Subgroups — #26126189

Order 12032.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245187	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245187	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₄₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
7	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₄×C₄₇⋊C₁₆	—	2	1	no	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
11	C₂³×C₈×C₄₇	—	2	1	no	—
12	C₂×C₄×D₄×C₄₇	C₄	1	1	yes	characteristic
13	C₈×D₄×C₄₇	—	2	1	no	—
14	C₈×D₄×C₄₇	—	2	1	no	—
15	C₈×D₄×C₄₇	C₄	1	2	yes	—
16	C₈×D₄×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	—	2	1	no	—
24	C₂²×C₈×C₄₇	—	2	1	no	—
25	C₂²×C₈×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	—	4	4	no	—
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄×D₄×C₄₇	—	2	1	no	—
31	C₄×D₄×C₄₇	—	2	1	no	—
32	C₄×D₄×C₄₇	C₈	1	2	yes	—
33	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₄₇	—	2	1	no	—
38	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂²⋊C₈×C₄₇	—	2	2	no	—
40	C₂²⋊C₈×C₄₇	—	2	2	no	—
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₂⁴×C₄₇	—	2	1	no	—
46	C₂×D₄×C₄₇	—	2	1	no	—
47	C₂×D₄×C₄₇	—	2	2	no	—
48	C₂×D₄×C₄₇	C₁₆	1	4	yes	complemented
49	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	—	4	1	no	—
61	C₂×C₈×C₄₇	—	4	4	no	—
62	C₂×C₈×C₄₇	—	4	2	no	—
63	C₂×C₈×C₄₇	—	4	4	no	—
64	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	1	no	—
69	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
70	C₂²⋊C₄×C₄₇	—	2	2	no	—
71	C₂²⋊C₄×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄₇⋊C₁₆	—	4	2	no	—
77	C₄₇⋊C₁₆	—	4	2	no	—
78	C₂³×C₄₇	—	2	4	no	—
79	C₂³×C₄₇	—	2	2	no	—
80	C₂³×C₄₇	—	4	2	no	—
81	C₂³×C₄₇	—	4	2	no	—
82	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
83	D₄×C₄₇	—	2	4	no	—
84	D₄×C₄₇	—	2	2	no	—
85	D₄×C₄₇	C₂×C₁₆	1	4	yes	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	4	1	no	—
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	—	4	2	no	—
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
98	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
102	C₈×C₄₇	—	2	1	no	—
103	C₈×C₄₇	—	2	1	no	—
104	C₈×C₄₇	—	2	1	no	—
105	C₈×C₄₇	—	4	2	no	—
106	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
107	#11629	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	—	2	1	no	—
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	4	8	no	—
112	C₂²×C₄₇	—	4	2	no	—
113	C₂²×C₄₇	—	4	2	no	—
114	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
115	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	—	4	2	no	—
120	C₄×C₄₇	C₂²⋊C₁₆	1	2	yes	—
121	C₄×C₄₇	#418	1	1	yes	characteristic
122	C₄×C₄₇	#421	1	1	yes	characteristic
123	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
124	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
125	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
126	#2162	—	47	1	no	—
127	#2162	—	47	1	no	—
128	D₄⋊C₁₆	—	47	4	no	—
129	C₂×C₄₇	—	2	2	no	—
130	C₂×C₄₇	—	4	4	no	—
131	C₂×C₄₇	#1478	1	1	yes	characteristic
132	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
133	C₂×C₄₇	#2117	1	1	yes	characteristic
134	C₂³×C₈	—	2	1	no	—
135	C₂×C₄×D₄	Dic₄₇	1	1	yes	characteristic
136	C₈×D₄	—	2	1	no	—
137	C₈×D₄	—	2	1	no	—
138	C₈×D₄	Dic₄₇	1	2	yes	—
139	C₈×D₄	C₂×D₄₇	1	2	yes	—
140	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
141	C₂×C₂²⋊C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
143	C₄⋊C₁₆	—	47	2	no	—
144	C₄⋊C₁₆	—	94	1	no	—
145	C₂²⋊C₁₆	—	94	1	no	—
146	C₂²⋊C₁₆	—	94	1	no	—
147	C₄×C₁₆	—	47	2	no	—
148	C₄×C₁₆	—	94	1	no	—
149	C₄₇	#11629	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	C₄₇⋊C₈	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	4	4	no	—
156	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₄₇⋊C₈	1	2	yes	—
161	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
162	C₂×C₄⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	—	2	1	no	—
164	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
165	C₂×C₁₆	—	94	2	no	—
166	C₂×C₁₆	—	94	2	no	—
167	C₄⋊C₈	—	2	1	no	—
168	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	C₂⁴	—	2	1	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₄₇⋊C₁₆	1	4	yes	complemented
179	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
186	C₂²×C₄	#10625329	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
203	C₄²	#10625329	1	1	yes	characteristic
204	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
205	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
206	C₁₆	—	188	2	no	—
207	C₁₆	—	188	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	2	no	—
212	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
213	D₄	—	2	2	no	—
214	D₄	—	2	4	no	—
215	D₄	C₂×C₄₇⋊C₁₆	1	4	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
225	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
228	C₂×C₄	#13093964	1	2	yes	—
229	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
231	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	#13094050	1	2	yes	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	4	8	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	2	yes	—
244	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	2	yes	—
250	C₄	#15689128	1	1	yes	characteristic
251	C₄	#15689140	1	1	yes	characteristic
252	C₄	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
253	C₄	C₂×#13093964	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	4	no	—
256	C₂	#20241779	1	1	yes	characteristic
257	C₂	C₂×C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
258	C₂	#20245016	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₄₇	C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#11629
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20245187	C₂	semidirect
2. #20245187	C₂	semidirect
3. D₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
10. C₄×C₄₇⋊C₁₆	C₂²	semidirect
48. C₂×D₄×C₄₇	C₁₆	semidirect
149. C₄₇	#11629	semidirect
178. C₂×D₄	C₄₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20245187	2
D₄⋊(C₄₇⋊C₁₆)	2
C₂×C₈×D₄×C₄₇	2
#11629	47