Subgroups — #26105745

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689179	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241223	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241222	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	C₄	1	2	yes	—
7	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
8	#15689179	—	2	4	no	—
9	#15689179	C₄	1	8	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
12	C₂×#13093934	—	2	2	no	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
15	C₄×C₈×D₄₇	—	2	4	no	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×D₄₇	Q₈	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₈×D₄₇	—	2	4	no	—
24	C₂×C₈×D₄₇	—	2	4	no	—
25	C₂×C₈×D₄₇	—	2	4	no	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
29	C₄²×D₄₇	C₂×C₄	1	4	yes	—
30	C₄²×D₄₇	D₄	1	4	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
33	(C₂×D₄₇)⋊C₈	—	2	8	no	—
34	#13093934	—	2	8	no	—
35	C₄×C₄₇⋊C₈	—	2	2	no	—
36	C₈×Dic₄₇	—	2	4	no	—
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₇	—	2	2	no	—
41	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₄₇	—	2	2	no	—
43	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×D₄₇	—	2	4	no	—
51	C₂×C₄×D₄₇	—	2	2	no	—
52	C₂×C₄×D₄₇	—	2	2	no	—
53	C₂×C₄×D₄₇	—	2	1	no	—
54	C₂×C₄×D₄₇	—	2	2	no	—
55	C₂×C₄×D₄₇	—	2	1	no	—
56	C₂×C₄×D₄₇	C₄²	1	2	yes	—
57	C₂×C₄×D₄₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	—	4	2	no	—
64	C₂×C₈×C₄₇	—	4	2	no	—
65	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
66	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₇	C₂×D₄	1	2	yes	—
68	C₄×Dic₄₇	C₄²	1	2	yes	—
69	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
70	C₄×Dic₄₇	C₂²⋊C₄	1	4	yes	—
71	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
72	C₂×C₄₇⋊C₈	—	2	2	no	—
73	C₂×C₄₇⋊C₈	—	2	2	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
78	C₈×D₄₇	—	2	16	no	—
79	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×D₄₇	—	2	2	no	—
81	C₂²×D₄₇	—	2	1	no	—
82	C₂²×D₄₇	—	2	1	no	—
83	C₂²×D₄₇	#95	1	2	yes	—
84	C₂²×D₄₇	#108	1	4	yes	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#135	1	2	yes	—
94	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
95	C₂×Dic₄₇	—	2	2	no	—
96	C₂×Dic₄₇	—	2	1	no	—
97	C₂×Dic₄₇	—	2	1	no	—
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	#95	1	2	yes	—
100	C₂×Dic₄₇	#95	1	2	yes	—
101	C₂×Dic₄₇	#95	1	2	yes	—
102	C₄×D₄₇	—	2	8	no	—
103	C₄×D₄₇	—	2	2	no	—
104	C₄×D₄₇	—	2	2	no	—
105	C₄×D₄₇	—	2	4	no	—
106	C₄×D₄₇	—	2	4	no	—
107	C₄×D₄₇	—	2	4	no	—
108	C₈×C₄₇	—	2	4	no	—
109	C₈×C₄₇	—	4	4	no	—
110	C₄₇⋊C₈	—	2	4	no	—
111	C₄₇⋊C₈	—	4	4	no	—
112	#7875	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
114	C₂²×C₄₇	#423	1	2	yes	—
115	C₂²×C₄₇	#427	1	1	yes	characteristic
116	C₂²×C₄₇	#427	1	1	yes	characteristic
117	C₂²×C₄₇	C₂×#108	1	2	yes	—
118	C₂×D₄₇	—	2	4	no	—
119	C₂×D₄₇	—	2	4	no	—
120	C₂×D₄₇	—	2	2	no	—
121	C₂×D₄₇	—	2	2	no	—
122	C₂×D₄₇	#341	1	4	yes	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
125	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₄₇	#407	1	2	yes	—
127	C₄×C₄₇	#416	1	2	yes	—
128	C₄×C₄₇	#416	1	2	yes	—
129	Dic₄₇	—	2	2	no	—
130	Dic₄₇	—	2	2	no	—
131	Dic₄₇	—	2	2	no	—
132	C₂×#432	—	47	2	no	—
133	C₂²×C₄×C₈	—	47	1	no	—
134	#1277	—	47	2	no	—
135	#1277	—	47	1	no	—
136	#1277	D₄₇	1	1	yes	characteristic, complemented
137	C₂×C₄₇	#1568	1	2	yes	—
138	C₂×C₄₇	#1568	1	2	yes	—
139	C₂×C₄₇	#1728	1	1	yes	characteristic
140	C₂×C₄₇	#1735	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
142	D₄₇	—	2	4	no	—
143	C₂³×C₈	—	47	2	no	—
144	C₂²×C₄²	—	47	1	no	—
145	#432	—	47	8	no	—
146	#432	—	94	4	no	—
147	C₂×C₄⋊C₈	—	47	2	no	—
