Subgroups — #26216690

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689160	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245793	C₂	1	2	yes	maximal
4	#20243551	C₂	1	4	yes	maximal, complemented
5	#20243548	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243544	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
9	C₂×#13093982	C₂²	1	4	yes	—
10	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
11	#15689160	—	2	16	no	—
12	C₂×#13093958	C₂²	1	4	yes	—
13	C₂×#13093955	C₂²	1	8	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
17	#15688993	C₂²	1	2	yes	—
18	#15688988	C₂²	1	4	yes	—
19	#15688980	—	2	4	no	—
20	C₄⋊C₄×Dic₄₇	—	2	4	no	—
21	C₂×#107×C₄₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	2	yes	—
24	C₂²×C₄×D₄₇	D₄	1	2	yes	—
25	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
27	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
28	C₂×C₈×D₄₇	D₄	1	8	yes	—
29	C₂×C₈×D₄₇	Q₈	1	8	yes	—
30	#13093982	—	2	16	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
37	C₄²×D₄₇	C₂×C₄	1	4	yes	—
38	C₄²×D₄₇	Q₈	1	4	yes	—
39	#13093958	—	2	8	no	—
40	#13093955	—	2	16	no	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
43	C₄₇⋊#95	—	2	8	no	—
44	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	#107×C₄₇	—	2	8	no	—
49	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	4	no	—
55	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
56	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×D₄₇	—	2	4	no	—
58	C₂×C₄×D₄₇	—	2	4	no	—
59	C₂×C₄×D₄₇	C₄⋊C₄	1	8	yes	—
60	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
62	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
63	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
67	C₄⋊C₄×C₄₇	—	2	8	no	—
68	C₄⋊C₄×C₄₇	—	4	8	no	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
71	(C₂×D₄₇)⋊C₄	—	4	8	no	—
72	C₄⋊Dic₄₇	—	2	8	no	—
73	C₄⋊Dic₄₇	—	4	8	no	—
74	C₄×Dic₄₇	—	4	8	no	—
75	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
76	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
77	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
78	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
80	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
81	C₈×D₄₇	C₄⋊C₄	1	32	yes	complemented
82	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂²×D₄₇	—	2	2	no	—
84	C₂²×D₄₇	—	2	2	no	—
85	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×D₄₇	#127	1	2	yes	—
87	C₂²×D₄₇	C₄⋊Q₈	1	2	yes	—
88	C₂×C₄×C₄₇	—	4	4	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₄₇	#135	1	4	yes	—
92	C₂×C₄×C₄₇	#135	1	4	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	4	8	no	—
101	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×Dic₄₇	#127	1	2	yes	—
104	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
105	C₄×D₄₇	—	2	16	no	—
106	C₄×D₄₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₄×D₄₇	C₄⋊Q₈	1	8	yes	—
108	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
109	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
110	#20797	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#135	1	2	yes	—
115	C₂²×C₄₇	C₂×#135	1	2	yes	—
116	C₂×D₄₇	—	2	2	no	—
117	C₂×D₄₇	—	2	2	no	—
118	C₂×D₄₇	—	2	4	no	—
119	C₂×D₄₇	—	2	4	no	—
120	C₂×D₄₇	#384	1	4	yes	—
121	C₄×C₄₇	—	8	4	no	—
122	C₄×C₄₇	#427	1	8	yes	complemented
123	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
124	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
125	Dic₄₇	—	2	4	no	—
126	Dic₄₇	—	8	4	no	—
127	Dic₄₇	#384	1	4	yes	—
128	C₂×#427	—	47	4	no	—
129	C₂²×C₄×C₈	—	47	1	no	—
130	#2285	—	47	2	no	—
131	#1846	—	47	4	no	—
132	#1845	—	47	1	no	—
133	#1845	D₄₇	1	1	yes	characteristic, complemented
134	#1844	—	47	2	no	—
135	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
136	C₂×C₄₇	C₂×#427	1	2	yes	—
137	C₂×C₄₇	#3147	1	1	yes	characteristic
138	C₂×C₄₇	#3147	1	1	yes	characteristic
139	C₂×C₄₇	#3160	1	2	yes	—
140	D₄₇	—	2	4	no	—
141	C₂³×C₈	—	47	2	no	—
142	C₂×#117	—	47	4	no	—
143	C₂²×C₄²	—	47	1	no	—
144	#427	—	94	16	no	—
145	C₂×#107	—	47	4	no	—
