Subgroups — #26225144

Order 12032.

662 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	#20253910	C₂	1	1	yes	maximal, characteristic
2	#20252322	C₂	1	1	yes	maximal, characteristic
3	#20244370	C₂	1	1	yes	maximal, characteristic
4	#20244368	C₂	1	1	yes	maximal, characteristic
5	#20244138	C₂	1	4	yes	maximal
6	#20243562	C₂	1	1	yes	maximal, characteristic
7	#20243268	C₂	1	4	yes	maximal
8	#20243074	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2960	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	1	yes	characteristic
11	#15689533	C₄	1	4	yes	complemented
12	#15689529	C₄	1	4	yes	complemented
13	#15689253	—	2	2	no	—
14	C₈×Dic₉₄	—	2	2	no	—
15	C₂×#13093959	C₂²	1	4	yes	—
16	#15689159	C₂²	1	4	yes	—
17	#15689149	C₂²	1	2	yes	—
18	#15689147	C₂²	1	1	yes	characteristic
19	#15689078	C₂²	1	2	yes	—
20	C₂×#13093935	C₂²	1	1	yes	characteristic
21	C₂×#13093934	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
23	#15688969	C₂²	1	1	yes	characteristic
24	#15688884	C₂²	1	4	yes	—
25	#15688882	C₂²	1	4	yes	—
26	C₄₇×#533	C₂²	1	1	yes	characteristic
27	C₄₇×#446	—	2	2	no	—
28	C₈×Q₈×C₄₇	—	2	2	no	—
29	C₄₇×#424	C₂²	1	2	yes	—
30	C₄₇×#423	C₂²	1	1	yes	characteristic
31	C₄₇×#405	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625329	C₂³	1	2	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	#13094024	C₂×C₄	1	2	yes	—
36	#13094016	C₂×C₄	1	2	yes	—
37	#13094003	—	2	2	no	—
38	C₄×Dic₉₄	—	2	2	no	—
39	#13093983	C₂³	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	#13093959	—	2	8	no	—
42	#13093957	C₂×C₄	1	8	yes	—
43	#13093955	C₂×C₄	1	8	yes	—
44	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
45	Dic₄₇⋊C₈	—	2	4	no	—
46	#13093935	C₂×C₄	1	2	yes	—
47	#13093935	C₂×C₄	1	2	yes	—
48	#13093934	—	2	2	no	—
49	#13093934	—	2	4	no	—
50	C₈×Dic₄₇	—	2	2	no	—
51	C₈×Dic₄₇	Q₈	1	8	yes	complemented
52	C₄₇⋊#95	—	2	2	no	—
53	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
57	#125×C₄₇	C₂×C₄	1	2	yes	—
58	C₂²⋊Q₈×C₄₇	—	2	2	no	—
59	C₄×Q₈×C₄₇	—	2	2	no	—
60	#117×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₄₇	—	2	4	no	—
63	C₄⋊C₈×C₄₇	—	2	2	no	—
64	#97×C₄₇	—	2	2	no	—
65	C₄×C₈×C₄₇	—	2	2	no	—
66	C₄×C₈×C₄₇	D₄	1	2	yes	—
67	C₄×C₈×C₄₇	D₄	1	2	yes	—
68	C₂×Q₈×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂×Dic₉₄	—	2	2	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	2	2	no	—
82	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₇	C₂×D₄	1	2	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	C₄×Dic₄₇	—	2	2	no	—
94	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
95	#10625329	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
99	C₂×C₄₇⋊C₈	#41	1	4	yes	—
100	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
101	Q₈×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	Dic₉₄	—	4	4	no	—
114	C₈×C₄₇	—	2	2	no	—
115	C₈×C₄₇	—	4	2	no	—
116	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
117	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
118	C₄₇⋊C₈	—	4	2	no	—
119	C₄₇⋊C₈	C₄×Q₈	1	8	yes	—
120	#21735	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₄₇	#533	1	2	yes	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	4	4	no	—
127	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
128	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₄₇	#532	1	1	yes	characteristic
130	Dic₄₇	—	4	4	no	—
131	Dic₄₇	—	4	4	no	—
132	#3162	—	47	1	no	—
133	#2960	—	47	1	no	—
134	#2960	D₄₇	1	1	yes	characteristic
135	#1995	—	47	1	no	—
136	#1995	—	47	1	no	—
137	#1946	—	47	4	no	—
138	#1849	—	47	1	no	—
139	#1772	—	47	4	no	—
140	#1726	—	47	1	no	—
141	C₂×C₄₇	—	2	2	no	—
142	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₄₇	#2955	1	2	yes	—
144	#533	—	47	1	no	—
