Subgroups — #26370713

Order 12032.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252965	C₂	1	2	yes	maximal, complemented
3	#20245481	C₂	1	4	yes	maximal, complemented
4	#20245476	C₂	1	2	yes	maximal, complemented
5	#20245471	C₂	1	2	yes	maximal, complemented
6	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245095	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
10	#15689630	—	2	2	no	—
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₄₇	—	2	4	no	—
13	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
14	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
15	#15689361	C₂²	1	4	yes	—
16	#15689356	C₂²	1	4	yes	—
17	#15689349	—	2	4	no	—
18	#15689161	C₂²	1	2	yes	—
19	#15689119	—	2	2	no	—
20	#15688951	C₂²	1	2	yes	complemented
21	#15688949	C₂²	1	2	yes	complemented
22	#15688927	C₂²	1	4	yes	complemented
23	#15688924	C₂²	1	4	yes	complemented
24	C₂×#135×C₄₇	C₂²	1	2	yes	—
25	C₄₇×#508	—	2	4	no	—
26	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	complemented
28	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₈×D₄₇	—	2	2	no	—
33	C₂×C₈×D₄₇	—	2	2	no	—
34	C₂×C₈×D₄₇	D₄	1	2	yes	—
35	C₂×C₈×D₄₇	C₂³	1	2	yes	—
36	#13094005	—	4	2	no	—
37	#13094000	—	2	4	no	—
38	#13093995	—	2	4	no	—
39	#13093992	—	4	2	no	—
40	C₄×C₂²⋊D₄₇	—	4	4	no	—
41	#13093983	—	2	2	no	—
42	C₂×C₄₇⋊C₁₆	—	2	2	no	—
43	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
44	C₁₆×D₄₇	—	2	4	no	—
45	C₁₆×D₄₇	—	2	4	no	—
46	#13093958	C₂³	1	4	yes	—
47	#13093956	—	2	2	no	—
48	#13093948	—	2	4	no	—
49	#13093945	—	2	4	no	—
50	C₂×#41×C₄₇	—	2	2	no	—
51	#135×C₄₇	—	2	4	no	—
52	#135×C₄₇	—	2	4	no	—
53	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
54	C₂×SD₁₆×C₄₇	—	2	2	no	—
55	C₂×D₈×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	Q₃₂×C₄₇	—	2	2	no	—
58	SD₃₂×C₄₇	—	2	4	no	—
59	D₁₆×C₄₇	—	2	2	no	—
60	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
61	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
63	C₂³×D₄₇	D₈	1	1	yes	characteristic
64	#41×C₄₇	—	4	4	no	—
65	#41×C₄₇	—	4	4	no	—
66	C₂×Q₈×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	—	2	2	no	—
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₄₇	—	4	4	no	—
72	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
73	C₂×C₄×D₄₇	—	2	2	no	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	D₈	1	2	yes	—
76	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
77	Q₁₆×C₄₇	—	2	2	no	—
78	Q₁₆×C₄₇	—	2	2	no	—
79	SD₁₆×C₄₇	—	4	4	no	—
80	D₈×C₄₇	—	2	2	no	—
81	D₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	#41	1	2	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₄₇	—	4	2	no	—
87	C₂²⋊Dic₄₇	—	4	4	no	—
88	(C₂×D₄₇)⋊C₄	—	4	4	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	Dic₄₇⋊C₄	—	4	4	no	—
91	C₄×Dic₄₇	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
94	C₈×D₄₇	—	2	4	no	—
95	C₈×D₄₇	—	2	2	no	—
96	C₈×D₄₇	—	2	2	no	—
97	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
98	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
99	C₄₇⋊C₁₆	—	2	2	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₄₇	—	2	2	no	—
103	C₂²×D₄₇	—	2	4	no	—
104	C₂²×D₄₇	C₂×D₈	1	2	yes	—
105	Q₈×C₄₇	—	4	2	no	—
106	Q₈×C₄₇	—	4	2	no	—
107	D₄×C₄₇	—	4	2	no	—
108	D₄×C₄₇	—	4	2	no	—
109	D₄×C₄₇	—	8	4	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	8	4	no	—
112	C₂×C₄×C₄₇	#121	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#135	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂²⋊D₄₇	—	8	8	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
120	C₄×D₄₇	—	2	2	no	—
121	C₄×D₄₇	—	2	2	no	—
122	C₄×D₄₇	—	2	4	no	—
123	C₈×C₄₇	#121	1	2	yes	—
124	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₄₇⋊C₈	—	2	2	no	—
127	#33600	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	8	4	no	—
130	C₂²×C₄₇	—	8	2	no	—
131	C₂²×C₄₇	#466	1	2	yes	—
132	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
133	C₂×D₄₇	—	2	8	no	—
134	C₂×D₄₇	—	2	2	no	—
135	C₂×D₄₇	—	2	2	no	—
136	C₂×D₄₇	#508	1	4	yes	complemented
137	C₄×C₄₇	—	8	2	no	—
138	C₄×C₄₇	#466	1	2	yes	—
139	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
141	Dic₄₇	—	2	2	no	—
142	Dic₄₇	—	8	4	no	—
143	C₂×#508	D₄₇	1	1	yes	characteristic, complemented
144	C₂³×C₁₆	—	47	1	no	—
145	#3050	—	47	2	no	—
146	#2217	—	47	4	no	—
147	#2216	—	47	2	no	—
148	#2215	—	47	2	no	—
149	#2158	—	47	1	no	—
150	#2138	—	47	2	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	—	16	2	no	—
153	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#508	1	2	yes	—
