Subgroups — #26216890

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252017	C₂	1	2	yes	maximal
2	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243561	C₂	1	2	yes	maximal, complemented
4	#20243527	C₂	1	2	yes	maximal, complemented
5	#20243430	C₂	1	1	yes	maximal, characteristic
6	#20243428	C₂	1	2	yes	maximal
7	#108×Dic₄₇	C₂	1	2	yes	maximal, complemented
8	#20243289	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1811	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15689177	—	2	4	no	—
13	#15689173	—	2	4	no	—
14	C₂×#13093960	C₂²	1	2	yes	—
15	C₂×#13093959	C₂²	1	4	yes	—
16	#15689146	C₂²	1	4	yes	—
17	#15689087	C₂²	1	2	yes	—
18	M₁₆×Dic₄₇	C₄	1	8	yes	—
19	M₁₆×Dic₄₇	C₂²	1	4	yes	—
20	C₂×#13093934	C₂²	1	1	yes	characteristic
21	C₂×#13093934	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688888	C₂²	1	2	yes	—
25	#15688887	C₂²	1	4	yes	—
26	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#407	C₂²	1	2	yes	—
29	C₄₇×#341	C₂²	1	2	yes	—
30	C₂³×Dic₄₇	Q₈	1	2	yes	—
31	C₂×#10625329	—	2	4	no	—
32	C₂×#10625329	C₂×C₄	1	4	yes	—
33	C₂×#10625329	C₂³	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
42	#13093960	—	2	4	no	—
43	#13093959	—	2	8	no	—
44	Dic₄₇⋊C₈	—	2	8	no	—
45	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
46	#13093934	—	2	4	no	—
47	#13093934	C₂×C₄	1	4	yes	—
48	#13093934	C₂×C₄	1	4	yes	—
49	C₈×Dic₄₇	—	2	4	no	—
50	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
51	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₄₇	—	2	4	no	—
54	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	#108×C₄₇	—	2	4	no	—
59	C₂²⋊C₈×C₄₇	—	2	4	no	—
60	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	4	4	no	—
68	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
70	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
72	M₁₆×C₄₇	—	2	8	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	M₁₆×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	4	yes	—
80	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
81	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
82	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
83	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
87	#10625329	—	2	8	no	—
88	#10625329	—	2	4	no	—
89	#10625329	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₄₇⋊C₈	#41	1	4	yes	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×Dic₄₇	—	2	8	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	2	no	—
112	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₈×C₄₇	—	2	2	no	—
116	C₈×C₄₇	—	4	4	no	—
117	C₈×C₄₇	#117	1	4	yes	—
118	C₄₇⋊C₈	—	2	2	no	—
119	C₄₇⋊C₈	—	4	4	no	—
120	C₄₇⋊C₈	#117	1	4	yes	—
121	#20827	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	4	4	no	—
126	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₇	#533	1	2	yes	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
131	C₄×C₄₇	#532	1	2	yes	—
132	Dic₄₇	—	2	4	no	—
133	Dic₄₇	—	2	2	no	—
134	Dic₄₇	#430	1	4	yes	—
135	#2920	—	47	2	no	—
136	C₂×C₄×M₁₆	—	47	1	no	—
137	#1849	—	47	2	no	—
138	#1838	—	47	2	no	—
139	#1811	—	47	2	no	—
140	#1811	—	47	1	no	—
141	#1811	D₄₇	1	1	yes	characteristic
142	C₄×#108	—	47	2	no	—
143	C₄×#108	—	47	2	no	—
144	C₂×C₄₇	—	2	2	no	—
145	C₂×C₄₇	—	4	2	no	—
146	C₂×C₄₇	#2285	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#430	1	2	yes	—
148	C₂²×M₁₆	—	47	1	no	—
149	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
150	C₂²×C₄²	—	47	1	no	—
151	#432	—	94	4	no	—
152	#431	—	94	4	no	—
153	C₂×#108	—	47	4	no	—
154	C₂×#108	—	47	2	no	—
155	C₂×#108	C₂×D₄₇	1	2	yes	—