145	#533	C₂×D₄₇	1	1	yes	characteristic
146	#501	—	47	4	no	—
147	#501	—	47	4	no	—
148	#446	—	2	2	no	—
149	#446	—	94	2	no	—
150	C₈×Q₈	—	2	2	no	—
151	C₈×Q₈	—	94	2	no	—
152	C₂×#108	—	47	4	no	—
153	#427	—	47	4	no	—
154	#424	—	47	2	no	—
155	#424	C₂×D₄₇	1	2	yes	—
156	#423	—	47	1	no	—
157	#423	C₂×D₄₇	1	1	yes	characteristic
158	#405	—	47	2	no	—
159	#405	C₂×D₄₇	1	2	yes	—
160	C₂×#97	—	47	1	no	—
161	C₂×#97	—	47	2	no	—
162	C₂×C₄×C₈	—	47	2	no	—
163	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
164	#376	—	47	1	no	—
165	#339	—	47	4	no	—
166	#339	—	47	4	no	—
167	C₄₇	#21735	1	1	yes	characteristic, complemented, Sylow
168	C₂×M₁₆	—	47	2	no	—
169	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
170	C₂²×C₈	—	47	2	no	—
171	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
173	C₄⋊Q₈	—	47	2	no	—
174	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
175	#125	—	47	2	no	—
176	#125	C₄×D₄₇	1	2	yes	—
177	C₂²⋊Q₈	—	2	2	no	—
178	C₂²⋊Q₈	—	94	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	—	94	2	no	—
181	#117	—	47	2	no	—
182	#117	C₂²×D₄₇	1	2	yes	—
183	C₂×C₄²	—	47	2	no	—
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	#108	—	94	8	no	—
186	#107	—	47	8	no	—
187	#106	—	47	8	no	—
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	94	4	no	—
191	C₄⋊C₈	—	94	2	no	—
192	#97	—	2	2	no	—
193	#97	—	47	2	no	—
194	#97	—	47	2	no	—
195	#97	—	94	4	no	—
196	#97	—	94	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	47	8	no	—
199	C₄×C₈	—	94	2	no	—
200	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
201	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
202	#95	—	94	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	94	2	no	—
205	C₂²×C₄	—	94	2	no	—
206	C₂²×C₄	#10625350	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	94	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	47	4	no	—
213	C₂×C₈	—	47	4	no	—
214	C₂×C₈	—	47	4	no	—
215	C₂×C₈	—	94	2	no	—
216	C₂×C₈	Q₈×D₄₇	1	4	yes	—
217	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	47	4	no	—
223	C₄⋊C₄	—	94	4	no	—
224	C₄⋊C₄	—	94	2	no	—
225	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	94	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	47	4	no	—
230	C₄²	—	94	2	no	—
231	C₄²	—	94	2	no	—
232	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
234	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
235	C₂³	C₂×#10625350	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	188	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	94	4	no	—
241	C₂×C₄	—	94	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
244	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
245	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
248	C₂×C₄	#41×D₄₇	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	47	8	no	—
252	C₈	—	188	2	no	—
253	C₈	#13094022	1	2	yes	—
254	C₈	#13094022	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂²	#15689677	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	188	4	no	—
262	C₄	—	188	4	no	—
263	C₄	C₄×Q₈×D₄₇	1	2	yes	—
264	C₄	C₂×#13094022	1	1	yes	characteristic
265	C₄	#15689669	1	1	yes	characteristic
266	C₂	—	2	2	no	—
267	C₂	C₄×#13094022	1	1	yes	characteristic
268	C₂	#20252289	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2960	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2960	D₄₇
hypocenter	C₄₇	#21735
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
11. #15689533	C₄	semidirect
12. #15689529	C₄	semidirect
51. C₈×Dic₄₇	Q₈	semidirect
167. C₄₇	#21735	semidirect

Maximal subgroups

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

subgroup	index
#20253910	2
#20252322	2
#20244370	2
#20244368	2
#20244138	2
#20243562	2
#20243268	2
#20243074	2
C₄₇×#2960	2
#21735	47