Subgroups — #26216686

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689158	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245793	C₂	1	2	yes	maximal
4	#20243549	C₂	1	4	yes	maximal, complemented
5	#20243547	C₂	1	2	yes	maximal, complemented
6	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1844	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	#15689158	—	2	16	no	—
12	C₂×#13093957	C₂²	1	8	yes	—
13	C₂×#13093956	C₂²	1	4	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×#106×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	#13093957	—	2	16	no	—
40	#13093956	—	2	8	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	#106×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	8	no	—
58	C₂×C₄×D₄₇	C₄⋊C₄	1	8	yes	—
59	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
61	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
62	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
66	C₄⋊C₄×C₄₇	—	2	8	no	—
67	C₄⋊C₄×C₄₇	—	4	8	no	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
70	(C₂×D₄₇)⋊C₄	—	4	8	no	—
71	C₄⋊Dic₄₇	—	2	8	no	—
72	C₄⋊Dic₄₇	—	4	8	no	—
73	C₄×Dic₄₇	—	4	8	no	—
74	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
75	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
76	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
77	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
78	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
79	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
80	C₈×D₄₇	C₄⋊C₄	1	32	yes	complemented
81	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	C₂²×D₄₇	—	2	4	no	—
83	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×D₄₇	#127	1	2	yes	—
85	C₂²×D₄₇	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	4	8	no	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₄₇	#135	1	8	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
94	C₂×Dic₄₇	—	2	4	no	—
95	C₂×Dic₄₇	—	4	4	no	—
96	C₂×Dic₄₇	—	4	8	no	—
97	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×Dic₄₇	#127	1	2	yes	—
100	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
101	C₄×D₄₇	—	2	16	no	—
102	C₄×D₄₇	C₂×C₄⋊C₄	1	8	yes	—
103	C₄×D₄₇	C₄⋊Q₈	1	8	yes	—
104	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
105	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
106	#20797	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×#135	1	4	yes	—
111	C₂×D₄₇	—	2	4	no	—
112	C₂×D₄₇	—	2	4	no	—
113	C₂×D₄₇	—	2	4	no	—
114	C₂×D₄₇	#384	1	4	yes	—
115	C₄×C₄₇	—	8	4	no	—
116	C₄×C₄₇	#427	1	8	yes	complemented
117	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
118	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
119	Dic₄₇	—	2	4	no	—
120	Dic₄₇	—	8	4	no	—
121	Dic₄₇	#384	1	4	yes	—
122	C₂×#427	—	47	4	no	—
123	C₂²×C₄×C₈	—	47	1	no	—
124	#2285	—	47	2	no	—
125	#1846	—	47	4	no	—
126	#1845	—	47	2	no	—
127	#1844	—	47	1	no	—
128	#1844	D₄₇	1	1	yes	characteristic, complemented
129	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
130	C₂×C₄₇	C₂×#427	1	2	yes	—
131	C₂×C₄₇	#3147	1	2	yes	—
132	C₂×C₄₇	#3160	1	2	yes	—
133	D₄₇	—	2	4	no	—
134	C₂³×C₈	—	47	2	no	—
135	C₂×#117	—	47	4	no	—
136	C₂²×C₄²	—	47	1	no	—
137	#427	—	94	16	no	—
138	C₂×#107	—	47	8	no	—
139	C₂×#106	—	47	4	no	—
140	C₂×#106	C₂×D₄₇	1	4	yes	—
141	C₂×C₄×C₈	—	47	2	no	—
142	C₂×C₄×C₈	—	47	1	no	—
143	C₂×C₄×C₈	—	47	8	no	—
144	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
145	#384	—	47	2	no	—
146	#384	C₂×D₄₇	1	2	yes	—
147	#382	—	47	4	no	—
148	#380	—	94	4	no	—
