Subgroups — #26185752

Order 12032.

678 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	#20252469	C₂	1	2	yes	maximal
2	#20242207	C₂	1	4	yes	maximal, complemented
3	Q₁₆×C₄₇⋊C₈	C₂	1	4	yes	maximal, complemented
4	#20242136	C₂	1	1	yes	maximal, characteristic
5	#20241783	C₂	1	2	yes	maximal
6	#20241671	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₄₇⋊C₈	—	2	4	no	—
9	Q₈×C₄₇⋊C₈	C₂²	1	8	yes	—
10	#15689176	—	2	4	no	—
11	#15689176	C₂²	1	2	yes	—
12	#15689176	C₂²	1	1	yes	characteristic
13	#15689176	C₂²	1	1	yes	characteristic
14	#15688876	C₂²	1	2	yes	—
15	#15688874	C₂²	1	2	yes	—
16	Q₈⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
17	#15688844	C₂²	1	2	yes	—
18	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₄₇	C₄	1	2	yes	—
22	C₄×Q₁₆×C₄₇	C₄	1	2	yes	—
23	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
25	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
26	C₂×#103×C₄₇	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
30	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
31	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
32	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
34	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
35	C₄×C₄₇⋊C₈	—	2	2	no	—
36	C₄×C₄₇⋊C₈	—	2	8	no	—
37	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×Q₁₆×C₄₇	C₈	1	8	yes	complemented
41	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₄₇	—	2	2	no	—
46	C₄×Q₈×C₄₇	—	2	2	no	—
47	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	#107×C₄₇	C₂×C₄	1	2	yes	—
54	#107×C₄₇	C₂×C₄	1	2	yes	—
55	#103×C₄₇	C₂×C₄	1	4	yes	—
56	#103×C₄₇	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	—	2	4	no	—
63	C₂×Q₈×C₄₇	C₂×C₈	1	8	yes	—
64	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
70	Q₁₆×C₄₇	C₂×C₈	1	8	yes	—
71	Q₁₆×C₄₇	C₂×C₈	1	8	yes	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	#41	1	1	yes	characteristic
87	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
91	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₄₇	—	2	4	no	—
93	Q₈×C₄₇	—	2	8	no	—
94	Q₈×C₄₇	C₂²×C₈	1	8	yes	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	4	1	no	—
100	C₂×C₄×C₄₇	—	8	2	no	—
101	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₄₇	#131	1	2	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₈×C₄₇	—	2	1	no	—
110	C₈×C₄₇	—	2	1	no	—
111	C₈×C₄₇	C₂²×C₈	1	2	yes	—
112	C₈×C₄₇	C₂²×C₈	1	2	yes	—
113	C₄₇⋊C₈	—	2	4	no	—
114	C₄₇⋊C₈	—	4	2	no	—
115	C₄₇⋊C₈	—	4	4	no	—
116	#17856	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	1	no	—
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	C₈×D₄	1	2	yes	—
120	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₄₇	—	2	8	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	4	4	no	—
126	C₄×C₄₇	C₈×D₄	1	2	yes	—
127	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
129	C₄×C₄₇	#578	1	2	yes	—
130	#2982	—	47	2	no	—
131	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
132	#1579	—	47	4	no	—
133	C₈×Q₁₆	—	47	4	no	—
134	#1564	—	47	1	no	—
135	#1479	—	47	2	no	—
136	#1450	—	47	1	no	—
137	C₂×C₄₇	—	2	2	no	—
138	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₄₇	#2950	1	1	yes	characteristic
140	C₂×C₄₇	#2957	1	1	yes	characteristic
141	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic
142	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
143	C₈×Q₈	—	47	8	no	—
144	C₈×Q₈	—	94	4	no	—
145	C₄×Q₁₆	Dic₄₇	1	2	yes	—
146	C₄×Q₁₆	Dic₄₇	1	2	yes	—
147	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
148	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
149	#431	—	47	1	no	—
150	#431	—	47	1	no	—
151	#431	—	47	2	no	—
152	#431	—	94	4	no	—
153	C₂×#107	Dic₄₇	1	1	yes	characteristic
154	C₂×#103	Dic₄₇	1	2	yes	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	#335	—	47	2	no	—
157	#334	—	47	2	no	—
158	Q₈⋊C₈	—	47	8	no	—
159	#322	—	47	2	no	—
160	C₈²	—	47	2	no	—
161	C₄₇	#17856	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
163	C₂×Q₁₆	C₄₇⋊C₈	1	8	yes	complemented
164	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
165	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
166	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
171	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	#107	C₂×Dic₄₇	1	2	yes	—
177	#107	C₂×Dic₄₇	1	2	yes	—
178	C₄⋊C₈	—	47	2	no	—
179	C₄⋊C₈	—	47	4	no	—
180	C₄⋊C₈	—	94	8	no	—
181	C₄⋊C₈	—	94	1	no	—
182	#103	C₂×Dic₄₇	1	4	yes	—
183	#103	C₂×Dic₄₇	1	4	yes	—
184	C₂²⋊C₈	—	47	2	no	—
185	C₂²⋊C₈	—	94	2	no	—
186	C₂²⋊C₈	—	94	1	no	—
187	C₄×C₈	—	47	2	no	—
188	C₄×C₈	—	47	2	no	—
189	C₄×C₈	—	94	8	no	—
190	C₄×C₈	—	94	2	no	—
191	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
192	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₂×C₄₇⋊C₈	1	8	yes	—
198	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	#10625352	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
204	Q₁₆	C₂×C₄₇⋊C₈	1	8	yes	—
205	Q₁₆	C₂×C₄₇⋊C₈	1	8	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	47	4	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
211	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	D₄×D₄₇	1	1	yes	characteristic
222	C₄²	#10625352	1	1	yes	characteristic
223	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
224	C₄²	C₂³×D₄₇	1	1	yes	characteristic
225	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	2	8	no	—
228	Q₈	C₂²×C₄₇⋊C₈	1	8	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
239	C₂×C₄	#13094031	1	2	yes	—
240	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	94	4	no	—
246	C₈	—	188	2	no	—
247	C₈	—	188	4	no	—
248	C₈	C₂²×C₄₇⋊C₈	1	2	yes	—
249	C₈	C₂²×C₄₇⋊C₈	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
253	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	8	no	—
258	C₄	—	4	4	no	—
259	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
260	C₄	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
261	C₄	C₂×#13094031	1	1	yes	characteristic
262	C₄	#15689936	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×D₄×C₄₇⋊C₈	1	1	yes	characteristic
265	C₂	#20252256	1	1	yes	characteristic
266	C₂	#20252305	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#17856
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20242207	C₂	semidirect
3. Q₁₆×C₄₇⋊C₈	C₂	semidirect
40. C₂×Q₁₆×C₄₇	C₈	semidirect
161. C₄₇	#17856	semidirect
163. C₂×Q₁₆	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20252469	2
#20242207	2
Q₁₆×C₄₇⋊C₈	2
#20242136	2
#20241783	2
#20241671	2
C₂×C₄×Q₁₆×C₄₇	2
#17856	47