Subgroups — #26216809

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245793	C₂	1	2	yes	maximal
3	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243515	C₂	1	2	yes	maximal, complemented
5	#20243410	C₂	1	4	yes	maximal
6	#106×Dic₄₇	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
9	C₂×#13093982	—	2	2	no	—
10	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
11	C₂×#13093956	C₂²	1	4	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
14	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
15	#15688993	C₂²	1	2	yes	—
16	#15688988	—	2	2	no	—
17	#15688980	—	2	2	no	—
18	#15688980	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₄₇	—	2	2	no	—
20	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
21	#15688885	C₂²	1	8	yes	—
22	C₂×#106×C₄₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#384	C₂²	1	2	yes	—
25	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
29	C₂×C₈×D₄₇	—	2	4	no	—
30	C₂×C₈×D₄₇	—	2	4	no	—
31	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
32	#13093982	—	4	8	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₄²×D₄₇	C₂×C₄	1	4	yes	—
42	C₄²×D₄₇	Q₈	1	4	yes	—
43	#13093956	—	2	8	no	—
44	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
45	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	#106×C₄₇	—	2	8	no	—
54	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
55	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×D₄₇	—	2	4	no	—
65	C₂×C₄×D₄₇	—	2	2	no	—
66	C₂×C₄×D₄₇	—	2	2	no	—
67	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
68	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×D₄₇	SD₁₆	1	8	yes	—
70	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₄₇	#41	1	4	yes	—
74	C₂×C₈×C₄₇	#41	1	4	yes	—
75	C₄⋊C₄×C₄₇	—	2	8	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
80	(C₂×D₄₇)⋊C₄	—	4	8	no	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	C₄×Dic₄₇	—	4	8	no	—
85	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
86	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
87	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
92	C₈×D₄₇	—	2	16	no	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂²×D₄₇	—	2	1	no	—
95	C₂²×D₄₇	—	2	1	no	—
96	C₂²×D₄₇	—	2	2	no	—
97	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂²×D₄₇	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
105	C₂×C₄×C₄₇	#135	1	4	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
110	C₂×Dic₄₇	—	2	1	no	—
111	C₂×Dic₄₇	—	2	1	no	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₄₇	C₂×SD₁₆	1	4	yes	—
120	C₄×D₄₇	—	2	4	no	—
121	C₄×D₄₇	—	2	4	no	—
122	C₄×D₄₇	—	2	8	no	—
123	C₄×D₄₇	#106	1	16	yes	complemented
124	C₈×C₄₇	#117	1	8	yes	complemented
125	C₄₇⋊C₈	—	2	4	no	—
126	#20816	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₇	#532	1	1	yes	characteristic
129	C₂²×C₄₇	#532	1	1	yes	characteristic
130	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
131	C₂²×C₄₇	C₂×#135	1	2	yes	—
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	2	4	no	—
134	C₂×D₄₇	—	2	4	no	—
135	C₂×D₄₇	—	2	2	no	—
136	C₂×D₄₇	C₂×#106	1	4	yes	—
137	C₄×C₄₇	—	8	4	no	—
138	C₄×C₄₇	C₂×#106	1	4	yes	—
139	C₄×C₄₇	#427	1	4	yes	—
140	C₄×C₄₇	C₂×#117	1	2	yes	—
141	C₄×C₄₇	#533	1	2	yes	—
142	Dic₄₇	—	2	2	no	—
143	Dic₄₇	—	2	2	no	—
144	Dic₄₇	—	8	4	no	—
145	Dic₄₇	C₂×#106	1	4	yes	—
146	C₂²×C₄×C₈	—	47	1	no	—
147	#2285	—	47	2	no	—
148	#1844	—	47	1	no	—
149	#1844	D₄₇	1	1	yes	characteristic, complemented
150	#1834	—	47	2	no	—
151	#1807	—	47	4	no	—
152	C₄×#106	—	47	4	no	—
153	C₂×C₄₇	#2285	1	1	yes	characteristic
154	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
156	C₂×C₄₇	#3167	1	2	yes	—
157	C₂×C₄₇	#3167	1	2	yes	—
158	D₄₇	—	2	4	no	—
159	C₂³×C₈	—	47	2	no	—
160	C₂×#117	—	94	2	no	—
161	C₂²×C₄²	—	47	1	no	—
162	C₂×#106	—	47	4	no	—
163	C₂×#106	C₂×D₄₇	1	4	yes	—
164	C₂×C₄×C₈	—	47	2	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	C₂×C₄×C₈	—	47	8	no	—
167	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
