Subgroups — #26225066

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×Dic₄₇	C₂	1	2	yes	maximal
2	#20244350	C₂	1	2	yes	maximal
3	#20244127	C₂	1	4	yes	maximal
4	#20243545	C₂	1	1	yes	maximal, characteristic
5	#20243287	C₂	1	4	yes	maximal
6	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₄₇	—	2	2	no	—
9	C₂×#13093956	C₂²	1	4	yes	—
10	C₂×#13093934	C₂²	1	2	yes	—
11	C₂×#13093934	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	#15689030	—	2	2	no	—
14	#15689022	—	2	2	no	—
15	#15689022	C₂²	1	4	yes	—
16	#15688993	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₄₇	—	2	2	no	—
18	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
19	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
20	#15688885	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
22	C₄₇×#500	C₄	1	8	yes	complemented
23	C₂×#107×C₄₇	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
26	Q₈×Dic₄₇	—	4	8	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×C₄×Dic₄₇	—	2	4	no	—
32	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
36	#13093956	—	2	8	no	—
37	#13093934	—	2	4	no	—
38	#13093934	Q₈	1	8	yes	complemented
39	C₄×C₄₇⋊C₈	—	2	2	no	—
40	C₄₇⋊#95	—	2	4	no	—
41	C₄₇⋊#95	—	2	4	no	—
42	C₂²×Q₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₄₇	—	2	4	no	—
45	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	#107×C₄₇	C₂×C₄	1	16	yes	—
50	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×Q₈×C₄₇	—	2	8	no	—
52	C₂×Q₈×C₄₇	—	4	4	no	—
53	C₂×Q₈×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
60	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₄₇	#41	1	4	yes	—
64	C₄⋊C₄×C₄₇	—	2	8	no	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	#41	1	2	yes	—
69	C₄⋊Dic₄₇	—	2	8	no	—
70	C₄⋊Dic₄₇	—	4	8	no	—
71	C₄⋊Dic₄₇	—	4	4	no	—
72	C₄⋊Dic₄₇	—	4	4	no	—
73	C₄×Dic₄₇	—	2	4	no	—
74	C₄×Dic₄₇	—	2	4	no	—
75	C₄×Dic₄₇	—	2	2	no	—
76	C₄×Dic₄₇	—	4	8	no	—
77	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	2	2	no	—
80	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
81	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₄₇	—	4	16	no	—
83	C₂×C₄×C₄₇	—	2	8	no	—
84	C₂×C₄×C₄₇	—	4	4	no	—
85	C₂×C₄×C₄₇	—	4	4	no	—
86	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
89	C₂×C₄×C₄₇	#137	1	4	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
94	C₂×Dic₄₇	—	2	2	no	—
95	C₂×Dic₄₇	—	2	2	no	—
96	C₂×Dic₄₇	—	2	2	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	4	4	no	—
101	C₂×Dic₄₇	—	4	4	no	—
102	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
103	C₈×C₄₇	C₄×Q₈	1	8	yes	complemented
104	C₄₇⋊C₈	—	2	4	no	—
105	#21726	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₄₇	#532	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₄₇	C₂×#137	1	2	yes	—
111	C₄×C₄₇	—	4	8	no	—
112	C₄×C₄₇	#441	1	4	yes	—
113	C₄×C₄₇	#442	1	4	yes	—
114	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
115	C₄×C₄₇	#533	1	2	yes	—
116	Dic₄₇	—	2	4	no	—
117	Dic₄₇	—	4	2	no	—
118	Dic₄₇	—	8	4	no	—
119	C₂×#500	D₄₇	1	1	yes	characteristic
120	C₄×C₄⋊Q₈	—	47	2	no	—
121	#1990	—	47	2	no	—
122	#1943	—	47	4	no	—
123	#1844	—	47	1	no	—
124	#1776	—	47	4	no	—
125	C₄×#97	—	47	1	no	—
126	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
128	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
129	C₂×C₄₇	#3165	1	2	yes	—
130	C₂×C₄₇	#3182	1	2	yes	—
131	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
132	C₂×C₄×Q₈	—	94	2	no	—
133	#500	Dic₄₇	1	8	yes	complemented
134	C₂×#107	C₂×D₄₇	1	4	yes	—
135	C₂×#106	—	47	4	no	—
136	C₂×#97	—	47	2	no	—
137	C₂×#97	—	47	2	no	—
138	C₂×C₄×C₈	—	47	1	no	—
