Subgroups — #26225122

Order 12032.

702 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	#20253898	C₂	1	1	yes	maximal, characteristic
2	#20252323	C₂	1	1	yes	maximal, characteristic
3	#20252318	C₂	1	1	yes	maximal, characteristic
4	#20244365	C₂	1	1	yes	maximal, characteristic
5	#20244135	C₂	1	4	yes	maximal
6	#20243559	C₂	1	1	yes	maximal, characteristic
7	#20243266	C₂	1	4	yes	maximal
8	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1994	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	1	yes	characteristic
11	#15689526	C₄	1	2	yes	complemented
12	#15689522	C₄	1	4	yes	complemented
13	#15689519	C₄	1	2	yes	complemented
14	#15689254	—	2	2	no	—
15	#15689250	—	2	2	no	—
16	Q₈×C₄₇⋊C₈	—	2	2	no	—
17	#15689247	—	2	2	no	—
18	C₂×#13093959	C₂²	1	4	yes	—
19	#15689159	C₂²	1	4	yes	—
20	#15689151	C₂²	1	2	yes	—
21	#15689150	C₂²	1	2	yes	—
22	#15689147	C₂²	1	1	yes	characteristic
23	#15689081	C₂²	1	2	yes	—
24	#15689079	C₂²	1	2	yes	—
25	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
29	#15688883	C₂²	1	4	yes	—
30	C₄₇×#533	C₂²	1	1	yes	characteristic
31	C₄₇×#423	C₂²	1	1	yes	characteristic
32	C₄³×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#339	C₂²	1	4	yes	—
34	C₂×#10625329	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	#13094024	C₂×C₄	1	2	yes	—
39	#13094016	C₂×C₄	1	2	yes	—
40	#13094003	—	2	2	no	—
41	C₄×Dic₉₄	—	2	2	no	—
42	#13093983	C₂³	1	2	yes	—
43	#13093959	—	2	8	no	—
44	#13093957	C₂×C₄	1	8	yes	—
45	#13093955	C₂×C₄	1	8	yes	—
46	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
48	Dic₄₇⋊C₈	—	2	4	no	—
49	#13093935	—	2	2	no	—
50	#13093934	—	2	2	no	—
51	C₄×C₄₇⋊C₈	—	2	2	no	—
52	C₄×C₄₇⋊C₈	—	2	4	no	—
53	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
55	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
57	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
58	C₄×C₄₇⋊C₈	Q₈	1	8	yes	complemented
59	C₈×Dic₄₇	—	2	2	no	—
60	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
62	#125×C₄₇	C₂×C₄	1	2	yes	—
63	C₂²⋊Q₈×C₄₇	—	2	2	no	—
64	C₄×Q₈×C₄₇	—	2	2	no	—
65	#117×C₄₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₇	—	2	2	no	—
67	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	—	2	2	no	—
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₉₄	—	2	2	no	—
75	M₁₆×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂²⋊C₄×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×D₄	1	2	yes	—
87	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
88	C₂²⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
91	Dic₄₇⋊C₄	—	2	4	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	#10625329	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
102	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
103	C₂×C₄₇⋊C₈	#41	1	4	yes	—
104	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
105	Q₈×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×Dic₄₇	—	2	4	no	—
117	Dic₉₄	—	4	4	no	—
118	C₈×C₄₇	—	4	2	no	—
119	C₄₇⋊C₈	—	2	2	no	—
120	C₄₇⋊C₈	—	4	2	no	—
121	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
122	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
123	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
124	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
125	#21732	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₄₇	#533	1	2	yes	—
129	C₄×C₄₇	—	2	8	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	4	4	no	—
132	C₄×C₄₇	#443	1	4	yes	—
133	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
134	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₄×C₄₇	#532	1	1	yes	characteristic
136	Dic₄₇	—	4	4	no	—
137	#3160	—	47	1	no	—
138	#2960	—	47	1	no	—
139	#2960	—	47	1	no	—
140	#1994	—	47	1	no	—
141	#1994	D₄₇	1	1	yes	characteristic
142	#1945	—	47	4	no	—
143	#1848	—	47	1	no	—
144	#1772	—	47	4	no	—
145	C₄²×C₈	—	47	1	no	—
146	C₂×C₄₇	—	2	2	no	—
147	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#443	1	2	yes	—
