Subgroups — #26115073

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243505	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
7	#15688878	—	2	2	no	—
8	#15688878	—	2	2	no	—
9	D₄⋊(C₄₇⋊C₈)	—	2	8	no	—
10	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₄₇	—	2	2	no	—
12	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#390	—	2	2	no	—
14	C₄₇×#390	C₄	1	2	yes	—
15	C₄₇×#384	C₄	1	1	yes	characteristic
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	D₄	1	4	yes	complemented
22	C₂²×D₄×C₄₇	—	2	2	no	—
23	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₇	—	2	2	no	—
25	#127×C₄₇	—	2	2	no	—
26	#127×C₄₇	C₈	1	4	yes	complemented
27	C₄×D₄×C₄₇	—	2	4	no	—
28	C₄×D₄×C₄₇	—	2	4	no	—
29	C₄×D₄×C₄₇	—	4	8	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂⁴×C₄₇	—	2	2	no	—
40	C₂×D₄×C₄₇	—	2	8	no	—
41	C₂×D₄×C₄₇	—	2	4	no	—
42	C₂×D₄×C₄₇	—	2	2	no	—
43	C₂×D₄×C₄₇	—	2	4	no	—
44	C₂×D₄×C₄₇	—	4	4	no	—
45	C₂×D₄×C₄₇	—	4	4	no	—
46	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	4	8	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄⋊C₄×C₄₇	—	4	4	no	—
58	C₂²⋊C₄×C₄₇	—	4	8	no	—
59	C₂²⋊C₄×C₄₇	—	4	4	no	—
60	C₄²×C₄₇	—	2	2	no	—
61	C₄²×C₄₇	—	2	4	no	—
62	C₄²×C₄₇	—	4	2	no	—
63	C₄²×C₄₇	C₂×C₈	1	2	yes	—
64	C₄²×C₄₇	M₁₆	1	2	yes	—
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	C₄²×C₄₇	#41	1	2	yes	—
67	C₂×C₄₇⋊C₈	—	2	4	no	—
68	C₂×C₄₇⋊C₈	—	4	4	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂×C₄₇⋊C₈	—	4	4	no	—
71	C₂³×C₄₇	—	2	4	no	—
72	C₂³×C₄₇	—	4	2	no	—
73	C₂³×C₄₇	—	4	2	no	—
74	C₂³×C₄₇	—	4	4	no	—
75	C₂³×C₄₇	—	4	4	no	—
76	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	D₄×C₄₇	—	2	8	no	—
78	D₄×C₄₇	—	4	16	no	—
79	D₄×C₄₇	—	4	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	8	4	no	—
89	C₂×C₄×C₄₇	—	8	4	no	—
90	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₄₇	#121	1	2	yes	—
93	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#131	1	2	yes	—
98	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
99	C₄₇⋊C₈	—	4	4	no	—
100	C₄₇⋊C₈	—	8	2	no	—
101	#9642	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	—	4	4	no	—
103	C₂²×C₄₇	—	4	8	no	—
104	C₂²×C₄₇	—	8	2	no	—
105	C₂²×C₄₇	—	8	2	no	—
106	C₂²×C₄₇	—	8	4	no	—
107	C₂²×C₄₇	#390	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₂²×C₄₇	#408	1	1	yes	characteristic
110	C₂²×C₄₇	#418	1	2	yes	—
111	C₂²×C₄₇	#421	1	2	yes	—
112	C₄×C₄₇	—	2	2	no	—
113	C₄×C₄₇	—	2	4	no	—
114	C₄×C₄₇	—	4	2	no	—
115	C₄×C₄₇	—	8	2	no	—
116	C₄×C₄₇	C₈×D₄	1	2	yes	—
117	C₄×C₄₇	#436	1	2	yes	—
118	C₄×C₄₇	#490	1	2	yes	—
119	C₄×C₄₇	#496	1	2	yes	—
120	C₄×#127	D₄₇	1	1	yes	characteristic
121	#1831	—	47	1	no	—
122	#1744	—	47	1	no	—
123	C₂×D₄⋊C₈	—	47	4	no	—
124	C₂×C₄₇	—	8	4	no	—
125	C₂×C₄₇	#1478	1	2	yes	—
126	C₂×C₄₇	#1496	1	2	yes	—
127	C₂×C₄₇	#1954	1	1	yes	characteristic
128	C₂×C₄₇	#1961	1	1	yes	characteristic
129	C₂×C₄₇	#1963	1	1	yes	characteristic
130	C₂×#127	Dic₄₇	1	1	yes	characteristic
131	C₂×C₄×D₄	—	2	2	no	—
132	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