155	D₄₇	—	2	4	no	—
156	C₂×#135	C₂×D₄₇	1	2	yes	—
157	C₂³×C₈	—	47	1	no	—
158	#525	—	94	2	no	—
159	#508	—	2	4	no	—
160	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic, complemented
161	C₂×SD₃₂	C₂×D₄₇	1	2	yes	complemented
162	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
163	C₂²×C₁₆	—	47	1	no	—
164	C₂²×C₁₆	—	47	2	no	—
165	C₂²×C₁₆	—	47	2	no	—
166	C₂²×C₁₆	—	94	4	no	—
167	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
168	#467	—	47	4	no	—
169	#466	—	47	4	no	—
170	#465	—	94	4	no	—
171	#427	—	47	2	no	—
172	#416	—	94	2	no	—
173	#367	—	47	2	no	—
174	#366	—	47	2	no	—
175	#358	—	47	4	no	—
176	#357	—	47	4	no	—
177	C₄₇	#33600	1	1	yes	characteristic, complemented, Sylow
178	C₂×#41	—	2	2	no	—
179	C₂³×C₄	—	47	1	no	—
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂²×D₄₇	1	2	yes	—
185	C₂²×C₈	—	47	1	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	94	2	no	—
189	C₂²×C₈	—	94	2	no	—
190	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	#123	—	188	2	no	—
192	C₂²⋊Q₈	—	94	4	no	—
193	#121	—	94	4	no	—
194	#120	—	188	2	no	—
195	C₄×D₄	—	188	4	no	—
196	#117	—	94	2	no	—
197	Q₃₂	—	2	2	no	—
198	SD₃₂	—	2	4	no	—
199	D₁₆	—	2	2	no	—
200	C₂×C₁₆	—	47	2	no	—
201	C₂×C₁₆	—	94	2	no	—
202	C₂×C₁₆	—	94	4	no	—
203	C₂×C₁₆	—	94	4	no	—
204	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
207	#107	—	47	4	no	—
208	#106	—	94	2	no	—
209	#103	—	94	4	no	—
210	#102	—	94	4	no	—
211	C₂⁴	—	47	1	no	—
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	188	4	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	47	1	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	94	2	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	D₈	—	2	2	no	—
229	D₈	—	2	2	no	—
230	C₂×C₈	—	47	2	no	—
231	C₂×C₈	—	94	2	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	2	no	—
235	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
236	C₂×C₈	#10625352	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
238	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
239	C₄⋊C₄	—	94	4	no	—
240	C₄⋊C₄	—	188	4	no	—
241	C₂²⋊C₄	—	188	4	no	—
242	C₂²⋊C₄	—	188	4	no	—
243	C₂²⋊C₄	—	188	2	no	—
244	C₄²	—	188	2	no	—
245	C₁₆	—	94	2	no	—
246	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
248	C₂³	—	4	2	no	—
249	C₂³	—	47	2	no	—
250	C₂³	—	94	4	no	—
251	C₂³	—	94	2	no	—
252	C₂³	D₈×D₄₇	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	376	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	47	2	no	—
262	C₂×C₄	—	94	2	no	—
263	C₂×C₄	—	94	4	no	—
264	C₂×C₄	—	94	2	no	—
265	C₂×C₄	—	94	2	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	#13093995	1	2	yes	—
268	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
269	C₂×C₄	#13094043	1	2	yes	—
270	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
271	C₈	—	94	2	no	—
272	C₈	#13093995	1	2	yes	—
273	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₈	C₂×#10625352	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	47	4	no	—
279	C₂²	—	94	8	no	—
280	C₂²	—	94	2	no	—
281	C₂²	—	94	2	no	—
282	C₂²	#15689356	1	2	yes	—
283	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
284	C₄	—	8	2	no	—
285	C₄	—	94	2	no	—
286	C₄	—	376	4	no	—
287	C₄	#15689356	1	2	yes	—
288	C₄	C₂×#13093995	1	1	yes	characteristic
289	C₄	C₂×#13094043	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	16	2	no	—
292	C₂	—	94	4	no	—
293	C₂	C₂×#15689356	1	1	yes	characteristic
294	C₂	#508×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#508	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×#508	D₄₇
hypocenter	C₄₇	#33600
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₄₇	C₂	semidirect
2. #20252965	C₂	semidirect
3. #20245481	C₂	semidirect
4. #20245476	C₂	semidirect
5. #20245471	C₂	semidirect
6. #20245172	C₂	semidirect
7. #20245095	C₂	semidirect
8. C₂×C₄₇×#508	C₂	semidirect
20. #15688951	C₂²	semidirect
21. #15688949	C₂²	semidirect
22. #15688927	C₂²	semidirect
23. #15688924	C₂²	semidirect
26. C₂×Q₃₂×C₄₇	C₂²	semidirect
27. C₂×SD₃₂×C₄₇	C₂²	semidirect
28. C₂×D₁₆×C₄₇	C₂²	semidirect
60. C₂×C₁₆×C₄₇	D₄	semidirect
136. C₂×D₄₇	#508	semidirect
143. C₂×#508	D₄₇	semidirect
160. C₂×Q₃₂	C₂×D₄₇	semidirect
161. C₂×SD₃₂	C₂×D₄₇	semidirect
162. C₂×D₁₆	C₂×D₄₇	semidirect
177. C₄₇	#33600	semidirect
204. C₂×C₁₆	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₄₇	2
#20252965	2
#20245481	2
#20245476	2
#20245471	2
#20245172	2
#20245095	2
C₂×C₄₇×#508	2
#33600	47