168	#384	—	47	2	no	—
169	#384	C₂×D₄₇	1	2	yes	—
170	#382	—	94	2	no	—
171	#380	—	47	4	no	—
172	#380	—	94	2	no	—
173	C₄×C₄⋊C₄	—	47	4	no	—
174	C₄×C₄⋊C₄	—	94	2	no	—
175	#340	—	47	8	no	—
176	C₄₇	#20816	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	—	47	1	no	—
178	C₂³×C₄	—	47	1	no	—
179	C₂³×C₄	—	47	1	no	—
180	C₂²×C₈	—	47	2	no	—
181	C₂²×C₈	—	47	8	no	—
182	C₂²×C₈	—	94	4	no	—
183	C₂²×C₈	—	94	4	no	—
184	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
185	#117	—	188	8	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	47	4	no	—
189	C₂×C₄⋊C₄	—	94	2	no	—
190	C₂×C₄⋊C₄	—	94	2	no	—
191	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
192	C₂×C₂²⋊C₄	—	94	4	no	—
193	C₂×C₄²	—	47	1	no	—
194	C₂×C₄²	—	47	1	no	—
195	C₂×C₄²	—	47	1	no	—
196	C₂×C₄²	—	47	4	no	—
197	C₂×C₄²	—	47	4	no	—
198	C₂×C₄²	—	94	4	no	—
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	#106	—	2	8	no	—
201	#106	—	94	8	no	—
202	C₄×C₈	—	47	8	no	—
203	C₄×C₈	—	47	4	no	—
204	C₄×C₈	C₄×D₄₇	1	4	yes	—
205	#95	—	94	4	no	—
206	#95	—	94	4	no	—
207	C₂⁴	—	47	1	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	47	1	no	—
210	C₂²×C₄	—	47	4	no	—
211	C₂²×C₄	—	47	8	no	—
212	C₂²×C₄	—	47	1	no	—
213	C₂²×C₄	—	47	1	no	—
214	C₂²×C₄	—	47	4	no	—
215	C₂²×C₄	—	47	2	no	—
216	C₂²×C₄	—	47	2	no	—
217	C₂²×C₄	—	94	4	no	—
218	C₂²×C₄	—	94	2	no	—
219	C₂²×C₄	—	94	2	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	C₂×C₈	—	47	4	no	—
225	C₂×C₈	—	94	2	no	—
226	C₂×C₈	—	94	2	no	—
227	C₂×C₈	—	94	16	no	—
228	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
229	C₂×C₈	#10625352	1	4	yes	—
230	C₂×C₈	#10625354	1	4	yes	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	94	8	no	—
235	C₄⋊C₄	—	188	4	no	—
236	C₄⋊C₄	—	188	4	no	—
237	C₂²⋊C₄	—	188	8	no	—
238	C₄²	—	47	4	no	—
239	C₄²	—	47	4	no	—
240	C₄²	—	47	2	no	—
241	C₄²	—	47	2	no	—
242	C₄²	—	188	8	no	—
243	C₄²	C₂×C₄×D₄₇	1	2	yes	—
244	C₄²	Q₈×D₄₇	1	2	yes	—
245	C₂³	—	47	2	no	—
246	C₂³	—	47	4	no	—
247	C₂³	—	94	2	no	—
248	C₂³	—	94	1	no	—
249	C₂³	—	94	1	no	—
250	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	47	2	no	—
255	C₂×C₄	—	47	4	no	—
256	C₂×C₄	—	47	16	no	—
257	C₂×C₄	—	47	2	no	—
258	C₂×C₄	—	47	2	no	—
259	C₂×C₄	—	94	2	no	—
260	C₂×C₄	—	94	1	no	—
261	C₂×C₄	—	94	1	no	—
262	C₂×C₄	—	94	4	no	—
263	C₂×C₄	—	94	8	no	—
264	C₂×C₄	—	94	4	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
269	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
270	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
271	C₂×C₄	#13094045	1	4	yes	—
272	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×#10625352	1	2	yes	—
275	C₂×C₄	C₂×#10625354	1	2	yes	—
276	C₈	—	94	4	no	—
277	C₈	#13093982	1	8	yes	complemented
278	C₂²	—	47	4	no	—
279	C₂²	—	94	2	no	—
280	C₂²	—	94	2	no	—
281	C₂²	—	94	4	no	—
282	C₂²	—	94	4	no	—
283	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
284	C₂²	#15689671	1	1	yes	characteristic
285	C₂²	#15689674	1	1	yes	characteristic
286	C₂²	C₂×SD₁₆×D₄₇	1	2	yes	—
287	C₂²	C₂×#13094045	1	2	yes	—
288	C₄	—	8	4	no	—
289	C₄	—	47	4	no	—
290	C₄	—	94	2	no	—
291	C₄	—	94	2	no	—
292	C₄	—	376	4	no	—
293	C₄	#106×D₄₇	1	4	yes	—
294	C₄	#15689158	1	4	yes	—
295	C₄	C₂×#13093982	1	2	yes	—
296	C₄	#15689678	1	2	yes	—
297	C₂	—	94	4	no	—
298	C₂	#20245793	1	1	yes	characteristic
299	C₂	C₂×#106×D₄₇	1	1	yes	characteristic
300	C₂	C₂×#15689158	1	1	yes	characteristic
301	C₂	#20253945	1	2	yes	—
302	C₂	#20253949	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#20816
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
3. #20243545	C₂	semidirect
4. #20243515	C₂	semidirect
6. #106×Dic₄₇	C₂	semidirect
7. C₄₇×#1844	C₂	semidirect
14. C₄×C₈×D₄₇	C₄	semidirect
123. C₄×D₄₇	#106	semidirect
124. C₈×C₄₇	#117	semidirect
149. #1844	D₄₇	semidirect
176. C₄₇	#20816	semidirect
277. C₈	#13093982	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₇	2
#20245793	2
#20243545	2
#20243515	2
#20243410	2
#106×Dic₄₇	2
C₄₇×#1844	2
#20816	47