Subgroups — #26109412

Order 12032.

466 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243526	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241813	C₂	1	2	yes	maximal
4	#20241285	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689147	C₂²	1	2	yes	—
8	#15688888	—	2	1	no	—
9	#15688888	—	2	1	no	—
10	#15688876	—	2	1	no	—
11	#15688874	—	2	1	no	—
12	D₄⋊(C₄₇⋊C₈)	—	2	4	no	—
13	C₂³×C₈×C₄₇	—	2	1	no	—
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₄₇	—	2	1	no	—
16	C₈×D₄×C₄₇	—	2	1	no	—
17	C₈×D₄×C₄₇	C₄	1	2	yes	—
18	C₈×D₄×C₄₇	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#10625329	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	—	2	1	no	—
30	C₂²×C₈×C₄₇	—	2	1	no	—
31	C₂²×C₈×C₄₇	—	2	2	no	—
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₈×C₄₇	—	4	4	no	—
34	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₄×D₄×C₄₇	—	2	1	no	—
37	C₄×D₄×C₄₇	—	2	1	no	—
38	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₄₇	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	—	2	1	no	—
44	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₂²⋊C₈×C₄₇	—	2	2	no	—
46	C₂²⋊C₈×C₄₇	—	2	2	no	—
47	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂⁴×C₄₇	—	2	1	no	—
52	C₂×D₄×C₄₇	—	2	4	no	—
53	C₂×D₄×C₄₇	—	2	1	no	—
54	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	2	2	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	4	4	no	—
67	C₂×C₈×C₄₇	—	4	4	no	—
68	C₂×C₈×C₄₇	—	4	2	no	—
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₄₇	—	2	1	no	—
74	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₄₇	—	2	2	no	—
76	C₂²⋊C₄×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	—	2	4	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	4	2	no	—
88	C₂³×C₄₇	—	4	2	no	—
89	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
90	D₄×C₄₇	—	2	4	no	—
91	D₄×C₄₇	—	2	2	no	—
92	D₄×C₄₇	#108	1	4	yes	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	4	1	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#135	1	2	yes	—
107	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₄₇	—	2	1	no	—
109	C₈×C₄₇	—	2	1	no	—
110	C₈×C₄₇	—	2	1	no	—
111	C₈×C₄₇	—	2	1	no	—
112	C₈×C₄₇	—	4	2	no	—
113	C₄₇⋊C₈	—	8	2	no	—
114	C₄₇⋊C₈	—	8	2	no	—
115	#8660	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	1	no	—
117	C₂²×C₄₇	—	2	1	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	4	8	no	—
120	C₂²×C₄₇	—	4	2	no	—
121	C₂²×C₄₇	—	4	2	no	—
122	C₂²×C₄₇	#380	1	1	yes	characteristic
123	C₂²×C₄₇	#478	1	2	yes	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	4	2	no	—
128	C₄×C₄₇	#416	1	1	yes	characteristic
129	C₄×C₄₇	#418	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#108	1	2	yes	—
131	C₄×C₄₇	#463	1	2	yes	—
132	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
133	#1837	—	47	1	no	—
134	#1564	—	47	1	no	—
135	#1486	—	47	2	no	—
136	#1301	—	47	2	no	—
137	C₂×C₄₇	—	2	2	no	—
138	C₂×C₄₇	—	4	4	no	—
139	C₂×C₄₇	#1804	1	1	yes	characteristic
140	C₂×C₄₇	#1810	1	1	yes	characteristic
141	C₂×C₄₇	#1814	1	1	yes	characteristic
142	C₂³×C₈	—	2	1	no	—
143	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
144	C₈×D₄	—	2	1	no	—
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	Dic₄₇	1	2	yes	—
147	C₈×D₄	Dic₄₇	1	2	yes	—
148	#431	—	47	2	no	—
149	#423	—	47	2	no	—
150	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	#341	—	94	1	no	—
154	#341	—	94	1	no	—
155	#335	—	94	1	no	—
156	#334	—	94	1	no	—
157	D₄⋊C₈	—	94	4	no	—
158	C₄₇	#8660	1	1	yes	characteristic, complemented, Sylow
159	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
160	C₂³×C₄	—	2	1	no	—
161	C₂×M₁₆	—	94	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	4	4	no	—
166	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	D₁₈₈	1	2	yes	—
171	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
172	C₂×C₄⋊C₄	Dic₉₄	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	94	2	no	—
177	C₄⋊C₈	—	94	2	no	—
178	C₄⋊C₈	—	94	2	no	—
179	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	94	2	no	—
183	C₄×C₈	—	94	2	no	—
184	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
185	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	C₄⋊Dic₄₇	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
198	C₂²×C₄	#10625350	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
200	M₁₆	—	188	2	no	—
201	M₁₆	—	188	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	188	2	no	—
209	C₂×C₈	—	188	2	no	—
210	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
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	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
219	C₄²	#10625350	1	1	yes	characteristic
220	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	#13094003	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	#13093960	1	4	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
239	C₂×C₄	#13093983	1	1	yes	characteristic
240	C₂×C₄	#13093998	1	1	yes	characteristic
241	C₂×C₄	#13093998	1	1	yes	characteristic
242	C₂×C₄	#13094003	1	1	yes	characteristic
243	C₂×C₄	#13094044	1	2	yes	—
244	C₂×C₄	#13094050	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	376	2	no	—
251	C₈	—	376	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#15688983	1	1	yes	characteristic
259	C₂²	#15689423	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	#15689119	1	1	yes	characteristic
265	C₄	#15689128	1	1	yes	characteristic
266	C₄	C₂×#13093960	1	2	yes	—
267	C₄	#15689338	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	#20243398	1	1	yes	characteristic
271	C₂	#20243426	1	1	yes	characteristic
272	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×D₄×C₄₇	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂²×D₄×C₄₇	Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#8660
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20243526	C₂	semidirect
2. #20242136	C₂	semidirect
4. #20241285	C₂	semidirect
158. C₄₇	#8660	semidirect

Maximal subgroups

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

subgroup	index
#20243526	2
#20242136	2
#20241813	2
#20241285	2
C₂×C₈×D₄×C₄₇	2
#8660	47