146	C₂×#107	C₂×D₄₇	1	4	yes	—
147	C₂×#106	—	47	8	no	—
148	C₂×C₄×C₈	—	47	2	no	—
149	C₂×C₄×C₈	—	47	1	no	—
150	C₂×C₄×C₈	—	47	8	no	—
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	#384	—	47	2	no	—
153	#384	C₂×D₄₇	1	2	yes	—
154	#382	—	47	4	no	—
155	#380	—	94	4	no	—
156	C₄×C₄⋊C₄	—	94	4	no	—
157	C₄₇	#20797	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	47	1	no	—
159	C₂³×C₄	—	47	2	no	—
160	C₂²×C₈	—	47	2	no	—
161	C₂²×C₈	—	47	8	no	—
162	C₂²×C₈	—	47	8	no	—
163	C₂²×C₈	—	47	8	no	—
164	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
165	#117	—	94	16	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	47	4	no	—
168	C₂×C₄⋊C₄	—	94	4	no	—
169	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
170	C₂×C₂²⋊C₄	—	94	4	no	—
171	C₂×C₄²	—	47	2	no	—
172	C₂×C₄²	—	47	1	no	—
173	C₂×C₄²	—	47	4	no	—
174	C₂×C₄²	—	47	4	no	—
175	C₂×C₄²	—	94	4	no	—
176	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
177	#107	—	2	8	no	—
178	#107	—	94	8	no	—
179	#106	—	94	16	no	—
180	C₄×C₈	—	47	8	no	—
181	C₄×C₈	—	47	4	no	—
182	C₄×C₈	C₄×D₄₇	1	4	yes	—
183	#95	—	94	8	no	—
184	C₂⁴	—	47	1	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	47	1	no	—
187	C₂²×C₄	—	47	2	no	—
188	C₂²×C₄	—	47	4	no	—
189	C₂²×C₄	—	47	2	no	—
190	C₂²×C₄	—	47	4	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	—	47	8	no	—
193	C₂²×C₄	—	94	4	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	D₄×D₄₇	1	2	yes	—
197	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
198	C₂×C₈	—	47	4	no	—
199	C₂×C₈	—	47	4	no	—
200	C₂×C₈	—	47	4	no	—
201	C₂×C₈	—	47	32	no	—
202	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
203	C₂×C₈	D₄×D₄₇	1	4	yes	—
204	C₂×C₈	Q₈×D₄₇	1	4	yes	—
205	C₄⋊C₄	—	2	8	no	—
206	C₄⋊C₄	—	4	8	no	—
207	C₄⋊C₄	—	94	8	no	—
208	C₄⋊C₄	—	188	8	no	—
209	C₂²⋊C₄	—	188	8	no	—
210	C₄²	—	47	8	no	—
211	C₄²	—	47	2	no	—
212	C₄²	—	47	2	no	—
213	C₄²	—	188	8	no	—
214	C₄²	C₂×C₄×D₄₇	1	2	yes	—
215	C₄²	Q₈×D₄₇	1	2	yes	—
216	C₂³	—	47	2	no	—
217	C₂³	—	47	2	no	—
218	C₂³	—	47	2	no	—
219	C₂³	—	94	2	no	—
220	C₂³	—	94	2	no	—
221	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	47	2	no	—
225	C₂×C₄	—	47	2	no	—
226	C₂×C₄	—	47	2	no	—
227	C₂×C₄	—	47	8	no	—
228	C₂×C₄	—	47	8	no	—
229	C₂×C₄	—	47	4	no	—
230	C₂×C₄	—	94	2	no	—
231	C₂×C₄	—	94	2	no	—
232	C₂×C₄	—	94	16	no	—
233	C₂×C₄	—	188	8	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
236	C₂×C₄	#13094043	1	4	yes	—
237	C₂×C₄	#13094047	1	4	yes	—
238	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
240	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
241	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
242	C₈	—	47	8	no	—
243	C₈	C₄⋊C₄×D₄₇	1	8	yes	—
244	C₂²	—	47	4	no	—
245	C₂²	—	94	4	no	—
246	C₂²	—	94	2	no	—
247	C₂²	—	94	2	no	—
248	C₂²	—	94	4	no	—
249	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
250	C₂²	#127×D₄₇	1	1	yes	characteristic
251	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
252	C₂²	C₂×#13094043	1	2	yes	—
253	C₂²	C₂×#13094047	1	2	yes	—
254	C₄	—	8	4	no	—
255	C₄	—	47	4	no	—
256	C₄	—	94	4	no	—
257	C₄	—	376	4	no	—
258	C₄	#15689160	1	8	yes	complemented
259	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
260	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
261	C₂	—	94	4	no	—
262	C₂	#384×D₄₇	1	1	yes	characteristic
263	C₂	C₂×#15689160	1	2	yes	—
264	C₂	#20253815	1	1	yes	characteristic
265	C₂	#20253817	1	1	yes	characteristic
266	C₂	#20253899	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1845	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#20797
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
4. #20243551	C₂	semidirect
5. #20243548	C₂	semidirect
6. #20243544	C₂	semidirect
7. C₄₇×#1845	C₂	semidirect
16. C₄×C₈×D₄₇	C₄	semidirect
81. C₈×D₄₇	C₄⋊C₄	semidirect
122. C₄×C₄₇	#427	semidirect
133. #1845	D₄₇	semidirect
157. C₄₇	#20797	semidirect
258. C₄	#15689160	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689160	2
C₂×C₄×C₈×D₄₇	2
#20245793	2
#20243551	2
#20243548	2
#20243544	2
C₄₇×#1845	2
#20797	47