139	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
140	#391	—	94	2	no	—
141	#389	—	47	4	no	—
142	#389	—	94	2	no	—
143	#384	—	47	2	no	—
144	C₄×C₄⋊C₄	—	47	4	no	—
145	C₄×C₄⋊C₄	—	94	2	no	—
146	C₄³	—	47	1	no	—
147	#340	—	47	8	no	—
148	C₄₇	#21726	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×C₈	—	47	2	no	—
151	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
154	C₄×Q₈	—	188	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	47	4	no	—
157	C₂×C₄⋊C₄	—	94	4	no	—
158	C₂×C₄⋊C₄	—	94	2	no	—
159	C₂×C₄⋊C₄	—	94	2	no	—
160	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
161	C₂×C₄²	—	47	1	no	—
162	C₂×C₄²	—	47	2	no	—
163	C₂×C₄²	—	47	2	no	—
164	C₂×C₄²	—	47	1	no	—
165	C₂×C₄²	—	94	4	no	—
166	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
167	#107	C₂×Dic₄₇	1	16	yes	—
168	#106	—	94	8	no	—
169	#97	—	47	8	no	—
170	#97	—	94	4	no	—
171	C₄×C₈	—	94	2	no	—
172	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
173	#95	—	94	4	no	—
174	#95	—	94	4	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	47	2	no	—
180	C₂²×C₄	—	47	2	no	—
181	C₂²×C₄	—	94	4	no	—
182	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
184	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
185	C₂×C₈	—	47	4	no	—
186	C₂×C₈	—	94	2	no	—
187	C₂×C₈	—	94	2	no	—
188	C₂×C₈	Q₈×D₄₇	1	4	yes	—
189	C₂×C₈	#10625354	1	4	yes	—
190	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	94	8	no	—
194	C₄⋊C₄	—	188	8	no	—
195	C₄⋊C₄	—	188	4	no	—
196	C₄⋊C₄	—	188	4	no	—
197	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
198	C₄²	—	47	4	no	—
199	C₄²	—	94	4	no	—
200	C₄²	—	94	4	no	—
201	C₄²	—	94	2	no	—
202	C₄²	—	188	8	no	—
203	C₄²	#10625352	1	2	yes	—
204	C₄²	C₂²×Dic₄₇	1	2	yes	—
205	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
206	Q₈	—	4	16	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	47	4	no	—
211	C₂×C₄	—	94	2	no	—
212	C₂×C₄	—	94	2	no	—
213	C₂×C₄	—	94	2	no	—
214	C₂×C₄	—	94	2	no	—
215	C₂×C₄	—	94	2	no	—
216	C₂×C₄	—	188	4	no	—
217	C₂×C₄	—	188	4	no	—
218	C₂×C₄	—	188	4	no	—
219	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
220	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
221	C₂×C₄	#13094048	1	4	yes	—
222	C₂×C₄	#13094056	1	4	yes	—
223	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
224	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
225	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
226	C₂×C₄	C₂×#10625354	1	2	yes	—
227	C₈	—	94	4	no	—
228	C₈	Q₈×Dic₄₇	1	8	yes	complemented
229	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
230	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
231	C₂²	#15689674	1	1	yes	characteristic
232	C₂²	C₂×#13094048	1	2	yes	—
233	C₂²	C₂×#13094056	1	2	yes	—
234	C₄	—	4	8	no	—
235	C₄	—	94	4	no	—
236	C₄	—	376	4	no	—
237	C₄	—	188	2	no	—
238	C₄	#15689230	1	4	yes	—
239	C₄	#15689235	1	4	yes	—
240	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
241	C₄	#15689678	1	2	yes	—
242	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
243	C₂	C₂×#15689230	1	1	yes	characteristic
244	C₂	C₂×#15689235	1	1	yes	characteristic
245	C₂	#20253931	1	2	yes	—
246	C₂	#20254108	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#21726
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
22. C₄₇×#500	C₄	semidirect
38. #13093934	Q₈	semidirect
103. C₈×C₄₇	C₄×Q₈	semidirect
133. #500	Dic₄₇	semidirect
148. C₄₇	#21726	semidirect
228. C₈	Q₈×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×Dic₄₇	2
#20244350	2
#20244127	2
#20243545	2
#20243287	2
C₄×#13093934	2
C₂×C₄₇×#500	2
#21726	47