Subgroups — #26226716

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252023	C₂	1	2	yes	maximal
2	C₂×#15689151	C₂	1	1	yes	maximal, characteristic
3	#20244162	C₂	1	4	yes	maximal, complemented
4	#20244160	C₂	1	4	yes	maximal, complemented
5	#20243430	C₂	1	1	yes	maximal, characteristic
6	#20243424	C₂	1	2	yes	maximal
7	C₄₇×#1811	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#13093983	C₂²	1	1	yes	characteristic
10	#15689181	—	2	8	no	—
11	C₂×#13093960	C₂²	1	2	yes	—
12	C₂×#13093959	C₂²	1	4	yes	—
13	#15689151	C₄	1	8	yes	—
14	#15689146	C₂²	1	4	yes	—
15	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
17	#15689087	C₂²	1	2	yes	—
18	M₁₆×Dic₄₇	C₂²	1	4	yes	—
19	#15688888	C₂²	1	2	yes	—
20	#15688887	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#108×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#407	C₂²	1	2	yes	—
24	C₄₇×#341	C₂²	1	2	yes	—
25	C₂×#10625329	C₂³	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
29	#13093983	C₂×C₄	1	4	yes	—
30	#13093983	D₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
32	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
34	#13093960	—	4	2	no	—
35	#13093959	—	4	4	no	—
36	Dic₄₇⋊C₈	—	2	8	no	—
37	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
38	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
40	#13093934	—	2	4	no	—
41	C₈×Dic₄₇	—	2	4	no	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	—	4	2	no	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	#108×C₄₇	—	4	2	no	—
49	C₂²⋊C₈×C₄₇	—	2	4	no	—
50	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
58	M₁₆×C₄₇	—	2	4	no	—
59	M₁₆×C₄₇	—	2	4	no	—
60	M₁₆×C₄₇	—	4	4	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
67	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
68	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
69	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
70	#10625329	—	2	4	no	—
71	#10625329	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	4	4	no	—
75	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	—	4	2	no	—
78	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
88	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
89	C₂×Dic₄₇	—	4	4	no	—
90	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
91	C₈×C₄₇	—	2	2	no	—
92	C₈×C₄₇	—	2	2	no	—
93	C₈×C₄₇	—	4	4	no	—
94	C₄₇⋊C₈	—	2	4	no	—
95	C₄₇⋊C₈	—	4	4	no	—
96	#21971	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	2	2	no	—
99	C₂²×C₄₇	—	2	4	no	—
100	C₂²×C₄₇	—	4	4	no	—
101	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄₇	#544	1	2	yes	—
103	C₄×C₄₇	—	2	2	no	—
104	C₄×C₄₇	#518	1	1	yes	characteristic
105	C₄×C₄₇	#519	1	1	yes	characteristic
106	C₄×C₄₇	#537	1	2	yes	—
107	Dic₄₇	—	2	8	no	—
108	#2921	—	47	2	no	—
109	C₂×#424	—	47	1	no	—
110	#1950	—	47	4	no	—
111	#1950	—	47	4	no	—
112	#1811	—	47	1	no	—
113	#1811	D₄₇	1	1	yes	characteristic
114	#1810	—	47	2	no	—
115	C₂×C₄₇	—	2	2	no	—
116	C₂×C₄₇	—	4	2	no	—
117	C₂×C₄₇	#2316	1	1	yes	characteristic
118	C₂×C₄₇	#2916	1	2	yes	—
119	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
120	C₂³×C₈	—	47	1	no	—
121	C₂×#117	—	47	1	no	—
122	#433	—	94	8	no	—
123	C₂×#108	—	47	4	no	—
124	C₂×#108	—	47	2	no	—
125	C₂×#108	C₂×D₄₇	1	2	yes	—
126	#424	—	47	8	no	—
127	#423	—	47	4	no	—
128	C₂×C₄⋊C₈	—	47	2	no	—
129	C₂×C₄⋊C₈	—	47	2	no	—
130	#407	—	47	2	no	—
131	#407	C₂×D₄₇	1	2	yes	—
132	C₄×M₁₆	—	47	4	no	—
133	#341	—	47	4	no	—
134	#341	—	47	2	no	—
135	#341	C₂×D₄₇	1	2	yes	—
136	C₄₇	#21971	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₂×M₁₆	—	4	2	no	—
139	C₂×M₁₆	—	47	4	no	—
140	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
141	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
142	C₂²×C₈	—	47	2	no	—
143	C₂²×C₈	—	47	4	no	—
144	C₂²×C₈	—	188	2	no	—
145	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
147	#117	—	47	4	no	—
148	#117	—	47	4	no	—
149	C₂×C₄⋊C₄	—	47	2	no	—
150	C₂×C₂²⋊C₄	—	47	2	no	—
151	C₂×C₄²	—	47	2	no	—
152	#108	—	4	2	no	—
153	#108	—	188	2	no	—
154	#108	—	188	4	no	—
155	C₄⋊C₈	—	47	8	no	—
156	C₄⋊C₈	—	47	8	no	—
157	C₄⋊C₈	—	94	8	no	—
158	C₂²⋊C₈	—	2	4	no	—
159	C₂²⋊C₈	—	94	4	no	—
160	#97	—	94	4	no	—
161	C₄×C₈	—	94	4	no	—
162	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	47	4	no	—
166	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
167	C₂²×C₄	D₄×D₄₇	1	2	yes	—
168	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
169	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
170	M₁₆	—	2	4	no	—
171	M₁₆	—	2	4	no	—
172	M₁₆	—	4	4	no	—
173	M₁₆	—	94	4	no	—
174	M₁₆	—	94	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	47	4	no	—
179	C₂×C₈	—	94	4	no	—
180	C₂×C₈	—	94	4	no	—
181	C₂×C₈	—	188	4	no	—
182	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
183	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
184	C₄⋊C₄	—	47	8	no	—
185	C₂²⋊C₄	—	47	8	no	—
186	C₄²	—	47	4	no	—
187	C₄²	—	47	4	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
191	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	47	8	no	—
197	C₂×C₄	—	188	4	no	—
198	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
199	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
201	C₂×C₄	#13094069	1	2	yes	—
202	C₂×C₄	#13094071	1	2	yes	—
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	94	4	no	—
207	C₈	—	188	4	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
213	C₂²	#15689752	1	2	yes	—
214	C₄	—	2	2	no	—
215	C₄	—	94	8	no	—
216	C₄	#15689596	1	1	yes	characteristic
217	C₄	#15689600	1	1	yes	characteristic
218	C₄	#15689700	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	—	4	2	no	—
221	C₂	#20246005	1	1	yes	characteristic
222	C₂	#20251998	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×#1811	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1811	D₄₇
hypocenter	C₄₇	#21971
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1811

Complements

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

N	H ≅ G/N	type
3. #20244162	C₂	semidirect
4. #20244160	C₂	semidirect
136. C₄₇	#21971	semidirect

Maximal subgroups

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

subgroup	index
#20252023	2
C₂×#15689151	2
#20244162	2
#20244160	2
#20243430	2
#20243424	2
C₄₇×#1811	2
#21971	47