Subgroups — #26216512

Order 12032.

778 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	#20251951	C₂	1	2	yes	maximal
2	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
3	#20243527	C₂	1	4	yes	maximal, complemented
4	#20243524	C₂	1	4	yes	maximal, complemented
5	#20243114	C₂	1	2	yes	maximal
6	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	#15689151	—	2	4	no	—
12	#15689149	—	2	4	no	—
13	#15689146	C₂²	1	8	yes	—
14	#15689078	C₄	1	4	yes	—
15	#15689078	C₄	1	4	yes	—
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	#15688888	C₂²	1	2	yes	—
19	#15688888	C₂²	1	2	yes	—
20	#15688887	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#341	C₂²	1	2	yes	—
24	C₄₇×#341	C₂²	1	2	yes	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	C₂×C₄	1	2	yes	—
30	C₂×#10625329	C₂×C₄	1	2	yes	—
31	C₂×#10625329	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	#13093983	C₂×C₄	1	2	yes	—
35	#13093983	C₂×C₄	1	2	yes	—
36	#13093983	D₄	1	2	yes	—
37	#13093983	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
39	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	Dic₄₇⋊C₈	—	2	16	no	—
42	#13093934	C₂×C₄	1	8	yes	—
43	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	4	no	—
47	C₂×M₁₆×C₄₇	—	2	4	no	—
48	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
49	C₂²×C₈×C₄₇	—	2	4	no	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	4	yes	—
62	C₂²×C₄×C₄₇	#41	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	4	4	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	4	4	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
77	C₂²⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
78	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
79	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
80	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
81	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	2	4	no	—
86	#10625329	—	4	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	4	4	no	—
90	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂³×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₇	#125	1	4	yes	—
95	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#117	1	2	yes	—
99	C₂×C₄×C₄₇	#117	1	2	yes	—
100	C₂×C₄×C₄₇	#117	1	2	yes	—
101	C₂×C₄×C₄₇	#117	1	2	yes	—
102	C₂×C₄×C₄₇	#124	1	4	yes	—
103	C₂×C₄×C₄₇	#124	1	4	yes	—
104	C₂×C₄×C₄₇	#125	1	4	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	—	2	2	no	—
113	C₈×C₄₇	—	4	4	no	—
114	C₄₇⋊C₈	—	2	4	no	—
115	C₄₇⋊C₈	—	4	4	no	—
116	#20768	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	4	4	no	—
121	C₂²×C₄₇	#382	1	2	yes	—
122	C₂²×C₄₇	#382	1	2	yes	—
123	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#125	1	2	yes	—
125	C₄×C₄₇	#382	1	2	yes	—
126	C₄×C₄₇	#382	1	2	yes	—
127	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
128	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
129	C₄×C₄₇	C₂×#124	1	2	yes	—
130	Dic₄₇	—	2	8	no	—
131	#2905	—	47	2	no	—
132	C₂×#405	—	47	1	no	—
133	#1838	—	47	4	no	—
134	#1837	—	47	4	no	—
135	#1738	—	47	2	no	—
136	C₂×#341	—	47	1	no	—
137	C₂×#341	D₄₇	1	1	yes	characteristic
138	C₂×C₄₇	—	2	2	no	—
139	C₂×C₄₇	—	2	2	no	—
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
142	C₂×C₄₇	#2916	1	2	yes	—
143	C₂²×M₁₆	—	47	1	no	—
144	C₂²×M₁₆	—	47	2	no	—
145	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
146	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
147	C₂×#117	—	47	1	no	—
148	#424	—	94	4	no	—
149	#424	—	94	4	no	—
150	#423	—	47	8	no	—
151	#405	—	47	4	no	—
152	#405	—	47	4	no	—
153	C₂×#97	—	47	2	no	—
154	C₂×C₄×C₈	—	47	2	no	—
155	#341	—	47	8	no	—
156	#341	—	47	2	no	—
157	#341	—	47	2	no	—
158	#341	C₂×D₄₇	1	2	yes	—
159	#341	C₂×D₄₇	1	2	yes	—
160	C₄₇	#20768	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	47	2	no	—
166	C₂×M₁₆	—	47	2	no	—
167	C₂×M₁₆	—	47	4	no	—
168	C₂×M₁₆	—	94	4	no	—
169	C₂×M₁₆	—	94	2	no	—
170	C₂×M₁₆	—	94	4	no	—
171	C₂×M₁₆	—	94	4	no	—
172	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	47	2	no	—
176	C₂²×C₈	—	94	2	no	—
177	C₂²×C₈	C₄×D₄₇	1	2	yes	—
178	C₂²×C₈	C₄×D₄₇	1	2	yes	—
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	#117	—	47	2	no	—
182	#117	—	47	2	no	—
183	#117	—	47	2	no	—
184	#117	—	47	2	no	—
185	C₂×C₄⋊C₄	—	47	2	no	—
186	C₂×C₂²⋊C₄	—	47	2	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₄⋊C₈	—	94	16	no	—
189	#97	—	47	8	no	—
190	C₄×C₈	—	47	8	no	—
191	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	—	47	4	no	—
193	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
195	C₂²×C₄	#10625350	1	4	yes	—
196	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	#10625352	1	4	yes	—
199	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	4	no	—
204	M₁₆	—	94	4	no	—
205	M₁₆	—	94	4	no	—
206	M₁₆	—	94	4	no	—
207	M₁₆	—	94	4	no	—
208	M₁₆	—	188	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	47	4	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	94	4	no	—
218	C₂×C₈	—	188	4	no	—
219	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
221	C₄⋊C₄	—	47	4	no	—
222	C₄⋊C₄	—	47	4	no	—
223	C₂²⋊C₄	—	47	4	no	—
224	C₂²⋊C₄	—	47	4	no	—
225	C₄²	—	47	4	no	—
226	C₄²	—	47	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
229	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
230	C₂³	#13094014	1	4	yes	—
231	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
232	C₂×C₄	—	47	8	no	—
233	C₂×C₄	—	188	4	no	—
234	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄	#13093980	1	2	yes	—
237	C₂×C₄	#13093980	1	2	yes	—
238	C₂×C₄	#13093981	1	2	yes	—
239	C₂×C₄	#13093981	1	2	yes	—
240	C₂×C₄	#13094010	1	4	yes	—
241	C₂×C₄	#13094010	1	4	yes	—
242	C₂×C₄	#13094014	1	4	yes	—
243	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#10625352	1	2	yes	—
246	C₂×C₄	C₂×#10625350	1	2	yes	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	4	no	—
250	C₈	—	94	4	no	—
251	C₈	—	188	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	#15688987	1	2	yes	—
257	C₂²	#15688987	1	2	yes	—
258	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
259	C₂²	C₂×#13094014	1	2	yes	—
260	C₄	—	94	8	no	—
261	C₄	#15688987	1	2	yes	—
262	C₄	#15688987	1	2	yes	—
263	C₄	C₂×#13093980	1	1	yes	characteristic
264	C₄	C₂×#13093981	1	1	yes	characteristic
265	C₄	C₂×#13094010	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	C₂×#15688987	1	1	yes	characteristic
270	C₂	#20251996	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#20768
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
3. #20243527	C₂	semidirect
4. #20243524	C₂	semidirect
160. C₄₇	#20768	semidirect

Maximal subgroups

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

subgroup	index
#20251951	2
C₂×#15689078	2
#20243527	2
#20243524	2
#20243114	2
C₂×#15688888	2
C₂×C₄₇×#341	2
#20768	47