Subgroups — #26109407

Order 12032.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	#20241817	C₂	1	2	yes	maximal
4	#20241285	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093960	—	2	1	no	—
8	C₂×#13093960	—	2	1	no	—
9	#15689147	C₂²	1	2	yes	—
10	#15688876	—	2	1	no	—
11	#15688874	—	2	1	no	—
12	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₄₇	—	2	1	no	—
15	C₈×Q₈×C₄₇	—	2	1	no	—
16	C₈×Q₈×C₄₇	C₄	1	2	yes	—
17	C₈×Q₈×C₄₇	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₇	—	2	1	no	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#10625329	—	2	2	no	—
23	#13093960	—	4	2	no	—
24	#13093960	—	4	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₄×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	—	2	1	no	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₄×Q₈×C₄₇	—	2	1	no	—
35	C₄×Q₈×C₄₇	—	2	1	no	—
36	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
37	C₄×Q₈×C₄₇	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	—	2	1	no	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₇	—	2	2	no	—
43	C₄⋊C₈×C₄₇	—	2	2	no	—
44	C₄⋊C₈×C₄₇	—	2	1	no	—
45	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₇	—	2	2	no	—
47	C₄×C₈×C₄₇	—	2	2	no	—
48	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	—	2	1	no	—
54	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
60	C₂×C₈×C₄₇	—	2	1	no	—
61	C₂×C₈×C₄₇	—	2	1	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	4	1	no	—
64	C₂×C₈×C₄₇	—	4	1	no	—
65	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
70	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
71	C₄⋊C₄×C₄₇	—	2	2	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	1	no	—
74	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₇	#41	1	1	yes	characteristic
81	#10625329	—	4	2	no	—
82	#10625329	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	4	2	no	—
85	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
86	Q₈×C₄₇	—	2	4	no	—
87	Q₈×C₄₇	—	2	2	no	—
88	Q₈×C₄₇	#108	1	4	yes	complemented
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	4	1	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
105	C₈×C₄₇	—	2	1	no	—
106	C₈×C₄₇	—	2	1	no	—
107	C₈×C₄₇	—	4	2	no	—
108	C₈×C₄₇	#103	1	2	yes	—
109	C₈×C₄₇	#103	1	2	yes	—
110	C₄₇⋊C₈	—	8	2	no	—
111	C₄₇⋊C₈	—	8	2	no	—
112	#8659	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	—	2	1	no	—
115	C₂²×C₄₇	#380	1	1	yes	characteristic
116	C₂²×C₄₇	#475	1	1	yes	characteristic
117	C₂²×C₄₇	#477	1	1	yes	characteristic
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	—	2	1	no	—
122	C₄×C₄₇	—	4	2	no	—
123	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
124	C₄×C₄₇	#417	1	1	yes	characteristic
125	C₄×C₄₇	C₂×#108	1	2	yes	—
126	C₄×C₄₇	#460	1	1	yes	characteristic
127	C₄×C₄₇	#462	1	1	yes	characteristic
128	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
129	#1848	—	47	1	no	—
130	#1564	—	47	1	no	—
131	#1487	—	47	2	no	—
132	#1301	—	47	2	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	1	no	—
135	C₂×C₄₇	#1803	1	1	yes	characteristic
136	C₂×C₄₇	#1810	1	1	yes	characteristic
137	C₂×C₄₇	#1814	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
139	C₈×Q₈	—	2	1	no	—
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	Dic₄₇	1	2	yes	—
142	C₈×Q₈	Dic₄₇	1	2	yes	—
143	#431	—	47	2	no	—
144	C₂×#108	—	94	1	no	—
145	C₂×#108	—	94	1	no	—
146	#423	—	47	2	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₂×C₄×C₈	—	2	1	no	—
150	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
151	#335	—	94	1	no	—
152	#334	—	94	1	no	—
153	Q₈⋊C₈	—	94	4	no	—
154	C₄₇	#8659	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
156	C₂×M₁₆	—	94	2	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	D₁₈₈	1	2	yes	—
163	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	Dic₉₄	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
168	#108	—	188	2	no	—
169	#108	—	188	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	94	2	no	—
174	C₄⋊C₈	—	94	2	no	—
175	C₄⋊C₈	—	94	2	no	—
176	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
177	C₂²⋊C₈	—	94	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	94	2	no	—
181	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
182	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×Q₈	C₄⋊Dic₄₇	1	2	yes	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
191	C₂²×C₄	#10625350	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	M₁₆	—	188	2	no	—
194	M₁₆	—	188	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	188	2	no	—
201	C₂×C₈	—	188	2	no	—
202	C₂×C₈	#10625336	1	2	yes	—
203	C₂×C₈	#10625337	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
215	C₄²	#10625350	1	1	yes	characteristic
216	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂³	#13094003	1	1	yes	characteristic
219	Q₈	—	2	4	no	—
220	Q₈	—	2	2	no	—
221	Q₈	#13093960	1	4	yes	complemented
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
230	C₂×C₄	#13093983	1	1	yes	characteristic
231	C₂×C₄	#13093998	1	1	yes	characteristic
232	C₂×C₄	#13093998	1	1	yes	characteristic
233	C₂×C₄	#13094003	1	1	yes	characteristic
234	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
235	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
236	C₂×C₄	#13094049	1	1	yes	characteristic
237	C₂×C₄	#13094054	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	376	2	no	—
242	C₈	—	376	2	no	—
243	C₈	#13093948	1	2	yes	—
244	C₈	#13093948	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#15688983	1	1	yes	characteristic
248	C₂²	#15689408	1	1	yes	characteristic
249	C₂²	#15689418	1	1	yes	characteristic
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	C₂×#13093948	1	1	yes	characteristic
256	C₄	#15689124	1	1	yes	characteristic
257	C₄	C₂×#13093960	1	2	yes	—
258	C₄	#15689323	1	1	yes	characteristic
259	C₄	#15689333	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#20243393	1	1	yes	characteristic
263	C₂	#20243426	1	1	yes	characteristic
264	C₂	#20243442	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	#8659
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
88. Q₈×C₄₇	#108	semidirect
154. C₄₇	#8659	semidirect
221. Q₈	#13093960	semidirect

Maximal subgroups

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

subgroup	index
#20243560	2
#20242136	2
#20241817	2
#20241285	2
C₂×C₈×Q₈×C₄₇	2
#8659	47