Subgroups — #26132777

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688948	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20243417	C₂	1	1	yes	maximal, characteristic
4	#20241479	C₂	1	2	yes	maximal, complemented
5	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	1	yes	characteristic
8	C₂×#13093958	C₂²	1	1	yes	characteristic
9	#15688983	—	2	1	no	—
10	#15688948	—	2	4	no	—
11	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	complemented
12	#15688886	—	2	1	no	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
15	C₂×#107×C₄₇	—	2	1	no	—
16	C₂×#107×C₄₇	—	2	4	no	—
17	C₂×#107×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₄₇	—	2	1	no	—
19	C₂×C₄⋊Dic₄₇	—	2	1	no	—
20	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
24	#13093958	—	2	2	no	—
25	#13093958	—	2	2	no	—
26	C₄₇⋊#95	—	4	1	no	—
27	C₂³×C₄×C₄₇	—	4	1	no	—
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
39	#107×C₄₇	—	2	4	no	—
40	#107×C₄₇	—	2	2	no	—
41	#107×C₄₇	D₄	1	4	yes	—
42	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	—	2	1	no	—
44	C₂²×C₄×C₄₇	—	2	1	no	—
45	C₂²×C₄×C₄₇	—	2	1	no	—
46	C₂²×C₄×C₄₇	—	2	1	no	—
47	C₂²×C₄×C₄₇	—	4	4	no	—
48	C₂²×C₄×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	—	8	2	no	—
50	C₂²×C₄×C₄₇	—	8	2	no	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	4	1	no	—
56	C₂²×Dic₄₇	—	4	1	no	—
57	C₂×C₈×C₄₇	—	2	1	no	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₄₇	#41	1	2	yes	—
66	C₄⋊C₄×C₄₇	—	2	4	no	—
67	C₄⋊C₄×C₄₇	—	4	4	no	—
68	C₄⋊C₄×C₄₇	—	4	2	no	—
69	C₂²⋊Dic₄₇	—	8	2	no	—
70	C₄⋊Dic₄₇	—	4	1	no	—
71	C₄⋊Dic₄₇	—	4	1	no	—
72	C₄⋊Dic₄₇	—	4	1	no	—
73	C₄⋊Dic₄₇	—	4	1	no	—
74	C₄₇⋊C₁₆	—	2	4	no	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	—	2	1	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	1	no	—
79	C₂³×C₄₇	#107	1	2	yes	—
80	C₂³×C₄₇	D₁₆	1	2	yes	—
81	C₂³×C₄₇	Q₃₂	1	2	yes	—
82	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	—	8	8	no	—
91	C₂×C₄×C₄₇	—	8	2	no	—
92	C₂×C₄×C₄₇	—	8	2	no	—
93	C₂×C₄×C₄₇	#107	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
100	C₂×Dic₄₇	—	8	1	no	—
101	C₂×Dic₄₇	—	8	1	no	—
102	C₂×Dic₄₇	—	8	1	no	—
103	C₂×Dic₄₇	—	8	1	no	—
104	C₂×Dic₄₇	—	8	1	no	—
105	C₂×Dic₄₇	—	8	1	no	—
106	C₈×C₄₇	—	2	2	no	—
107	C₈×C₄₇	C₄×D₄	1	2	yes	—
108	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	#12919	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	#366	1	4	yes	complemented
117	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
118	C₂²×C₄₇	#447	1	1	yes	characteristic
119	C₂²×C₄₇	#480	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₄₇	#509	1	1	yes	characteristic
123	C₂²×C₄₇	#510	1	1	yes	characteristic
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	8	4	no	—
126	C₄×C₄₇	#385	1	1	yes	characteristic
127	C₄×C₄₇	#428	1	1	yes	characteristic
128	C₄×C₄₇	#451	1	1	yes	characteristic
129	C₄×C₄₇	#484	1	1	yes	characteristic
130	Dic₄₇	—	16	1	no	—
131	Dic₄₇	—	16	1	no	—
132	C₂²×#107	D₄₇	1	1	yes	characteristic
133	C₂×#366	—	47	2	no	—
134	C₂×C₂²⋊C₁₆	—	47	1	no	—
135	#1808	—	47	1	no	—
136	#1380	—	47	2	no	—
137	C₂×C₄₇	—	2	4	no	—
138	C₂×C₄₇	#1852	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#366	1	1	yes	characteristic
140	C₂×C₄₇	#2159	1	1	yes	characteristic
141	C₂×C₄₇	#2184	1	1	yes	characteristic
142	C₂×C₄₇	#2188	1	1	yes	characteristic
143	C₂×C₄₇	#2232	1	1	yes	characteristic
144	C₂×C₄₇	#2236	1	1	yes	characteristic
145	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
146	C₂²×C₄⋊C₄	—	2	1	no	—
147	C₂²×C₁₆	—	47	2	no	—
148	C₂×#107	—	2	1	no	—
149	C₂×#107	—	2	4	no	—