156	#423	—	47	4	no	—
157	#407	—	47	2	no	—
158	#407	C₂×D₄₇	1	2	yes	—
159	C₄×M₁₆	—	47	8	no	—
160	C₄×M₁₆	—	47	4	no	—
161	C₂×#97	—	47	1	no	—
162	C₂×#97	—	47	1	no	—
163	C₂×C₄×C₈	—	47	1	no	—
164	C₂×C₄×C₈	—	47	1	no	—
165	#341	—	47	4	no	—
166	#341	—	47	2	no	—
167	#341	C₂×D₄₇	1	2	yes	—
168	C₄₇	#20827	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	47	2	no	—
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	47	4	no	—
174	C₂×M₁₆	—	94	4	no	—
175	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
176	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
177	C₂²×C₈	—	47	1	no	—
178	C₂²×C₈	—	47	1	no	—
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	—	47	4	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	2	no	—
184	C₂×C₄²	—	47	2	no	—
185	C₂×C₄²	—	47	1	no	—
186	C₂×C₄²	—	47	1	no	—
187	#108	—	2	4	no	—
188	#108	—	94	8	no	—
189	#108	—	94	4	no	—
190	C₄⋊C₈	—	94	8	no	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₂²⋊C₈	—	94	4	no	—
193	#97	—	47	4	no	—
194	#97	—	47	4	no	—
195	#97	—	94	4	no	—
196	C₄×C₈	—	47	4	no	—
197	C₄×C₈	—	47	4	no	—
198	C₄×C₈	—	94	4	no	—
199	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	47	4	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	47	4	no	—
205	C₂²×C₄	—	47	2	no	—
206	C₂²×C₄	—	188	4	no	—
207	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
208	C₂²×C₄	D₄×D₄₇	1	2	yes	—
209	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
211	M₁₆	—	2	8	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	94	8	no	—
215	M₁₆	—	94	4	no	—
216	M₁₆	—	94	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	47	2	no	—
220	C₂×C₈	—	47	4	no	—
221	C₂×C₈	—	47	2	no	—
222	C₂×C₈	—	94	2	no	—
223	C₂×C₈	—	94	4	no	—
224	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
226	C₂×C₈	#10625352	1	4	yes	—
227	C₄²	—	47	4	no	—
228	C₄²	—	47	4	no	—
229	C₄²	—	47	2	no	—
230	C₄²	—	47	2	no	—
231	C₄²	—	47	1	no	—
232	C₄²	—	47	1	no	—
233	C₄²	—	47	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
237	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	47	2	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	94	4	no	—
246	C₂×C₄	—	94	8	no	—
247	C₂×C₄	—	188	2	no	—
248	C₂×C₄	—	188	4	no	—
249	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
250	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
252	C₂×C₄	#41×D₄₇	1	2	yes	—
253	C₂×C₄	C₂×#10625352	1	2	yes	—
254	C₈	—	2	2	no	—
255	C₈	—	4	4	no	—
256	C₈	—	47	4	no	—
257	C₈	—	94	2	no	—
258	C₈	—	188	4	no	—
259	C₈	#13093980	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
265	C₂²	#15689675	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	47	4	no	—
268	C₄	—	94	2	no	—
269	C₄	—	94	4	no	—
270	C₄	#117×D₄₇	1	1	yes	characteristic
271	C₄	C₂×#13093980	1	1	yes	characteristic
272	C₄	#15689668	1	2	yes	—
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	#20245790	1	1	yes	characteristic
276	C₂	#430×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1811	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1811	D₄₇
hypocenter	C₄₇	#20827
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1811

Complements

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

N	H ≅ G/N	type
3. #20243561	C₂	semidirect
4. #20243527	C₂	semidirect
7. #108×Dic₄₇	C₂	semidirect
8. #20243289	C₂	semidirect
168. C₄₇	#20827	semidirect

Maximal subgroups

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

subgroup	index
#20252017	2
C₂×M₁₆×Dic₄₇	2
#20243561	2
#20243527	2
#20243430	2
#20243428	2
#108×Dic₄₇	2
#20243289	2
C₄₇×#1811	2
#20827	47