149	#533	—	47	1	no	—
150	#533	C₂×D₄₇	1	1	yes	characteristic
151	#500	—	47	2	no	—
152	#499	—	47	4	no	—
153	#498	—	47	2	no	—
154	#446	—	94	2	no	—
155	#446	—	94	2	no	—
156	C₈×Q₈	—	94	2	no	—
157	C₈×Q₈	—	94	2	no	—
158	C₂×#108	—	47	4	no	—
159	#427	—	47	4	no	—
160	#424	—	47	2	no	—
161	#424	—	47	2	no	—
162	#423	—	47	1	no	—
163	#423	C₂×D₄₇	1	1	yes	characteristic
164	#405	—	47	2	no	—
165	#405	—	47	2	no	—
166	C₂×C₄×C₈	—	47	2	no	—
167	C₂×C₄×C₈	—	47	1	no	—
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	—	47	2	no	—
170	C₄³	C₂×D₄₇	1	1	yes	characteristic
171	#339	—	47	4	no	—
172	#339	C₂×D₄₇	1	4	yes	—
173	C₄₇	#21732	1	1	yes	characteristic, complemented, Sylow
174	C₂×M₁₆	—	47	2	no	—
175	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
176	C₂²×C₈	—	47	1	no	—
177	C₂²×C₈	—	47	2	no	—
178	C₂²×C₈	—	47	1	no	—
179	C₄⋊Q₈	—	47	2	no	—
180	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
181	#125	—	47	2	no	—
182	#125	C₄×D₄₇	1	2	yes	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	—	94	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	—	94	2	no	—
187	#117	—	47	2	no	—
188	#117	C₂²×D₄₇	1	2	yes	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	#108	—	94	8	no	—
193	#107	—	47	8	no	—
194	#106	—	47	8	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	94	4	no	—
197	C₄⋊C₈	—	94	4	no	—
198	C₄⋊C₈	—	94	2	no	—
199	#97	—	94	2	no	—
200	#97	—	94	2	no	—
201	C₄×C₈	—	47	1	no	—
202	C₄×C₈	—	47	2	no	—
203	C₄×C₈	—	47	1	no	—
204	C₄×C₈	—	47	2	no	—
205	C₄×C₈	—	47	2	no	—
206	C₄×C₈	—	47	8	no	—
207	C₄×C₈	—	94	4	no	—
208	C₄×C₈	—	94	2	no	—
209	C₄×C₈	—	94	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	94	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	#10625352	1	2	yes	—
214	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	94	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	47	2	no	—
219	C₂×C₈	—	47	2	no	—
220	C₂×C₈	—	47	1	no	—
221	C₂×C₈	—	47	1	no	—
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	47	4	no	—
224	C₂×C₈	—	47	4	no	—
225	C₂×C₈	—	47	2	no	—
226	C₂×C₈	—	94	2	no	—
227	C₂×C₈	—	94	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	47	4	no	—
231	C₄⋊C₄	—	94	4	no	—
232	C₄⋊C₄	—	94	2	no	—
233	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	94	2	no	—
236	C₄²	—	2	8	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	94	2	no	—
240	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₄²	D₄×D₄₇	1	2	yes	—
243	C₄²	Q₈×D₄₇	1	4	yes	—
244	C₂³	C₂×#10625352	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	188	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
253	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
254	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
257	C₂×C₄	#41×D₄₇	1	2	yes	—
258	C₈	—	4	2	no	—
259	C₈	—	47	4	no	—
260	C₈	—	47	4	no	—
261	C₈	—	47	2	no	—
262	C₈	—	47	2	no	—
263	C₈	—	94	2	no	—
264	C₈	—	188	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
267	C₂²	#15689675	1	2	yes	—
268	C₄	—	2	8	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	4	no	—
271	C₄	—	188	4	no	—
272	C₄	#15689237	1	4	yes	—
273	C₄	C₄×Q₈×D₄₇	1	2	yes	—
274	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
275	C₄	#15689672	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	#20245957	1	1	yes	characteristic
278	C₂	C₂×#15689237	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1994	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1994	D₄₇
hypocenter	C₄₇	#21732
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1994

Complements

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

N	H ≅ G/N	type
11. #15689526	C₄	semidirect
12. #15689522	C₄	semidirect
13. #15689519	C₄	semidirect
58. C₄×C₄₇⋊C₈	Q₈	semidirect
173. C₄₇	#21732	semidirect

Maximal subgroups

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

subgroup	index
#20253898	2
#20252323	2
#20252318	2
#20244365	2
#20244135	2
#20243559	2
#20243266	2
C₄²×C₄₇⋊C₈	2
C₄₇×#1994	2
#21732	47