Subgroups — #26225082

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×Dic₄₇	C₂	1	2	yes	maximal
2	#20244354	C₂	1	2	yes	maximal
3	#20244132	C₂	1	2	yes	maximal
4	#20244127	C₂	1	2	yes	maximal
5	#20243552	C₂	1	1	yes	maximal, characteristic
6	#20243288	C₂	1	2	yes	maximal
7	#20243287	C₂	1	2	yes	maximal
8	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#499	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093958	C₂²	1	2	yes	—
11	C₂×#13093956	C₂²	1	2	yes	—
12	C₂×#13093934	C₂²	1	1	yes	characteristic
13	C₂×#13093934	C₂²	1	1	yes	characteristic
14	C₂×#13093934	C₂²	1	1	yes	characteristic
15	C₂×#13093934	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15689022	C₂²	1	4	yes	—
18	#15689009	—	2	4	no	—
19	#15688989	C₂²	1	2	yes	—
20	C₄⋊C₄×Dic₄₇	—	2	4	no	—
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15688885	C₂²	1	4	yes	—
24	#15688885	C₂²	1	4	yes	—
25	C₂×#125×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#499	C₄	1	8	yes	complemented
27	C₂×#107×C₄₇	C₂²	1	2	yes	—
28	C₂×#106×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
38	#13093958	—	2	4	no	—
39	#13093956	—	2	4	no	—
40	#13093934	—	2	2	no	—
41	#13093934	—	2	2	no	—
42	#13093934	Q₈	1	4	yes	—
43	#13093934	Q₈	1	4	yes	—
44	C₄×C₄₇⋊C₈	—	2	2	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	#125×C₄₇	—	2	4	no	—
51	#125×C₄₇	C₂×C₄	1	8	yes	—
52	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	#107×C₄₇	C₂×C₄	1	8	yes	—
56	#106×C₄₇	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
63	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₄₇	#41	1	2	yes	—
69	C₂×C₈×C₄₇	#41	1	2	yes	—
70	C₄⋊C₄×C₄₇	—	2	16	no	—
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	#41	1	2	yes	—
75	C₄⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	2	4	no	—
77	C₄⋊Dic₄₇	—	4	8	no	—
78	C₄×Dic₄₇	—	2	1	no	—
79	C₄×Dic₄₇	—	2	1	no	—
80	C₄×Dic₄₇	—	4	4	no	—
81	C₄×Dic₄₇	—	4	8	no	—
82	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
83	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	C₂×C₄₇⋊C₈	—	2	1	no	—
86	C₂×C₄₇⋊C₈	—	2	1	no	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₄₇	—	2	4	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	4	4	no	—
106	C₂×Dic₄₇	—	4	4	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
109	C₈×C₄₇	C₄×Q₈	1	4	yes	—
110	C₈×C₄₇	C₄×Q₈	1	4	yes	—
111	C₄₇⋊C₈	—	2	2	no	—
112	C₄₇⋊C₈	—	2	2	no	—
113	#21728	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄₇	#532	1	1	yes	characteristic
117	C₂²×C₄₇	#575	1	4	yes	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	4	8	no	—
120	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₄₇	#533	1	1	yes	characteristic
123	C₄×C₄₇	#533	1	1	yes	characteristic
124	Dic₄₇	—	2	4	no	—
125	Dic₄₇	—	4	2	no	—
126	Dic₄₇	—	8	4	no	—
127	C₂×#499	D₄₇	1	1	yes	characteristic
128	C₄×#125	—	47	2	no	—
129	#1991	—	47	2	no	—
130	#1944	—	47	2	no	—
131	#1943	—	47	2	no	—
132	#1846	—	47	1	no	—
133	#1776	—	47	2	no	—
134	#1776	—	47	2	no	—
135	C₄×#97	—	47	1	no	—
136	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₄₇	#2943	1	2	yes	—
138	C₂×C₄₇	#3166	1	2	yes	—
139	C₂×C₄₇	#3183	1	2	yes	—
140	C₂×#125	C₂×D₄₇	1	2	yes	—
141	#499	Dic₄₇	1	8	yes	complemented
142	C₂×#107	—	47	2	no	—
143	C₂×#107	C₂×D₄₇	1	2	yes	—
144	C₂×#106	—	47	2	no	—
145	C₂×#106	C₂×D₄₇	1	2	yes	—
146	C₂×#97	—	47	1	no	—
147	C₂×#97	—	47	1	no	—
148	C₂×#97	—	47	1	no	—
149	C₂×#97	—	47	1	no	—
150	C₂×C₄×C₈	—	47	1	no	—
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	#389	—	47	4	no	—
