Subgroups — #26133178

Order 12032.

488 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	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15688907	C₂	1	2	yes	maximal, complemented
3	#20244999	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241477	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2929	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	—	2	4	no	—
7	C₂×#13093964	—	2	2	no	—
8	C₂×#13093964	C₂²	1	2	yes	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15688907	—	2	4	no	—
11	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
12	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#525	C₄	1	1	yes	characteristic
15	C₄₇×#435	—	2	2	no	—
16	C₈×D₄×C₄₇	—	2	2	no	—
17	C₄₇×#424	C₄	1	1	yes	characteristic
18	C₄₇×#408	C₄	1	1	yes	characteristic
19	C₄₇×#407	C₄	1	1	yes	characteristic
20	C₄₇×#405	C₂²	1	1	yes	characteristic
21	#13093964	—	2	8	no	—
22	#13093964	—	2	2	no	—
23	#13093964	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	C₂×C₄₇⋊C₁₆	—	2	2	no	—
27	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
28	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
30	#131×C₄₇	—	2	2	no	—
31	#131×C₄₇	—	2	2	no	—
32	C₂×M₁₆×C₄₇	—	2	1	no	—
33	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	—	2	1	no	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	D₄	1	2	yes	—
40	C₂²×C₈×C₄₇	D₄	1	2	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	#123×C₄₇	—	2	1	no	—
43	C₂²⋊Q₈×C₄₇	C₈	1	2	yes	—
44	#121×C₄₇	C₈	1	2	yes	—
45	#120×C₄₇	—	2	1	no	—
46	C₄×D₄×C₄₇	—	2	2	no	—
47	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₄₇	—	2	2	no	—
49	C₂²⋊C₈×C₄₇	—	2	2	no	—
50	C₂²⋊C₈×C₄₇	—	2	2	no	—
51	#97×C₄₇	—	2	1	no	—
52	C₄×C₈×C₄₇	—	2	1	no	—
53	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
54	#41×C₄₇	—	2	2	no	—
55	#41×C₄₇	—	2	2	no	—
56	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₄₇	—	2	2	no	—
58	C₂×D₄×C₄₇	—	2	1	no	—
59	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₄₇	—	2	2	no	—
68	M₁₆×C₄₇	—	2	2	no	—
69	M₁₆×C₄₇	—	4	1	no	—
70	M₁₆×C₄₇	—	4	1	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	—	2	1	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	#41	1	2	yes	—
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₄₇	—	2	2	no	—
86	C₂²⋊C₄×C₄₇	—	2	1	no	—
87	C₂²⋊C₄×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	1	no	—
89	C₄₇⋊C₁₆	—	2	4	no	—
90	C₄₇⋊C₁₆	—	4	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
95	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
96	Q₈×C₄₇	—	2	2	no	—
97	D₄×C₄₇	—	2	2	no	—
98	D₄×C₄₇	—	4	4	no	—
99	D₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#131	1	2	yes	—
113	C₈×C₄₇	—	2	2	no	—
114	C₈×C₄₇	—	4	1	no	—
115	C₈×C₄₇	—	4	1	no	—
116	C₈×C₄₇	#120	1	1	yes	characteristic
117	C₈×C₄₇	#121	1	2	yes	—
118	C₈×C₄₇	#123	1	1	yes	characteristic
119	#12984	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	4	4	no	—
125	C₂²×C₄₇	—	4	2	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₄₇	#435	1	2	yes	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	4	2	no	—
131	C₄×C₄₇	—	4	1	no	—
132	C₄×C₄₇	#386	1	1	yes	characteristic
133	C₄×C₄₇	#408	1	1	yes	characteristic
134	C₄×C₄₇	C₈×D₄	1	2	yes	—
135	C₂²×M₃₂	—	47	1	no	—
136	#2929	D₄₇	1	1	yes	characteristic
137	C₂×#350	—	47	2	no	—
138	#2112	—	47	1	no	—
139	#1379	—	47	2	no	—
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	—	4	2	no	—
142	C₂×C₄₇	—	8	1	no	—
143	C₂×C₄₇	#1869	1	1	yes	characteristic
144	C₂×C₄₇	#2117	1	2	yes	—
145	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
146	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
147	#525	Dic₄₇	1	1	yes	characteristic
148	C₂×M₃₂	—	47	2	no	—
149	C₂×M₃₂	—	94	4	no	—
150	C₂×M₃₂	—	94	2	no	—
151	C₂²×C₁₆	—	47	2	no	—
152	#435	—	2	2	no	—
153	C₈×D₄	—	2	2	no	—
154	#424	Dic₄₇	1	1	yes	characteristic
155	#408	Dic₄₇	1	1	yes	characteristic
156	#407	Dic₄₇	1	1	yes	characteristic
157	#405	C₂×D₄₇	1	1	yes	characteristic
158	#350	—	94	4	no	—
159	C₂²⋊C₁₆	—	94	2	no	—
160	C₄₇	#12984	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
162	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
163	#131	—	2	2	no	—
164	#131	—	2	2	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
173	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	#123	—	2	1	no	—
176	C₂²⋊Q₈	C₄₇⋊C₈	1	2	yes	—
177	#121	C₄₇⋊C₈	1	2	yes	—
178	#120	—	2	1	no	—
179	C₄×D₄	—	2	2	no	—
180	#117	C₂×Dic₄₇	1	1	yes	characteristic
181	M₃₂	—	94	8	no	—
182	M₃₂	—	94	2	no	—
183	M₃₂	—	94	2	no	—
184	C₂×C₁₆	—	47	4	no	—
185	C₂×C₁₆	—	94	2	no	—
186	C₂×C₁₆	—	94	2	no	—
187	C₂×C₁₆	—	94	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	2	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	#97	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₂⁴	#10625329	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
204	C₂²×C₄	#10625329	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
219	C₂×C₈	D₄×D₄₇	1	2	yes	—
220	C₂×C₈	#10625352	1	2	yes	—
221	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₁₆	—	94	4	no	—
230	C₁₆	—	188	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	2	no	—
234	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
235	C₂³	C₂×#10625329	1	1	yes	characteristic
236	Q₈	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	4	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
251	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
252	C₂×C₄	#13094031	1	2	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	4	1	no	—
256	C₈	#13093992	1	1	yes	characteristic
257	C₈	#13093996	1	2	yes	—
258	C₈	#13094005	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
267	C₂²	#15689195	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	#15689003	1	1	yes	characteristic
272	C₄	#15689090	1	1	yes	characteristic
273	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	4	2	no	—
276	C₂	—	8	1	no	—
277	C₂	#20243681	1	1	yes	characteristic
278	C₂	#20245016	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2929	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₄₇×#525	C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2929	D₄₇
hypocenter	C₄₇	#12984
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093964	C₂	semidirect
2. C₂×#15688907	C₂	semidirect
3. #20244999	C₂	semidirect
4. #20241477	C₂	semidirect
27. C₂×C₄₇⋊C₁₆	D₄	semidirect
160. C₄₇	#12984	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093964	2
C₂×#15688907	2
#20244999	2
#20241477	2
C₄₇×#2929	2
#12984	47