150	C₂×#107	—	47	1	no	—
151	C₂×#107	—	47	1	no	—
152	C₂×#107	C₂×D₄₇	1	2	yes	—
153	#380	—	94	1	no	—
154	#366	—	94	4	no	—
155	C₂²⋊C₁₆	—	47	4	no	—
156	#340	—	94	1	no	—
157	C₄₇	#12919	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	4	1	no	—
159	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	Dic₉₄	1	2	yes	—
163	C₂²×C₈	C₄×D₄₇	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	4	2	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₄⋊C₄	—	94	1	no	—
171	C₂×C₄⋊C₄	—	94	1	no	—
172	C₂×C₂²⋊C₄	—	188	1	no	—
173	C₂×C₁₆	—	47	4	no	—
174	C₂×C₁₆	—	94	2	no	—
175	C₂×C₁₆	—	94	2	no	—
176	#107	—	2	4	no	—
177	#107	—	2	2	no	—
178	#107	—	94	2	no	—
179	#107	—	94	2	no	—
180	#107	C₂²⋊D₄₇	1	4	yes	—
181	#95	—	188	1	no	—
182	C₂⁴	#10625334	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	8	2	no	—
190	C₂²×C₄	—	8	2	no	—
191	C₂²×C₄	—	188	1	no	—
192	C₂²×C₄	—	188	1	no	—
193	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
194	C₂²×C₄	#10625334	1	1	yes	characteristic
195	C₂²×C₄	#10625337	1	2	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
202	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
203	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₈	#10625350	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	188	1	no	—
210	C₄⋊C₄	—	188	1	no	—
211	C₄⋊C₄	—	188	1	no	—
212	C₄⋊C₄	—	188	1	no	—
213	C₂²⋊C₄	—	376	2	no	—
214	C₁₆	—	94	4	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#13093957	1	2	yes	—
220	C₂³	#13093965	1	2	yes	—
221	C₂³	#13093970	1	2	yes	—
222	C₂³	C₂×#10625334	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	376	1	no	—
234	C₂×C₄	—	376	1	no	—
235	C₂×C₄	—	376	1	no	—
236	C₂×C₄	—	376	1	no	—
237	C₂×C₄	—	376	1	no	—
238	C₂×C₄	—	376	1	no	—
239	C₂×C₄	#13093957	1	2	yes	—
240	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
241	C₂×C₄	#13093993	1	1	yes	characteristic
242	C₂×C₄	#13094007	1	1	yes	characteristic
243	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
244	C₂×C₄	#13094050	1	1	yes	characteristic
245	C₂×C₄	#13094055	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
248	C₈	#13094003	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#15688948	1	4	yes	complemented
256	C₂²	C₂×#13093957	1	1	yes	characteristic
257	C₂²	#15689257	1	1	yes	characteristic
258	C₂²	#15689436	1	1	yes	characteristic
259	C₂²	C₂×#13093965	1	1	yes	characteristic
260	C₂²	C₂×#13093970	1	1	yes	characteristic
261	C₂²	#15689559	1	1	yes	characteristic
262	C₂²	#15689564	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	8	4	no	—
265	C₄	—	752	1	no	—
266	C₄	—	752	1	no	—
267	C₄	#15688996	1	1	yes	characteristic
268	C₄	#15689163	1	1	yes	characteristic
269	C₄	#15689279	1	1	yes	characteristic
270	C₄	#15689458	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#20243576	1	1	yes	characteristic
273	C₂	C₂×#15688948	1	1	yes	characteristic
274	C₂	#20245174	1	1	yes	characteristic
275	C₂	#20245300	1	1	yes	characteristic
276	C₂	#20245322	1	1	yes	characteristic
277	C₂	#20245568	1	1	yes	characteristic
278	C₂	#20245590	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	#12919
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
1. C₂×#15688948	C₂	semidirect
4. #20241479	C₂	semidirect
11. C₂²⋊(C₄₇⋊C₁₆)	C₄	semidirect
23. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
116. C₂²×C₄₇	#366	semidirect
157. C₄₇	#12919	semidirect
255. C₂²	#15688948	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688948	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20243417	2
#20241479	2
C₂²×#107×C₄₇	2
#12919	47