#1848	—	47	1	no	—
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	C₂×D₄₇	1	2	yes	—
114	#500	Dic₄₇	1	2	yes	complemented
115	#499	Dic₄₇	1	4	yes	complemented
116	#498	Dic₄₇	1	2	yes	complemented
117	#446	—	94	4	no	—
118	C₈×Q₈	—	94	4	no	—
119	C₂×#108	—	47	4	no	—
120	#427	C₂×D₄₇	1	4	yes	—
121	#424	—	47	4	no	—
122	#423	—	47	2	no	—
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	—	47	4	no	—
132	C₂²×C₈	—	47	1	no	—
133	C₂²×C₈	—	47	1	no	—
134	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
135	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
136	#125	C₂×Dic₄₇	1	4	yes	—
137	C₂²⋊Q₈	—	2	4	no	—
138	C₄×Q₈	—	2	4	no	—
139	#117	C₂²×D₄₇	1	4	yes	—
140	C₂×C₄²	—	47	2	no	—
141	C₂×C₄²	—	94	2	no	—
142	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	#108	—	94	8	no	—
144	#107	C₂×Dic₄₇	1	8	yes	—
145	#106	C₂×Dic₄₇	1	8	yes	—
146	C₄⋊C₈	—	94	8	no	—
147	C₄⋊C₈	—	94	4	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	C₂×Dic₄₇	1	1	yes	characteristic
154	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
155	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
156	C₂×Q₈	—	2	4	no	—
157	C₂²×C₄	—	94	2	no	—
158	C₂²×C₄	#10625352	1	2	yes	—
159	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
160	M₁₆	—	94	8	no	—
161	C₂×C₈	—	47	2	no	—
162	C₂×C₈	—	47	4	no	—
163	C₂×C₈	—	47	2	no	—
164	C₂×C₈	—	94	2	no	—
165	C₂×C₈	—	94	4	no	—
166	C₂×C₈	Q₈×D₄₇	1	4	yes	—
167	C₂×C₈	#10625354	1	4	yes	—
168	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
169	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
170	C₄⋊C₄	—	2	8	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	47	4	no	—
176	C₄²	—	94	2	no	—
177	C₄²	—	188	4	no	—
178	C₄²	D₄×D₄₇	1	2	yes	—
179	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
180	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
181	C₂³	C₂×#10625352	1	1	yes	characteristic
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	94	4	no	—
186	C₂×C₄	—	188	4	no	—
187	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
188	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
189	C₂×C₄	C₂³×Dic₄₇	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×#10625354	1	2	yes	—
193	C₈	—	47	4	no	—
194	C₈	—	94	2	no	—
195	C₈	—	188	4	no	—
196	C₈	Q₈×Dic₄₇	1	4	yes	—
197	C₈	Q₈×Dic₄₇	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
200	C₂²	#15689678	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	4	8	no	—
203	C₄	—	188	4	no	—
204	C₄	#15689245	1	4	yes	—
205	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
206	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
207	C₄	#15689674	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
210	C₂	C₂×#15689245	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×#3160	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3160	D₄₇
hypocenter	C₄₇	#21734
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
20. C₄₇×#500	C₄	semidirect
21. C₄₇×#499	C₄	semidirect
22. C₄₇×#498	C₄	semidirect
35. #13093934	Q₈	semidirect
114. #500	Dic₄₇	semidirect
115. #499	Dic₄₇	semidirect
116. #498	Dic₄₇	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
#20252323	2
#20244366	2
#20244142	2
#20243560	2
#20243274	2
C₄×#13093934	2
C₄₇×#3160	2
#21734	47