Subgroups — #26370573

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689564	C₂	1	2	yes	maximal
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
3	#20245435	C₂	1	4	yes	maximal, complemented
4	#20245430	C₂	1	4	yes	maximal, complemented
5	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245108	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	1	yes	characteristic
9	C₄×#10625350	—	2	1	no	—
10	#15689564	—	2	8	no	—
11	C₂×#13093970	C₂²	1	4	yes	—
12	C₂×#13093967	C₂²	1	4	yes	—
13	C₂×#13093964	C₂²	1	2	yes	—
14	C₄×Dic₁₈₈	C₂²	1	2	yes	—
15	C₄×#10625325	—	2	2	no	—
16	C₄×D₃₇₆	C₂²	1	2	yes	—
17	#15689161	C₂²	1	1	yes	characteristic
18	#15689120	—	2	1	no	—
19	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
20	#15688927	C₂²	1	8	yes	—
21	C₂²×Q₁₆×C₄₇	—	2	1	no	—
22	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
24	C₄₇×#494	—	2	2	no	—
25	C₄₇×#494	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625350	—	2	1	no	—
28	#13094046	—	2	2	no	—
29	#13094046	D₄	1	4	yes	—
30	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
31	C₂×#10625325	—	2	1	no	—
32	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
33	C₄×Dic₉₄	—	2	2	no	—
34	C₄×D₁₈₈	—	2	2	no	—
35	C₄×C₂²⋊D₄₇	—	4	2	no	—
36	#13093983	—	2	1	no	—
37	#13093981	—	4	1	no	—
38	C₄²×D₄₇	—	4	1	no	—
39	#13093970	—	2	8	no	—
40	#13093967	—	2	8	no	—
41	#13093964	—	2	4	no	—
42	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
43	#13093958	C₂³	1	2	yes	—
44	#13093956	—	2	1	no	—
45	#13093949	—	2	2	no	—
46	#13093946	—	2	2	no	—
47	C₂²×Q₈×C₄₇	—	2	2	no	—
48	C₂×Q₁₆×C₄₇	—	2	4	no	—
49	C₂×Q₁₆×C₄₇	—	2	2	no	—
50	C₂×Q₁₆×C₄₇	—	2	4	no	—
51	C₂×Q₁₆×C₄₇	—	2	4	no	—
52	C₂×Q₁₆×C₄₇	—	4	4	no	—
53	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₄₇	—	2	4	no	—
57	C₄⋊Q₈×C₄₇	—	4	1	no	—
58	C₄⋊Q₈×C₄₇	—	4	1	no	—
59	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
60	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₄₇	D₄	1	2	yes	—
62	C₄×C₈×C₄₇	C₂³	1	2	yes	—
63	C₂×Q₈×C₄₇	—	2	4	no	—
64	C₂×Q₈×C₄₇	—	4	2	no	—
65	C₂×Q₈×C₄₇	—	4	4	no	—
66	C₂×Q₈×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	D₈	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₇	—	4	1	no	—
71	#10625350	—	2	4	no	—
72	#10625350	—	4	2	no	—
73	C₂×D₁₈₈	—	2	1	no	—
74	C₂×C₄×D₄₇	—	4	1	no	—
75	C₂×C₄×D₄₇	—	4	2	no	—
76	C₂×Dic₉₄	—	2	1	no	—
77	Q₁₆×C₄₇	—	2	4	no	—
78	Q₁₆×C₄₇	—	2	4	no	—
79	Q₁₆×C₄₇	—	4	8	no	—
80	Q₁₆×C₄₇	—	4	4	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	#41	1	2	yes	—
86	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	—	4	4	no	—
88	C₄⋊C₄×C₄₇	—	4	4	no	—
89	C₄²×C₄₇	D₈	1	1	yes	characteristic
90	C₄²×C₄₇	D₈	1	1	yes	characteristic
91	C₄²×C₄₇	C₂×D₄	1	2	yes	—
92	C₂²⋊Dic₄₇	—	4	1	no	—
93	(C₂×D₄₇)⋊C₄	—	4	2	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	Dic₄₇⋊C₄	—	4	2	no	—
96	C₄×Dic₄₇	—	4	1	no	—
97	C₄×Dic₄₇	—	4	2	no	—
98	Dic₁₈₈	—	2	1	no	—
99	Dic₁₈₈	C₂×D₄	1	2	yes	—
100	D₃₇₆	—	2	1	no	—
101	D₃₇₆	C₂×D₄	1	2	yes	—
102	#10625325	—	2	4	no	—
103	C₄₇⋊C₁₆	—	2	4	no	—
104	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
105	C₂²×D₄₇	—	4	1	no	—
106	Q₈×C₄₇	—	4	4	no	—
107	Q₈×C₄₇	—	4	4	no	—
108	Q₈×C₄₇	—	8	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	8	4	no	—
112	C₂×C₄×C₄₇	#121	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	1	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	D₁₈₈	—	2	2	no	—
122	D₁₈₈	—	4	1	no	—
123	C₄×D₄₇	—	4	4	no	—
124	C₄×D₄₇	—	8	4	no	—
125	Dic₉₄	—	2	2	no	—
126	Dic₉₄	—	4	1	no	—
127	C₈×C₄₇	—	2	2	no	—
128	C₈×C₄₇	#121	1	2	yes	—
129	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
131	#33583	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	#459	1	2	yes	—
134	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
135	C₂×D₄₇	—	4	2	no	—
136	C₂×D₄₇	—	8	2	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	8	4	no	—
139	C₄×C₄₇	#459	1	2	yes	—
140	C₄×C₄₇	#510	1	4	yes	complemented
141	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
142	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
143	Dic₄₇	—	4	2	no	—
144	Dic₄₇	—	8	2	no	—
145	C₂×#510	—	47	2	no	—
146	C₂×#494	D₄₇	1	1	yes	characteristic, complemented
147	C₄×#135	—	47	1	no	—
148	#2209	—	47	4	no	—
149	#2208	—	47	4	no	—
150	#2150	—	47	1	no	—
151	#2141	—	47	2	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#510	1	2	yes	—
