Subgroups — #26216914

Order 12032.

770 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	#20252019	C₂	1	2	yes	maximal
2	C₂×#97×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243561	C₂	1	2	yes	maximal, complemented
4	#20243527	C₂	1	2	yes	maximal, complemented
5	#20243425	C₂	1	2	yes	maximal
6	#20243423	C₂	1	2	yes	maximal
7	#20243295	C₂	1	2	yes	maximal, complemented
8	#20243294	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	#15689178	—	2	4	no	—
14	#15689174	—	2	4	no	—
15	C₂×#13093960	C₂²	1	2	yes	—
16	C₂×#13093959	C₂²	1	4	yes	—
17	#15689146	C₂²	1	4	yes	—
18	#15689086	C₂²	1	4	yes	—
19	C₄×#10625329	C₂²	1	2	yes	—
20	C₂×#13093935	C₂²	1	1	yes	characteristic
21	C₂×#13093934	C₂²	1	1	yes	characteristic
22	C₂×#13093934	C₂²	1	1	yes	characteristic
23	#97×D₄₇	C₄	1	8	yes	—
24	#15688888	C₂²	1	2	yes	—
25	#15688887	C₂²	1	4	yes	—
26	C₂×#108×C₄₇	C₂²	1	2	yes	—
27	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#341	C₂²	1	2	yes	—
30	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
35	C₂×C₈×D₄₇	—	2	4	no	—
36	C₂×C₈×D₄₇	—	2	4	no	—
37	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₄²×D₄₇	C₂×C₄	1	4	yes	—
42	C₄²×D₄₇	D₄	1	4	yes	—
43	#13093960	—	2	4	no	—
44	#13093959	—	2	8	no	—
45	Dic₄₇⋊C₈	—	2	8	no	—
46	(C₂×D₄₇)⋊C₈	—	2	8	no	—
47	#13093935	—	2	2	no	—
48	#13093935	C₂×C₄	1	4	yes	—
49	#13093934	C₂×C₄	1	4	yes	—
50	#13093934	C₂×C₄	1	4	yes	—
51	C₄×C₄₇⋊C₈	—	2	2	no	—
52	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#108×C₄₇	—	2	4	no	—
57	#97×C₄₇	—	2	2	no	—
58	#97×C₄₇	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₄₇	—	2	2	no	—
60	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
65	C₂×C₄×D₄₇	—	2	2	no	—
66	C₂×C₄×D₄₇	—	2	2	no	—
67	C₂×C₄×D₄₇	—	4	4	no	—
68	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
70	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	#41	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	C₂×D₄	1	2	yes	—
81	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
82	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
83	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
84	#10625329	—	2	4	no	—
85	#10625329	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₄₇⋊C₈	#41	1	4	yes	—
91	C₈×D₄₇	—	2	8	no	—
92	C₈×D₄₇	—	2	8	no	—
93	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
94	C₂²×D₄₇	—	2	2	no	—
95	C₂²×D₄₇	—	2	4	no	—
96	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
107	C₄×D₄₇	—	2	8	no	—
108	C₄×D₄₇	—	2	2	no	—
109	C₄×D₄₇	—	2	2	no	—
110	C₄×D₄₇	—	2	4	no	—
111	C₄×D₄₇	—	4	4	no	—
112	C₈×C₄₇	—	2	2	no	—
113	C₈×C₄₇	—	4	4	no	—
114	C₈×C₄₇	#117	1	4	yes	—
115	C₄₇⋊C₈	—	2	2	no	—
116	C₄₇⋊C₈	—	4	4	no	—
117	C₄₇⋊C₈	#117	1	4	yes	—
118	#20830	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₇	#533	1	2	yes	—
122	C₂×D₄₇	—	2	8	no	—
123	C₂×D₄₇	—	2	2	no	—
124	C₂×D₄₇	—	2	2	no	—
125	C₂×D₄₇	#430	1	4	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
128	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
129	C₄×C₄₇	#532	1	2	yes	—
130	Dic₄₇	—	2	4	no	—
131	Dic₄₇	—	2	2	no	—
132	#2920	—	47	2	no	—
133	C₂²×#97	—	47	1	no	—
134	#1849	—	47	2	no	—
135	#1838	—	47	2	no	—
136	#1810	—	47	2	no	—
137	#1810	—	47	2	no	—
138	#1778	—	47	2	no	—
139	#1778	—	47	1	no	—
140	#1778	D₄₇	1	1	yes	characteristic, complemented
141	C₂×C₄₇	—	2	2	no	—
142	C₂×C₄₇	#2285	1	1	yes	characteristic
143	C₂×C₄₇	C₂×#430	1	2	yes	—
144	D₄₇	—	2	4	no	—
145	C₂³×C₈	—	47	1	no	—
146	C₂³×C₈	—	47	1	no	—
147	C₂²×C₄²	—	47	1	no	—
148	#432	—	94	4	no	—
149	#431	—	94	4	no	—
150	C₂×#108	—	47	4	no	—
151	C₂×#108	—	47	2	no	—
152	C₂×#108	C₂×D₄₇	1	2	yes	—
