Subgroups — #26225126

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 218 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	#20244137	C₂	1	4	yes	maximal
4	#20243560	C₂	1	1	yes	maximal, characteristic
5	#20243269	C₂	1	4	yes	maximal
6	C₄×C₈×Dic₄₇	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×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
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	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
36	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
37	C₈×Dic₄₇	—	2	4	no	—
38	C₈×Dic₄₇	Q₈	1	8	yes	complemented
39	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
41	#125×C₄₇	C₂×C₄	1	4	yes	—
42	C₂²⋊Q₈×C₄₇	—	2	4	no	—
43	C₄×Q₈×C₄₇	—	2	4	no	—
44	#117×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	#107×C₄₇	C₂×C₄	1	8	yes	—
47	#106×C₄₇	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	#41	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₄₇	#41	1	4	yes	—
59	C₄⋊C₄×C₄₇	—	2	8	no	—
60	C₄⋊C₄×C₄₇	—	2	4	no	—
61	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
62	C₂²⋊C₄×C₄₇	—	2	4	no	—
63	C₄²×C₄₇	—	2	4	no	—
64	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₄₇	C₂×D₄	1	2	yes	—
67	C₄×Dic₄₇	—	2	8	no	—
68	C₄×Dic₄₇	—	2	2	no	—
69	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
70	#10625329	—	2	8	no	—
71	C₂×C₄₇⋊C₈	—	2	2	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
74	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
77	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
78	Q₈×C₄₇	—	4	8	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
87	C₂×Dic₄₇	—	2	8	no	—
88	C₂×Dic₄₇	—	2	4	no	—
89	C₈×C₄₇	C₄×Q₈	1	4	yes	—
90	C₈×C₄₇	C₄×Q₈	1	4	yes	—
91	C₄₇⋊C₈	—	2	2	no	—
92	C₄₇⋊C₈	—	4	4	no	—
93	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
94	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
95	#21732	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂²×C₄₇	#533	1	2	yes	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	—	4	8	no	—
101	C₄×C₄₇	#443	1	4	yes	—
102	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
103	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₄×C₄₇	#532	1	1	yes	characteristic
105	Dic₄₇	—	2	8	no	—
106	#3160	D₄₇	1	1	yes	characteristic
107	#2960	—	47	2	no	—
108	#1994	—	47	2	no	—
109	#1945	—	47	4	no	—
110	#1848	—	47	1	no	—
111	#1772	—	47	4	no	—
112	C₄²×C₈	—	47	1	no	—
113	C₂×C₄₇	—	2	2	no	—
114	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄₇	C₂×#443	1	2	yes	—
116	#533	C₂×D₄₇	1	2	yes	—
117	#500	Dic₄₇	1	2	yes	complemented
118	#499	Dic₄₇	1	4	yes	complemented
119	#498	Dic₄₇	1	2	yes	complemented
120	#446	—	94	4	no	—
121	C₈×Q₈	—	94	4	no	—
122	C₂×#108	—	47	4	no	—
123	#427	C₂×D₄₇	1	4	yes	—
124	#424	—	47	4	no	—
125	#423	—	47	2	no	—
126	#405	—	47	4	no	—
127	C₂×C₄×C₈	—	47	2	no	—
128	C₂×C₄×C₈	—	47	1	no	—
129	C₂×C₄×C₈	—	47	2	no	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	C₄³	—	47	1	no	—
132	#339	—	47	8	no	—
133	C₄₇	#21732	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	47	4	no	—
135	C₂²×C₈	—	47	1	no	—
136	C₂²×C₈	—	47	1	no	—
137	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
138	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
139	#125	C₂×Dic₄₇	1	4	yes	—
140	C₂²⋊Q₈	—	2	4	no	—
141	C₄×Q₈	—	2	4	no	—
142	#117	C₂²×D₄₇	1	4	yes	—
143	C₂×C₄²	—	47	2	no	—
144	C₂×C₄²	—	94	2	no	—
145	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
146	#108	—	94	8	no	—
147	#107	C₂×Dic₄₇	1	8	yes	—
148	#106	C₂×Dic₄₇	1	8	yes	—
149	C₄⋊C₈	—	94	8	no	—
150	C₄⋊C₈	—	94	4	no	—
151	#97	—	94	4	no	—
152	C₄×C₈	—	47	8	no	—
153	C₄×C₈	—	47	2	no	—
154	C₄×C₈	—	47	2	no	—
155	C₄×C₈	—	94	4	no	—
156	C₄×C₈	—	94	4	no	—
157	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
158	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
159	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
160	C₂×Q₈	—	2	4	no	—
161	C₂²×C₄	—	94	2	no	—
162	C₂²×C₄	#10625352	1	2	yes	—
163	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
164	M₁₆	—	94	8	no	—
165	C₂×C₈	—	47	1	no	—
166	C₂×C₈	—	47	1	no	—
167	C₂×C₈	—	47	4	no	—
168	C₂×C₈	—	47	2	no	—
169	C₂×C₈	—	94	2	no	—
170	C₂×C₈	—	94	4	no	—
171	C₂×C₈	Q₈×D₄₇	1	4	yes	—
172	C₂×C₈	#10625354	1	4	yes	—
173	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
174	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
175	C₄⋊C₄	—	2	8	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	47	4	no	—
181	C₄²	—	94	8	no	—
182	C₄²	—	94	2	no	—
183	C₄²	D₄×D₄₇	1	2	yes	—
184	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
185	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
186	C₂³	C₂×#10625352	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	94	8	no	—
191	C₂×C₄	—	94	4	no	—
192	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
193	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
194	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
197	C₂×C₄	C₂×#10625354	1	2	yes	—
198	C₈	—	47	2	no	—
199	C₈	—	47	2	no	—
200	C₈	—	94	2	no	—
201	C₈	—	188	4	no	—
202	C₈	Q₈×Dic₄₇	1	4	yes	—
203	C₈	Q₈×Dic₄₇	1	4	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
206	C₂²	#15689678	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	4	8	no	—
209	C₄	—	94	8	no	—
210	C₄	#15689240	1	4	yes	—
211	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
212	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
213	C₄	#15689674	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
216	C₂	C₂×#15689240	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	#21732
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
38. C₈×Dic₄₇	Q₈	semidirect
117. #500	Dic₄₇	semidirect
118. #499	Dic₄₇	semidirect
119. #498	Dic₄₇	semidirect
133. C₄₇	#21732	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
#20244137	2
#20243560	2
#20243269	2
C₄×C₈×Dic₄₇	2
C₄₇×#3160	2
#21732	47