155	D₄₇	—	8	2	no	—
156	C₂×#135	—	47	1	no	—
157	C₂²×Q₁₆	—	2	1	no	—
158	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
159	C₂×C₄⋊Q₈	—	2	1	no	—
160	C₄×#41	—	94	1	no	—
161	#510	—	94	8	no	—
162	C₂×Q₃₂	—	47	4	no	—
163	C₂×SD₃₂	—	47	4	no	—
164	C₂×M₃₂	—	47	2	no	—
165	#494	—	2	2	no	—
166	#494	C₂×D₄₇	1	4	yes	—
167	C₄×Q₁₆	—	47	2	no	—
168	C₄×SD₁₆	—	94	2	no	—
169	C₄×D₈	—	47	2	no	—
170	#427	—	47	1	no	—
171	#416	—	94	1	no	—
172	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
173	C₄⋊C₁₆	—	47	4	no	—
174	#358	—	47	8	no	—
175	C₄₇	#33583	1	1	yes	characteristic, complemented, Sylow
176	C₂×#41	—	94	1	no	—
177	C₂²×Q₈	—	2	2	no	—
178	#135	—	47	4	no	—
179	#135	—	94	2	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	2	4	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	4	4	no	—
185	C₂×Q₁₆	—	47	1	no	—
186	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
187	C₂×SD₁₆	—	94	1	no	—
188	C₂×D₈	—	47	1	no	—
189	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	4	1	no	—
193	C₄⋊Q₈	—	4	1	no	—
194	C₄×Q₈	—	94	2	no	—
195	C₄×D₄	—	94	2	no	—
196	C₄×D₄	—	188	2	no	—
197	#117	—	94	1	no	—
198	#117	—	188	1	no	—
199	C₂×C₄⋊C₄	—	4	2	no	—
200	C₂×C₄²	—	188	1	no	—
201	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
202	Q₃₂	—	94	8	no	—
203	SD₃₂	—	94	8	no	—
204	M₃₂	—	94	4	no	—
205	C₂×C₁₆	—	47	4	no	—
206	#107	—	47	2	no	—
207	#106	—	94	1	no	—
208	#103	—	94	2	no	—
209	#102	—	94	2	no	—
210	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
211	C₄×C₈	C₂²×D₄₇	1	2	yes	—
212	#41	—	94	4	no	—
213	#41	—	188	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	94	1	no	—
219	C₂×D₄	—	94	1	no	—
220	C₂×D₄	—	188	1	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	188	2	no	—
223	C₂²×C₄	—	188	1	no	—
224	C₂²×C₄	#10625334	1	2	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	Q₁₆	—	2	4	no	—
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	4	8	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	—	47	2	no	—
231	Q₁₆	—	94	1	no	—
232	SD₁₆	—	94	4	no	—
233	D₈	—	47	2	no	—
234	D₈	—	94	1	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	D₄×D₄₇	1	2	yes	—
237	C₂×C₈	#10625352	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	94	2	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₂²⋊C₄	—	188	1	no	—
246	C₂²⋊C₄	—	188	2	no	—
247	C₄²	—	188	1	no	—
248	C₄²	—	188	2	no	—
249	C₄²	#10625334	1	1	yes	characteristic
250	C₄²	#10625334	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
252	C₁₆	—	94	4	no	—
253	C₂³	—	188	1	no	—
254	C₂³	C₂×#10625334	1	1	yes	characteristic
255	Q₈	—	4	4	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	94	2	no	—
259	Q₈	—	188	1	no	—
260	D₄	—	94	2	no	—
261	D₄	—	188	4	no	—
262	D₄	—	188	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	—	376	4	no	—
267	C₂×C₄	—	188	2	no	—
268	C₂×C₄	—	188	1	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	#13093995	1	2	yes	—
271	C₂×C₄	C₂×#10625334	1	2	yes	—
272	C₂×C₄	C₂×#10625334	1	2	yes	—
273	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
274	C₂×C₄	#13094050	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	#13093995	1	2	yes	—
278	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
279	C₈	C₂×#10625352	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	376	2	no	—
282	C₂²	—	188	2	no	—
283	C₂²	#15689315	1	2	yes	—
284	C₂²	C₂²×#10625334	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	8	4	no	—
287	C₄	—	376	2	no	—
288	C₄	—	188	2	no	—
289	C₄	#15689315	1	2	yes	—
290	C₄	#15689564	1	4	yes	complemented
291	C₄	C₂×#13093995	1	1	yes	characteristic
292	C₄	C₂×#13094050	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	376	2	no	—
295	C₂	C₂×#15689315	1	1	yes	characteristic
296	C₂	C₂×#15689564	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#13094046	C₂²
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	#33583
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #20245435	C₂	semidirect
4. #20245430	C₂	semidirect
5. #20245144	C₂	semidirect
6. #20245108	C₂	semidirect
7. C₂×C₄₇×#494	C₂	semidirect
140. C₄×C₄₇	#510	semidirect
146. C₂×#494	D₄₇	semidirect
175. C₄₇	#33583	semidirect
290. C₄	#15689564	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689564	2
C₄×#13094046	2
#20245435	2
#20245430	2
#20245144	2
#20245108	2
C₂×C₄₇×#494	2
#33583	47