153	#423	—	47	4	no	—
154	#407	—	47	4	no	—
155	C₄×M₁₆	—	47	2	no	—
156	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
157	C₂×#97	—	47	1	no	—
158	C₂×#97	—	47	1	no	—
159	C₂×#97	—	47	1	no	—
160	C₂×#97	—	47	8	no	—
161	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
162	#341	—	47	4	no	—
163	#341	—	47	2	no	—
164	#341	C₂×D₄₇	1	2	yes	—
165	C₄₇	#20830	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	—	47	1	no	—
167	C₂³×C₄	—	47	2	no	—
168	C₂×M₁₆	—	47	4	no	—
169	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
170	C₂²×C₈	—	47	1	no	—
171	C₂²×C₈	—	47	1	no	—
172	C₂²×C₈	—	47	4	no	—
173	C₂²×C₈	—	47	4	no	—
174	C₂²×C₈	—	94	4	no	—
175	C₂²×C₈	—	94	4	no	—
176	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	47	1	no	—
180	C₂×C₄²	—	47	4	no	—
181	C₂×C₄²	—	47	4	no	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	#108	—	2	4	no	—
184	#108	—	94	8	no	—
185	#108	—	94	4	no	—
186	C₄⋊C₈	—	94	8	no	—
187	C₂²⋊C₈	—	94	8	no	—
188	#97	—	2	2	no	—
189	#97	—	47	4	no	—
190	#97	—	47	4	no	—
191	#97	—	47	4	no	—
192	#97	—	94	2	no	—
193	#97	C₄×D₄₇	1	4	yes	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	94	2	no	—
196	C₂⁴	—	47	1	no	—
197	C₂²×C₄	—	47	1	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	47	2	no	—
201	C₂²×C₄	—	47	2	no	—
202	C₂²×C₄	—	47	4	no	—
203	C₂²×C₄	—	94	2	no	—
204	C₂²×C₄	—	94	2	no	—
205	C₂²×C₄	—	188	4	no	—
206	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	94	4	no	—
211	M₁₆	—	94	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	47	2	no	—
215	C₂×C₈	—	47	4	no	—
216	C₂×C₈	—	47	2	no	—
217	C₂×C₈	—	94	8	no	—
218	C₂×C₈	—	94	2	no	—
219	C₂×C₈	—	94	8	no	—
220	C₂×C₈	—	94	4	no	—
221	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
223	C₂×C₈	#10625350	1	4	yes	—
224	C₄²	—	47	8	no	—
225	C₄²	—	47	2	no	—
226	C₄²	—	47	2	no	—
227	C₄²	C₂×C₄×D₄₇	1	2	yes	—
228	C₄²	D₄×D₄₇	1	2	yes	—
229	C₂³	—	47	2	no	—
230	C₂³	—	94	4	no	—
231	C₂³	—	94	2	no	—
232	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	47	2	no	—
235	C₂×C₄	—	47	4	no	—
236	C₂×C₄	—	94	8	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	4	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	2	no	—
243	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
244	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
246	C₂×C₄	#41×D₄₇	1	2	yes	—
247	C₂×C₄	C₂×#10625350	1	2	yes	—
248	C₈	—	2	2	no	—
249	C₈	—	4	4	no	—
250	C₈	—	47	4	no	—
251	C₈	—	94	2	no	—
252	C₈	—	188	4	no	—
253	C₈	#13093981	1	4	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	47	4	no	—
256	C₂²	—	94	8	no	—
257	C₂²	—	94	2	no	—
258	C₂²	—	94	2	no	—
259	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
260	C₂²	#15689677	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	94	2	no	—
263	C₄	—	94	4	no	—
264	C₄	#117×D₄₇	1	1	yes	characteristic
265	C₄	C₂×#13093981	1	1	yes	characteristic
266	C₄	#15689670	1	2	yes	—
267	C₂	—	2	2	no	—
268	C₂	—	94	4	no	—
269	C₂	#20245792	1	1	yes	characteristic
270	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×#1778	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1778	D₄₇
hypocenter	C₄₇	#20830
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

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. #20243295	C₂	semidirect
8. #20243294	C₂	semidirect
9. C₄₇×#1778	C₂	semidirect
140. #1778	D₄₇	semidirect
165. C₄₇	#20830	semidirect

Maximal subgroups

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

subgroup	index
#20252019	2
C₂×#97×D₄₇	2
#20243561	2
#20243527	2
#20243425	2
#20243423	2
#20243295	2
#20243294	2
C₄₇×#1778	2
#20830	47