Subgroups — #26220880

Order 12032.

1166 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20244358	C₂	1	4	yes	maximal, complemented
4	#20244354	C₂	1	4	yes	maximal, complemented
5	#20243107	C₂	1	4	yes	maximal
6	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₄×#10625329	—	2	4	no	—
10	C₂×#13093935	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15688993	C₂²	1	2	yes	—
13	#15688989	C₂²	1	2	yes	—
14	#15688983	—	2	2	no	—
15	C₄×C₄⋊Dic₄₇	—	2	2	no	—
16	#15688885	C₂²	1	16	yes	—
17	C₂×#125×C₄₇	—	2	2	no	—
18	C₂×#125×C₄₇	—	2	4	no	—
19	C₂×#125×C₄₇	C₄	1	8	yes	complemented
20	C₂×#125×C₄₇	C₂²	1	2	yes	—
21	C₂×#125×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	Q₈	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	#13093983	—	2	8	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	#13093935	—	2	4	no	—
35	C₄×C₄₇⋊C₈	—	2	4	no	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₂³×C₄×C₄₇	—	2	2	no	—
38	C₂³×C₄×C₄₇	D₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	#125×C₄₇	—	2	16	no	—
41	#125×C₄₇	—	2	4	no	—
42	#125×C₄₇	—	2	4	no	—
43	#125×C₄₇	C₂×C₄	1	8	yes	—
44	#125×C₄₇	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	D₄	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	—	4	8	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	2	16	no	—
73	C₄⋊C₄×C₄₇	—	2	16	no	—
74	C₄⋊C₄×C₄₇	—	2	8	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	C₂×D₄	1	2	yes	—
81	C₄²×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²⋊Dic₄₇	—	4	4	no	—
83	C₄⋊Dic₄₇	—	2	8	no	—
84	C₄⋊Dic₄₇	—	4	8	no	—
85	C₄×Dic₄₇	—	4	4	no	—
86	#10625329	—	2	16	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂³×C₄₇	#124	1	2	yes	—
93	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
94	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	4	8	no	—
99	C₂×C₄×C₄₇	—	4	16	no	—
100	C₂×C₄×C₄₇	—	4	8	no	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
106	C₂×C₄×C₄₇	#124	1	2	yes	—
107	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	8	no	—
114	C₄₇⋊C₈	—	2	8	no	—
115	#21325	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	#391	1	4	yes	—
120	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₄₇	#575	1	4	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	2	4	no	—
126	C₄×C₄₇	—	4	8	no	—
127	C₄×C₄₇	#391	1	4	yes	—
128	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
129	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
130	Dic₄₇	—	8	4	no	—
131	C₂²×#125	D₄₇	1	1	yes	characteristic
132	C₂×C₄×M₁₆	—	47	1	no	—
133	#2285	—	47	1	no	—
134	#1992	—	47	4	no	—
135	#1991	—	47	4	no	—
136	#1736	—	47	4	no	—
137	C₂×C₄₇	—	2	4	no	—
138	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
139	C₂×C₄₇	#2896	1	2	yes	—
140	C₂×C₄₇	#3142	1	2	yes	—
141	C₂×C₄₇	#3166	1	2	yes	—
142	C₂²×M₁₆	—	47	2	no	—
143	C₂×#125	—	2	2	no	—
144	C₂×#125	—	2	4	no	—
145	C₂×#125	Dic₄₇	1	8	yes	complemented
146	C₂×#125	C₂×D₄₇	1	2	yes	—
147	C₂×#125	C₂×D₄₇	1	2	yes	—
148	C₂×#117	—	47	2	no	—
149	C₂²×C₄⋊C₄	—	2	2	no	—
150	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
151	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
152	C₄×M₁₆	—	94	4	no	—
153	C₂×#97	—	47	2	no	—
154	C₂×C₄×C₈	—	47	2	no	—
155	#384	—	47	2	no	—
156	#382	—	47	2	no	—
157	#380	—	94	2	no	—
158	C₄×C₄⋊C₄	—	94	2	no	—
159	#340	—	47	16	no	—
160	C₄₇	#21325	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
164	C₂×M₁₆	—	47	8	no	—
165	C₂×M₁₆	—	94	4	no	—
166	C₂×M₁₆	—	94	4	no	—
167	C₂²×C₈	—	47	4	no	—
168	#125	—	2	16	no	—
169	#125	—	2	4	no	—
170	#125	—	2	4	no	—
171	#125	C₂×Dic₄₇	1	8	yes	—
172	#125	C₂×Dic₄₇	1	8	yes	—
173	#117	—	94	8	no	—
174	C₂×C₄⋊C₄	—	2	16	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	47	4	no	—
180	C₂×C₄⋊C₄	—	94	4	no	—
181	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
182	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
183	C₂×C₂²⋊C₄	—	94	2	no	—
184	C₂×C₄²	—	94	2	no	—
185	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
186	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
188	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
190	#97	—	94	4	no	—
191	C₄×C₈	—	94	4	no	—
192	#95	—	94	4	no	—
193	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	—	4	8	no	—
198	C₂²×C₄	—	94	4	no	—
199	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
200	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
201	C₂²×C₄	#10625354	1	4	yes	—
202	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	M₁₆	—	94	16	no	—
209	C₂×C₈	—	47	8	no	—
210	C₂×C₈	—	94	4	no	—
211	C₂×C₈	—	94	4	no	—
212	C₄⋊C₄	—	2	16	no	—
213	C₄⋊C₄	—	2	16	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	94	8	no	—
217	C₄⋊C₄	—	188	8	no	—
218	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
219	C₂²⋊C₄	—	188	4	no	—
220	C₄²	—	188	4	no	—
221	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
222	C₄²	C₂²×Dic₄₇	1	2	yes	—
223	C₄²	C₂²×Dic₄₇	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
225	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
228	C₂³	#13094013	1	2	yes	—
229	C₂³	#13094022	1	2	yes	—
230	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	16	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	188	4	no	—
238	C₂×C₄	—	188	8	no	—
239	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
241	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
242	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
243	C₂×C₄	#13094000	1	8	yes	—
244	C₂×C₄	#13094013	1	2	yes	—
245	C₂×C₄	#13094022	1	2	yes	—
246	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
249	C₂×C₄	C₂×#10625354	1	2	yes	—
250	C₈	—	94	8	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#15689030	1	4	yes	—
255	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
256	C₂²	C₂×#13094013	1	1	yes	characteristic
257	C₂²	C₂×#13094022	1	1	yes	characteristic
258	C₂²	#15689921	1	4	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	376	4	no	—
263	C₄	#15689030	1	4	yes	—
264	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
265	C₄	C₂×#13094000	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	C₂×#15689030	1	1	yes	characteristic
268	C₂	#20251907	1	2	yes	—
269	C₂	#20253791	1	2	yes	—
270	C₂	#20253937	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	#21325
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
3. #20244358	C₂	semidirect
4. #20244354	C₂	semidirect
19. C₂×#125×C₄₇	C₄	semidirect
145. C₂×#125	Dic₄₇	semidirect
160. C₄₇	#21325	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
#20245795	2
#20244358	2
#20244354	2
#20243107	2
C₂²×#125×C₄₇	2
#21325	47