149	C₄×C₄⋊C₄	—	94	4	no	—
150	C₄₇	#20797	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	47	1	no	—
152	C₂³×C₄	—	47	2	no	—
153	C₂²×C₈	—	47	2	no	—
154	C₂²×C₈	—	47	8	no	—
155	C₂²×C₈	—	47	8	no	—
156	C₂²×C₈	—	47	8	no	—
157	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
158	#117	—	94	16	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	47	4	no	—
161	C₂×C₄⋊C₄	—	94	4	no	—
162	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
163	C₂×C₂²⋊C₄	—	94	4	no	—
164	C₂×C₄²	—	47	2	no	—
165	C₂×C₄²	—	47	1	no	—
166	C₂×C₄²	—	47	4	no	—
167	C₂×C₄²	—	47	4	no	—
168	C₂×C₄²	—	94	4	no	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#107	—	94	16	no	—
171	#106	—	2	8	no	—
172	#106	—	94	8	no	—
173	C₄×C₈	—	47	8	no	—
174	C₄×C₈	—	47	4	no	—
175	C₄×C₈	C₄×D₄₇	1	4	yes	—
176	#95	—	94	8	no	—
177	C₂⁴	—	47	1	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	47	1	no	—
180	C₂²×C₄	—	47	2	no	—
181	C₂²×C₄	—	47	4	no	—
182	C₂²×C₄	—	47	2	no	—
183	C₂²×C₄	—	47	4	no	—
184	C₂²×C₄	—	47	2	no	—
185	C₂²×C₄	—	47	8	no	—
186	C₂²×C₄	—	94	8	no	—
187	C₂²×C₄	—	94	4	no	—
188	C₂²×C₄	D₄×D₄₇	1	2	yes	—
189	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
190	C₂×C₈	—	47	4	no	—
191	C₂×C₈	—	47	4	no	—
192	C₂×C₈	—	47	4	no	—
193	C₂×C₈	—	47	32	no	—
194	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
195	C₂×C₈	D₄×D₄₇	1	4	yes	—
196	C₂×C₈	Q₈×D₄₇	1	4	yes	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	4	8	no	—
199	C₄⋊C₄	—	94	8	no	—
200	C₄⋊C₄	—	188	8	no	—
201	C₂²⋊C₄	—	188	8	no	—
202	C₄²	—	47	8	no	—
203	C₄²	—	47	2	no	—
204	C₄²	—	47	2	no	—
205	C₄²	—	188	8	no	—
206	C₄²	C₂×C₄×D₄₇	1	2	yes	—
207	C₄²	Q₈×D₄₇	1	2	yes	—
208	C₂³	—	47	2	no	—
209	C₂³	—	47	2	no	—
210	C₂³	—	47	2	no	—
211	C₂³	—	94	4	no	—
212	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	47	2	no	—
216	C₂×C₄	—	47	2	no	—
217	C₂×C₄	—	47	2	no	—
218	C₂×C₄	—	47	8	no	—
219	C₂×C₄	—	47	8	no	—
220	C₂×C₄	—	47	4	no	—
221	C₂×C₄	—	94	4	no	—
222	C₂×C₄	—	94	16	no	—
223	C₂×C₄	—	188	8	no	—
224	C₂×C₄	—	188	4	no	—
225	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
226	C₂×C₄	#13094045	1	8	yes	—
227	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
229	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
230	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
231	C₈	—	47	8	no	—
232	C₈	C₄⋊C₄×D₄₇	1	8	yes	—
233	C₂²	—	47	4	no	—
234	C₂²	—	94	4	no	—
235	C₂²	—	94	4	no	—
236	C₂²	—	94	4	no	—
237	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
238	C₂²	#127×D₄₇	1	1	yes	characteristic
239	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
240	C₂²	C₂×#13094045	1	4	yes	—
241	C₄	—	8	4	no	—
242	C₄	—	47	4	no	—
243	C₄	—	94	4	no	—
244	C₄	—	376	4	no	—
245	C₄	#15689158	1	8	yes	complemented
246	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
247	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
248	C₂	—	94	4	no	—
249	C₂	#384×D₄₇	1	1	yes	characteristic
250	C₂	C₂×#15689158	1	2	yes	—
251	C₂	#20253811	1	2	yes	—
252	C₂	#20253897	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#20797
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
4. #20243549	C₂	semidirect
5. #20243547	C₂	semidirect
6. #20243545	C₂	semidirect
7. C₄₇×#1844	C₂	semidirect
16. C₄×C₈×D₄₇	C₄	semidirect
80. C₈×D₄₇	C₄⋊C₄	semidirect
116. C₄×C₄₇	#427	semidirect
128. #1844	D₄₇	semidirect
150. C₄₇	#20797	semidirect
245. C₄	#15689158	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689158	2
C₂×C₄×C₈×D₄₇	2
#20245793	2
#20243549	2
#20243547	2
#20243545	2
C₄₇×#1844	2
#20797	47