Subgroups — #26220522

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245866	C₂	1	2	yes	maximal
2	#20244260	C₂	1	2	yes	maximal
3	#20244127	C₂	1	2	yes	maximal
4	#20244080	C₂	1	4	yes	maximal
5	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
6	C₄×#13093948	C₂	1	2	yes	maximal
7	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093956	C₂²	1	2	yes	—
10	C₂×#13093948	C₂²	1	4	yes	—
11	C₂×#13093934	C₂²	1	2	yes	—
12	C₂×#13093934	C₂²	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15689030	C₂²	1	4	yes	—
15	#15689022	C₂²	1	4	yes	—
16	#15689009	—	2	4	no	—
17	C₄×C₄⋊Dic₄₇	—	2	2	no	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	#15688969	—	2	2	no	—
20	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688886	C₂²	1	2	yes	—
22	#15688885	C₂²	1	4	yes	—
23	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₄₇	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#103×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
36	#13093956	—	2	4	no	—
37	#13093948	—	2	8	no	—
38	#13093934	—	2	4	no	—
39	#13093934	—	2	4	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₄₇⋊#95	—	2	4	no	—
43	C₄₇⋊#95	—	2	4	no	—
44	C₄₇⋊#95	—	2	4	no	—
45	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₄₇	—	2	4	no	—
47	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₄₇	—	2	4	no	—
51	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
52	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#107×C₄₇	C₂×C₄	1	4	yes	—
57	#103×C₄₇	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×Q₈×C₄₇	—	2	2	no	—
60	C₂×Q₈×C₄₇	—	2	4	no	—
61	C₂×Q₈×C₄₇	—	2	4	no	—
62	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
70	Q₁₆×C₄₇	—	2	8	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	#41	1	4	yes	—
75	C₄⋊C₄×C₄₇	—	2	8	no	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₄²×C₄₇	—	2	8	no	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	#41	1	2	yes	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	4	8	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₄×Dic₄₇	—	2	2	no	—
87	C₄×Dic₄₇	—	4	2	no	—
88	C₄×Dic₄₇	—	4	4	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	#41	1	4	yes	—
92	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₄₇	—	2	4	no	—
94	Q₈×C₄₇	—	2	4	no	—
95	Q₈×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	#137	1	4	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₈×C₄₇	—	2	2	no	—
115	C₈×C₄₇	#117	1	4	yes	—
116	C₄₇⋊C₈	—	2	4	no	—
117	#21278	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₄₇	#532	1	2	yes	—
120	C₂²×C₄₇	C₂×#137	1	2	yes	—
121	C₂²×C₄₇	#575	1	2	yes	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	4	4	no	—
124	C₄×C₄₇	—	4	4	no	—
125	C₄×C₄₇	#441	1	4	yes	—
126	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
127	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
128	C₄×C₄₇	#529	1	2	yes	—
129	Dic₄₇	—	4	4	no	—
130	Dic₄₇	—	8	4	no	—
131	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
132	#2295	—	47	2	no	—
133	#1971	—	47	2	no	—
134	#1943	—	47	2	no	—
135	#1935	—	47	4	no	—
136	C₄×#106	—	47	1	no	—
137	C₄×#103	—	47	2	no	—
138	C₄×#97	—	47	1	no	—
139	C₂×C₄₇	#2293	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
141	C₂×C₄₇	#2943	1	1	yes	characteristic
142	C₂×C₄₇	#3172	1	4	yes	—
143	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
144	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
145	C₄×Q₁₆	Dic₄₇	1	8	yes	complemented
146	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
147	C₂×#106	—	47	2	no	—
148	C₂×#103	—	47	4	no	—
149	C₂×#103	C₂×D₄₇	1	2	yes	—
150	C₂×#97	—	47	2	no	—
151	C₂×#97	—	47	2	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
154	#391	—	47	4	no	—
155	#389	—	47	4	no	—
156	#387	—	94	4	no	—
157	C₄×C₄⋊C₄	—	47	2	no	—
158	C₄×C₄⋊C₄	—	94	2	no	—
159	#376	—	94	2	no	—
160	C₄³	—	47	1	no	—
161	#340	—	47	2	no	—
162	#340	—	47	4	no	—
163	C₄₇	#21278	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
167	C₂²×C₈	—	47	2	no	—
168	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
170	C₄×Q₈	—	2	4	no	—
171	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	47	4	no	—
174	C₂×C₄⋊C₄	—	94	4	no	—
175	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	47	2	no	—
180	C₂×C₄²	—	94	2	no	—
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	#107	C₂×Dic₄₇	1	4	yes	—
183	#106	—	94	4	no	—
184	#103	—	94	8	no	—
185	#103	C₂×Dic₄₇	1	8	yes	—
186	#97	—	94	4	no	—
187	#97	—	94	4	no	—
188	C₄×C₈	—	94	2	no	—
189	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
190	#95	—	94	4	no	—
191	#95	—	94	4	no	—
192	#95	—	94	4	no	—
193	#95	—	94	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	47	4	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
205	Q₁₆	—	2	8	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	47	4	no	—
208	C₂×C₈	—	94	2	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	#10625350	1	4	yes	—
211	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
212	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	94	4	no	—
216	C₄⋊C₄	—	94	4	no	—
217	C₄⋊C₄	—	188	8	no	—
218	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	94	4	no	—
221	C₄²	—	94	2	no	—
222	C₄²	—	94	2	no	—
223	C₄²	—	188	2	no	—
224	C₄²	—	188	4	no	—
225	C₄²	#10625352	1	2	yes	—
226	C₄²	C₂²×Dic₄₇	1	2	yes	—
227	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	2	4	no	—
230	Q₈	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	94	4	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	—	94	4	no	—
239	C₂×C₄	—	188	4	no	—
240	C₂×C₄	—	188	4	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
243	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
244	C₂×C₄	#13094056	1	4	yes	—
245	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	#41×D₄₇	1	2	yes	—
247	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
248	C₂×C₄	C₂×#10625350	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	94	4	no	—
251	C₈	#13093983	1	4	yes	—
252	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
253	C₂²	#15689670	1	2	yes	—
254	C₂²	C₂×#13094056	1	2	yes	—
255	C₂²	#15689920	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	—	376	4	no	—
260	C₄	—	188	4	no	—
261	C₄	#15689230	1	4	yes	—
262	C₄	C₂×#13093983	1	1	yes	characteristic
263	C₄	#41×Dic₄₇	1	1	yes	characteristic
264	C₄	#15689655	1	2	yes	—
265	C₂	#20245853	1	1	yes	characteristic
266	C₂	C₂×#15689230	1	1	yes	characteristic
267	C₂	#20252189	1	1	yes	characteristic
268	C₂	#20253992	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#21278
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₄₇	C₄	semidirect
145. C₄×Q₁₆	Dic₄₇	semidirect
163. C₄₇	#21278	semidirect

Maximal subgroups

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

subgroup	index
#20245866	2
#20244260	2
#20244127	2
#20244080	2
C₄×#13093956	2
C₄×#13093948	2
C₄×#13093934	2
C₂×C₄×Q₁₆×C₄₇	2
#21278	47