153	#387	—	94	4	no	—
154	#382	—	47	2	no	—
155	C₄×C₄⋊C₄	—	47	4	no	—
156	C₄×C₄⋊C₄	—	94	4	no	—
157	C₄³	—	47	1	no	—
158	#340	—	47	4	no	—
159	#340	—	47	4	no	—
160	C₄₇	#21728	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₈	—	47	1	no	—
162	C₂²×C₈	—	47	1	no	—
163	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
165	#125	—	2	4	no	—
166	#125	C₂×Dic₄₇	1	8	yes	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	47	4	no	—
169	C₂×C₄⋊C₄	—	94	4	no	—
170	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
171	C₂×C₄²	—	47	1	no	—
172	C₂×C₄²	—	47	1	no	—
173	C₂×C₄²	—	47	4	no	—
174	C₂×C₄²	—	94	4	no	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	#107	—	94	4	no	—
177	#107	C₂×Dic₄₇	1	8	yes	—
178	#106	—	94	4	no	—
179	#106	C₂×Dic₄₇	1	8	yes	—
180	#97	—	47	4	no	—
181	#97	—	47	4	no	—
182	#97	—	94	2	no	—
183	#97	—	94	2	no	—
184	C₄×C₈	—	94	2	no	—
185	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
186	#95	—	94	4	no	—
187	#95	—	94	4	no	—
188	#95	—	94	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	—	94	4	no	—
193	C₂²×C₄	D₄×D₄₇	1	2	yes	—
194	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
195	C₂×C₈	—	47	2	no	—
196	C₂×C₈	—	47	2	no	—
197	C₂×C₈	—	94	1	no	—
198	C₂×C₈	—	94	1	no	—
199	C₂×C₈	—	94	1	no	—
200	C₂×C₈	—	94	1	no	—
201	C₂×C₈	Q₈×D₄₇	1	2	yes	—
202	C₂×C₈	Q₈×D₄₇	1	2	yes	—
203	C₂×C₈	#10625354	1	2	yes	—
204	C₂×C₈	#10625354	1	2	yes	—
205	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
206	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	94	4	no	—
210	C₄⋊C₄	—	94	4	no	—
211	C₄⋊C₄	—	188	8	no	—
212	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
213	C₄²	—	47	2	no	—
214	C₄²	—	47	2	no	—
215	C₄²	—	94	1	no	—
216	C₄²	—	94	1	no	—
217	C₄²	—	188	4	no	—
218	C₄²	—	188	8	no	—
219	C₄²	#10625352	1	2	yes	—
220	C₄²	C₂²×Dic₄₇	1	2	yes	—
221	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	47	4	no	—
227	C₂×C₄	—	94	4	no	—
228	C₂×C₄	—	94	4	no	—
229	C₂×C₄	—	94	2	no	—
230	C₂×C₄	—	188	4	no	—
231	C₂×C₄	—	188	4	no	—
232	C₂×C₄	—	188	4	no	—
233	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
234	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
238	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
239	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
240	C₈	—	94	2	no	—
241	C₈	—	94	2	no	—
242	C₈	Q₈×Dic₄₇	1	4	yes	—
243	C₈	Q₈×Dic₄₇	1	4	yes	—
244	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
246	C₂²	#15689674	1	1	yes	characteristic
247	C₂²	#15689920	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	4	8	no	—
250	C₄	—	94	4	no	—
251	C₄	—	376	4	no	—
252	C₄	—	188	2	no	—
253	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
254	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
255	C₄	#15689678	1	1	yes	characteristic
256	C₄	#15689678	1	1	yes	characteristic
257	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
258	C₂	#20252189	1	2	yes	—
259	C₂	#20253936	1	2	yes	—
260	C₂	#20254119	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#499	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#499	D₄₇
hypocenter	C₄₇	#21728
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
26. C₄₇×#499	C₄	semidirect
141. #499	Dic₄₇	semidirect
160. C₄₇	#21728	semidirect

Maximal subgroups

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

subgroup	index
#125×Dic₄₇	2
#20244354	2
#20244132	2
#20244127	2
#20243552	2
#20243288	2
#20243287	2
C₄×#13093934	2
C₂×C₄₇×#499	2
#21728	47