133	C₂×C₄⋊C₈	—	47	2	no	—
134	C₂×C₄×C₈	—	47	2	no	—
135	#390	—	2	2	no	—
136	#390	Dic₄₇	1	2	yes	—
137	#384	Dic₄₇	1	1	yes	characteristic
138	C₄³	C₂×D₄₇	1	1	yes	characteristic
139	#336	—	94	2	no	—
140	#336	—	94	2	no	—
141	D₄⋊C₈	—	94	8	no	—
142	C₄₇	#9642	1	1	yes	characteristic, complemented, Sylow
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
145	C₂³×C₄	—	2	2	no	—
146	C₂²×C₈	—	94	2	no	—
147	C₂²×C₈	—	94	2	no	—
148	#127	—	2	2	no	—
149	#127	C₄₇⋊C₈	1	4	yes	complemented
150	C₄×D₄	—	2	4	no	—
151	C₄×D₄	—	2	4	no	—
152	C₄×D₄	—	4	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
155	C₂×C₂²⋊C₄	—	2	4	no	—
156	C₂×C₂²⋊C₄	—	2	2	no	—
157	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
161	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
162	C₄⋊C₈	—	94	4	no	—
163	C₄×C₈	—	47	4	no	—
164	C₄×C₈	—	94	2	no	—
165	C₂⁴	—	2	2	no	—
166	C₂×D₄	—	2	8	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	4	4	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	C₂×C₄₇⋊C₈	1	4	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×C₈	—	94	4	no	—
183	C₂×C₈	—	188	2	no	—
184	C₂×C₈	—	188	4	no	—
185	C₂×C₈	—	188	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₂²⋊C₄	—	4	8	no	—
189	C₂²⋊C₄	—	4	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	4	2	no	—
193	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
194	C₄²	#10625329	1	2	yes	—
195	C₄²	D₄×D₄₇	1	2	yes	—
196	C₄²	#10625352	1	2	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
203	D₄	—	2	8	no	—
204	D₄	—	4	16	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
217	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
218	C₂×C₄	#13093995	1	2	yes	—
219	C₂×C₄	#13094011	1	1	yes	characteristic
220	C₂×C₄	#13094020	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
222	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
223	C₂×C₄	#13094031	1	2	yes	—
224	C₂×C₄	#13094050	1	4	yes	—
225	C₈	—	376	2	no	—
226	C₈	—	188	4	no	—
227	C₂²	—	4	8	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	#15689026	1	1	yes	characteristic
233	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
234	C₂²	#15689090	1	1	yes	characteristic
235	C₂²	#15689128	1	2	yes	—
236	C₂²	#15689140	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	8	2	no	—
241	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
242	C₄	#15689201	1	2	yes	—
243	C₄	#15689487	1	2	yes	—
244	C₄	#15689510	1	2	yes	—
245	C₂	—	8	4	no	—
246	C₂	#20241779	1	2	yes	—
247	C₂	#20241857	1	2	yes	—
248	C₂	#20244183	1	1	yes	characteristic
249	C₂	#20244213	1	1	yes	characteristic
250	C₂	#20244221	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#9642
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20243505	C₂	semidirect
2. #20243134	C₂	semidirect
3. C₂×D₄⋊(C₄₇⋊C₈)	C₂	semidirect
21. C₄×C₄₇⋊C₈	D₄	semidirect
26. #127×C₄₇	C₈	semidirect
142. C₄₇	#9642	semidirect
149. #127	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
#20243134	2
C₂×D₄⋊(C₄₇⋊C₈)	2
C₄×#127×C₄₇	2
#9642	47