Subgroups — #26221177

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245786	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20244158	C₂	1	4	yes	maximal, complemented
4	#20243589	C₂	1	4	yes	maximal
5	#20243285	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂×#13093958	C₂²	1	4	yes	—
9	C₂×#13093956	C₂²	1	4	yes	—
10	#15688991	—	2	1	no	—
11	#15688989	C₂²	1	2	yes	—
12	#15688986	—	2	1	no	—
13	#15688983	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
18	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
19	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
20	#15688969	—	2	1	no	—
21	#15688969	—	2	1	no	—
22	C₄²×Dic₄₇	—	2	1	no	—
23	#15688886	C₂²	1	8	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#432	C₄	1	8	yes	complemented
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	2	yes	—
29	#13093983	—	2	8	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	2	no	—
38	#13093958	—	2	8	no	—
39	#13093956	—	2	8	no	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₄₇⋊#95	—	2	2	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	2	no	—
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
58	#97×C₄₇	C₂×C₄	1	8	yes	—
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
78	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂²⋊Dic₄₇	—	2	8	no	—
85	C₂²⋊Dic₄₇	—	4	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	#41	1	8	yes	—
90	C₄×Dic₄₇	—	2	8	no	—
91	C₄×Dic₄₇	—	4	4	no	—
92	C₄×Dic₄₇	—	4	4	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₄₇	C₂²×D₄	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	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
116	C₂×Dic₄₇	—	2	8	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₈×C₄₇	—	2	8	no	—
123	#21366	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	4	4	no	—
128	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₄₇	#525	1	2	yes	—
130	C₂²×C₄₇	#575	1	2	yes	—
131	C₂²×C₄₇	#575	1	2	yes	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	C₄×#41	1	2	yes	—
138	C₄×C₄₇	#533	1	2	yes	—
139	Dic₄₇	—	4	8	no	—
140	Dic₄₇	—	8	4	no	—
141	C₂×#432	D₄₇	1	1	yes	characteristic
142	#2283	—	47	1	no	—
143	C₄×#117	—	47	1	no	—
144	#1949	—	47	4	no	—
145	#1854	—	47	4	no	—
146	#1776	—	47	4	no	—
147	C₂×C₄₇	—	4	2	no	—
148	C₂×C₄₇	#2296	1	1	yes	characteristic
149	C₂×C₄₇	#2911	1	1	yes	characteristic
150	C₂×C₄₇	#2911	1	1	yes	characteristic
151	C₂×C₄₇	#3111	1	4	yes	—
152	C₂×#117	—	47	2	no	—
153	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
154	#432	Dic₄₇	1	8	yes	complemented
155	C₂×#107	—	47	4	no	—
156	C₂×#106	—	47	4	no	—
157	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
158	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
159	C₂×#97	C₂×D₄₇	1	2	yes	—
160	#383	—	94	1	no	—
161	#382	—	47	2	no	—
162	#381	—	94	1	no	—
163	#380	—	47	2	no	—
164	C₄×C₄⋊C₄	—	47	2	no	—
165	C₄×C₄⋊C₄	—	47	2	no	—
166	C₄×C₄⋊C₄	—	47	2	no	—
167	C₄×C₂²⋊C₄	—	94	1	no	—
168	C₄×C₂²⋊C₄	—	94	1	no	—
169	C₄×C₂²⋊C₄	—	94	1	no	—
170	#376	—	94	1	no	—
171	#376	—	94	1	no	—
172	C₄³	—	94	1	no	—
173	#340	—	47	8	no	—
174	C₄₇	#21366	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
176	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
179	#117	—	94	8	no	—
180	C₂×C₄⋊C₄	—	47	4	no	—
181	C₂×C₄⋊C₄	—	47	4	no	—
182	C₂×C₂²⋊C₄	—	94	2	no	—
183	C₂×C₂²⋊C₄	—	94	2	no	—
184	C₂×C₄²	—	94	2	no	—
185	C₂×C₄²	—	94	2	no	—
186	C₂×C₄²	—	94	2	no	—
187	C₂×C₄²	—	94	2	no	—
188	C₂×C₄²	Dic₉₄	1	4	yes	—
189	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
190	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
192	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
194	#107	—	94	8	no	—
195	#106	—	94	8	no	—
196	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
197	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
198	#97	C₂×Dic₄₇	1	8	yes	—
199	#95	—	94	2	no	—
200	#95	—	94	2	no	—
201	#95	—	94	2	no	—
202	#95	—	94	2	no	—
203	#95	—	94	2	no	—
204	#95	—	94	2	no	—
205	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
212	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
213	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
214	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
224	C₄⋊C₄	—	47	8	no	—
225	C₄⋊C₄	—	94	4	no	—
226	C₄⋊C₄	—	94	4	no	—
227	C₄⋊C₄	—	94	4	no	—
228	C₂²⋊C₄	—	94	8	no	—
229	C₂²⋊C₄	—	188	4	no	—
230	C₄²	—	94	8	no	—
231	C₄²	—	188	4	no	—
232	C₄²	—	188	4	no	—
233	C₄²	—	188	4	no	—
234	C₄²	C₄⋊Dic₄₇	1	4	yes	—
235	C₄²	C₄⋊Dic₄₇	1	4	yes	—
236	C₄²	C₂×Dic₉₄	1	2	yes	—
237	C₄²	C₂×Dic₉₄	1	2	yes	—
238	C₄²	C₂²×Dic₄₇	1	2	yes	—
239	C₄²	C₂²×Dic₄₇	1	2	yes	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
244	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	94	8	no	—
255	C₂×C₄	—	188	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	—	188	4	no	—
259	C₂×C₄	—	188	4	no	—
260	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
265	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
267	C₂×C₄	#41×D₄₇	1	4	yes	—
268	C₈	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
274	C₂²	#15689635	1	2	yes	—
275	C₂²	#15689923	1	2	yes	—
276	C₂²	#15689923	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	376	4	no	—
283	C₄	—	188	8	no	—
284	C₄	#41×Dic₄₇	1	2	yes	—
285	C₄	#15689676	1	2	yes	—
286	C₂	—	4	2	no	—
287	C₂	#20245872	1	1	yes	characteristic
288	C₂	#20251978	1	1	yes	characteristic
289	C₂	#20251978	1	1	yes	characteristic
290	C₂	#20253499	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#21366
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20244158	C₂	semidirect
5. #20243285	C₂	semidirect
25. C₄₇×#432	C₄	semidirect
154. #432	Dic₄₇	semidirect
174. C₄₇	#21366	semidirect

Maximal subgroups

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

subgroup	index
#20245786	2
C₄×#13093983	2
#20244158	2
#20243589	2
#20243285	2
C₂×C₄₇×#432	2
#21366	47