148	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
149	C₂×C₂²⋊C₈	—	94	2	no	—
150	C₂×#97	—	94	2	no	—
151	C₂×C₄×C₈	—	47	2	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₂×C₄×C₈	—	94	4	no	—
154	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₄₇	#7875	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₄	—	47	1	no	—
157	C₂³×C₄	—	47	1	no	—
158	C₂³×C₄	—	47	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	47	2	no	—
161	C₂²×C₈	—	94	4	no	—
162	C₂²×C₈	—	94	4	no	—
163	C₂²×C₈	—	94	4	no	—
164	C₂²×C₈	—	94	2	no	—
165	C₂²×C₈	C₄×D₄₇	1	2	yes	—
166	C₂×C₄²	—	47	1	no	—
167	C₂×C₄²	—	47	1	no	—
168	C₂×C₄²	—	47	1	no	—
169	C₂×C₄²	—	47	4	no	—
170	C₂×C₄²	—	47	4	no	—
171	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	47	4	no	—
174	C₄⋊C₈	—	94	2	no	—
175	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
176	C₂²⋊C₈	—	94	8	no	—
177	#97	—	94	8	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	94	4	no	—
180	C₄×C₈	—	94	2	no	—
181	C₂⁴	—	47	1	no	—
182	C₂²×C₄	—	47	1	no	—
183	C₂²×C₄	—	47	1	no	—
184	C₂²×C₄	—	47	1	no	—
185	C₂²×C₄	—	47	2	no	—
186	C₂²×C₄	—	47	4	no	—
187	C₂²×C₄	—	47	2	no	—
188	C₂²×C₄	—	47	4	no	—
189	C₂²×C₄	—	94	4	no	—
190	C₂²×C₄	—	94	2	no	—
191	C₂²×C₄	—	94	2	no	—
192	C₂²×C₄	—	94	1	no	—
193	C₂²×C₄	—	94	2	no	—
194	C₂²×C₄	—	94	1	no	—
195	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	—	94	16	no	—
205	C₂×C₈	—	94	2	no	—
206	C₂×C₈	—	94	2	no	—
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	188	2	no	—
209	C₂×C₈	—	188	2	no	—
210	C₂×C₈	#10625324	1	4	yes	—
211	C₄²	—	47	2	no	—
212	C₄²	—	47	2	no	—
213	C₄²	—	47	4	no	—
214	C₄²	—	47	4	no	—
215	C₄²	C₂×C₄×D₄₇	1	2	yes	—
216	C₄²	D₄×D₄₇	1	2	yes	—
217	C₂³	—	47	2	no	—
218	C₂³	—	47	4	no	—
219	C₂³	—	94	2	no	—
220	C₂³	—	94	1	no	—
221	C₂³	—	94	1	no	—
222	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	47	2	no	—
225	C₂×C₄	—	47	2	no	—
226	C₂×C₄	—	47	2	no	—
227	C₂×C₄	—	94	8	no	—
228	C₂×C₄	—	94	2	no	—
229	C₂×C₄	—	94	1	no	—
230	C₂×C₄	—	94	1	no	—
231	C₂×C₄	—	94	2	no	—
232	C₂×C₄	—	94	2	no	—
233	C₂×C₄	—	94	2	no	—
234	C₂×C₄	—	94	4	no	—
235	C₂×C₄	—	94	4	no	—
236	C₂×C₄	—	94	4	no	—
237	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
240	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
241	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
242	C₂×C₄	C₂×#10625324	1	2	yes	—
243	C₂×C₄	#13094043	1	1	yes	characteristic
244	C₂×C₄	#13094045	1	2	yes	—
245	C₂×C₄	#13094047	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	—	4	4	no	—
248	C₈	—	94	4	no	—
249	C₈	—	188	4	no	—
250	C₂²	—	47	4	no	—
251	C₂²	—	94	2	no	—
252	C₂²	—	94	2	no	—
253	C₂²	—	94	4	no	—
254	C₂²	—	94	4	no	—
255	C₂²	#95×D₄₇	1	1	yes	characteristic
256	C₂²	#15689145	1	2	yes	—
257	C₂²	#15689158	1	1	yes	characteristic
258	C₂²	#15689160	1	1	yes	characteristic
259	C₂²	#108×D₄₇	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	94	2	no	—
262	C₄	—	94	2	no	—
263	C₄	—	94	2	no	—
264	C₄	C₄×#10625324	1	1	yes	characteristic
265	C₄	#15689075	1	1	yes	characteristic
266	C₄	#15689085	1	2	yes	—
267	C₄	#15689116	1	2	yes	—
268	C₄	#15689118	1	2	yes	—
269	C₂	—	94	4	no	—
270	C₂	#20242148	1	2	yes	—
271	C₂	#20242150	1	2	yes	—
272	C₂	#20243080	1	1	yes	characteristic
273	C₂	#20243103	1	1	yes	characteristic
274	C₂	#341×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#7875
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
3. #20241223	C₂	semidirect
4. #20241222	C₂	semidirect
5. C₄₇×#1277	C₂	semidirect
136. #1277	D₄₇	semidirect
155. C₄₇	#7875	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689179	2
C₂×C₄×C₈×D₄₇	2
#20241223	2
#20241222	2
C₄₇×#1277	2
#7875	47