Subgroups — #26131641

Order 12032.

326 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688943	C₂	1	1	yes	maximal, characteristic
2	#20244987	C₂	1	1	yes	maximal, characteristic
3	#20244985	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688898	C₂	1	1	yes	maximal, characteristic
5	#20243268	C₂	1	1	yes	maximal, characteristic
6	#20241476	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1379	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093964	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#13093959	C₄	1	1	yes	characteristic
11	#15689159	C₄	1	1	yes	characteristic
12	#15689078	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15688943	C₄	1	4	yes	complemented
15	#15688899	—	2	1	no	—
16	#15688898	—	2	1	no	—
17	#15688898	C₄	1	4	yes	complemented
18	C₄×C₄₇⋊C₁₆	—	2	1	no	—
19	C₁₆×Dic₄₇	—	2	1	no	—
20	#15688884	C₄	1	1	yes	characteristic
21	#15688882	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#405	C₂²	1	1	yes	characteristic
25	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	#13093983	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	#13093964	—	2	1	no	—
30	#13093964	C₂×C₄	1	2	yes	—
31	C₂×C₄₇⋊C₁₆	—	2	1	no	—
32	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
34	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
35	C₂×C₄₇⋊C₁₆	Q₈	1	2	yes	—
36	#13093959	C₈	1	4	yes	—
37	#13093957	C₈	1	2	yes	—
38	#13093955	C₈	1	2	yes	—
39	#13093934	—	2	1	no	—
40	C₈×Dic₄₇	—	2	1	no	—
41	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
42	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₄₇	—	2	1	no	—
47	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₁₆×C₄₇	—	2	1	no	—
49	C₂×C₁₆×C₄₇	—	2	1	no	—
50	C₂×C₁₆×C₄₇	—	2	1	no	—
51	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
52	#97×C₄₇	—	2	1	no	—
53	C₄×C₈×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	1	no	—
56	M₁₆×C₄₇	—	2	1	no	—
57	M₁₆×C₄₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
64	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
66	C₂²⋊C₄×C₄₇	—	2	1	no	—
67	C₄²×C₄₇	—	2	1	no	—
68	C₂²⋊Dic₄₇	—	2	1	no	—
69	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
70	C₄×Dic₄₇	—	2	1	no	—
71	C₄×Dic₄₇	—	2	1	no	—
72	C₄×Dic₄₇	C₄²	1	2	yes	—
73	#10625329	—	2	1	no	—
74	#10625329	C₂×C₈	1	2	yes	—
75	C₂×C₄₇⋊C₈	—	2	1	no	—
76	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
77	C₂×C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
80	C₁₆×C₄₇	—	2	2	no	—
81	C₁₆×C₄₇	—	2	2	no	—
82	C₄₇⋊C₁₆	—	2	2	no	—
83	C₄₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
84	C₄₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₇	#131	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
93	C₂×Dic₄₇	—	2	2	no	—
94	C₂×Dic₄₇	—	2	2	no	—
95	C₂×Dic₄₇	—	4	2	no	—
96	C₈×C₄₇	—	2	2	no	—
97	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₈×C₄₇	#117	1	1	yes	characteristic
99	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
100	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
101	C₄₇⋊C₈	—	2	2	no	—
102	C₄₇⋊C₈	C₄×C₈	1	2	yes	—
103	C₄₇⋊C₈	C₄×C₈	1	2	yes	—
104	#12665	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	#405	1	1	yes	characteristic
107	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
108	C₂²×C₄₇	C₈×Q₈	1	1	yes	characteristic
109	C₄×C₄₇	—	4	2	no	—
110	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
112	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
113	C₄×C₄₇	C₈×Q₈	1	1	yes	characteristic
114	Dic₄₇	—	4	2	no	—
115	Dic₄₇	—	4	2	no	—
116	C₂×#364	—	47	1	no	—
117	#2108	—	47	1	no	—
118	#2108	—	47	1	no	—
119	C₂×#346	—	47	1	no	—
120	#1772	—	47	1	no	—
121	#1379	—	47	1	no	—
122	#1379	D₄₇	1	1	yes	characteristic
123	C₂×C₄₇	—	2	1	no	—
124	C₂×C₄₇	—	2	1	no	—
125	C₂×C₄₇	C₈×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄₇	#2171	1	2	yes	—
127	C₂×M₃₂	—	47	1	no	—
128	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
129	C₂²×C₁₆	—	47	1	no	—
130	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
131	C₂×#108	—	47	1	no	—
132	#427	—	47	1	no	—
133	#405	—	47	1	no	—
134	#405	C₂×D₄₇	1	1	yes	characteristic
135	C₂×C₄×C₈	—	47	1	no	—
136	#364	—	47	4	no	—
137	#346	—	47	4	no	—
138	#346	—	94	1	no	—
139	#346	—	94	1	no	—
140	C₄×C₁₆	—	94	1	no	—
141	C₄×C₁₆	—	94	1	no	—
142	#339	—	47	1	no	—
143	#339	—	47	1	no	—
144	C₄₇	#12665	1	1	yes	characteristic, complemented, Sylow
145	C₂×M₁₆	—	47	1	no	—
146	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
147	C₂²×C₈	—	47	1	no	—
148	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
149	#117	—	47	1	no	—
150	#117	C₄×D₄₇	1	1	yes	characteristic
151	C₂×C₄²	—	47	1	no	—
152	M₃₂	—	2	1	no	—
153	M₃₂	—	47	2	no	—
154	M₃₂	—	94	1	no	—
155	M₃₂	C₄×D₄₇	1	2	yes	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	47	2	no	—
160	C₂×C₁₆	—	47	1	no	—
161	C₂×C₁₆	—	47	1	no	—
162	C₂×C₁₆	—	47	2	no	—
163	C₂×C₁₆	—	94	1	no	—
164	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
165	#108	—	47	4	no	—
166	#107	—	47	2	no	—
167	#106	—	47	2	no	—
168	#97	—	2	1	no	—
169	#97	—	94	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	47	2	no	—
172	C₄×C₈	—	47	2	no	—
173	C₄×C₈	—	94	1	no	—
174	C₂²×C₄	—	94	1	no	—
175	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
176	M₁₆	—	2	1	no	—
177	M₁₆	—	47	2	no	—
178	M₁₆	—	94	1	no	—
179	M₁₆	C₈×D₄₇	1	2	yes	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	47	2	no	—
182	C₂×C₈	—	47	1	no	—
183	C₂×C₈	—	47	1	no	—
184	C₂×C₈	—	47	2	no	—
185	C₂×C₈	—	94	1	no	—
186	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
187	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
188	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
189	C₂×C₈	#10625352	1	1	yes	characteristic
190	C₂×C₈	#10625352	1	1	yes	characteristic
191	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
192	C₄⋊C₄	—	47	2	no	—
193	C₄⋊C₄	C₈×D₄₇	1	2	yes	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	94	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	47	2	no	—
198	C₄²	—	94	1	no	—
199	C₄²	—	94	1	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	—	47	2	no	—
203	C₁₆	—	47	2	no	—
204	C₁₆	—	94	2	no	—
205	C₂³	#13094032	1	1	yes	characteristic
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	94	2	no	—
208	C₂×C₄	—	94	2	no	—
209	C₂×C₄	—	188	2	no	—
210	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
211	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
212	C₂×C₄	#13093989	1	1	yes	characteristic
213	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
215	C₂×C₄	#13094032	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	47	2	no	—
218	C₈	—	47	2	no	—
219	C₈	—	94	2	no	—
220	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
221	C₈	#13093980	1	1	yes	characteristic
222	C₈	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
223	C₈	#13093989	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	#15689080	1	1	yes	characteristic
226	C₂²	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
227	C₂²	#15689247	1	1	yes	characteristic
228	C₄	—	4	2	no	—
229	C₄	—	188	2	no	—
230	C₄	—	188	2	no	—
231	C₄	#15688974	1	1	yes	characteristic
232	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
233	C₄	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
234	C₄	#15689247	1	1	yes	characteristic
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	#20243249	1	1	yes	characteristic
238	C₂	#20245227	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1379	C₂
socle	C₂²×C₄₇	C₈×D₄
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1379	D₄₇
hypocenter	C₄₇	#12665
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
14. #15688943	C₄	semidirect
17. #15688898	C₄	semidirect
144. C₄₇	#12665	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688943	2
#20244987	2
#20244985	2
C₂×#15688898	2
#20243268	2
#20241476	2
C₄₇×#1379	2
#12665	47