Subgroups — #16857498

Order 3968.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521866	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521856	C₂	1	2	yes	maximal, complemented
3	#14521854	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521433	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521324	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521319	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521313	C₂	1	2	yes	maximal, complemented
8	#14521228	C₂	1	2	yes	maximal, complemented
9	#14521223	C₂	1	2	yes	maximal, complemented
10	#14521167	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#11758567	C₂²	1	2	yes	complemented
13	#11758566	C₂²	1	2	yes	—
14	C₃₁×#135	C₂²	1	1	yes	characteristic
15	C₃₁×#131	C₂²	1	1	yes	characteristic
16	C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
17	#11758528	C₂²	1	2	yes	—
18	#11758526	C₂²	1	2	yes	complemented
19	#11758495	—	2	2	no	—
20	#11758493	—	2	2	no	—
21	SD₁₆×D₃₁	—	2	2	no	—
22	#11758488	—	2	2	no	—
23	#11758486	—	2	2	no	—
24	#11758486	C₂²	1	2	yes	complemented
25	#11758485	—	2	2	no	—
26	#11758485	C₂²	1	2	yes	—
27	C₂×Dic₁₂₄	—	2	1	no	—
28	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic, complemented
29	#11758481	—	2	1	no	—
30	#11758481	—	2	1	no	—
31	#11758481	C₂²	1	1	yes	characteristic
32	C₂×D₂₄₈	—	2	1	no	—
33	C₂×D₂₄₈	C₂²	1	1	yes	characteristic
34	C₂×#92470	C₂²	1	2	yes	complemented
35	#11758478	C₂²	1	2	yes	complemented
36	#11758452	C₂²	1	2	yes	complemented
37	C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#104	C₂²	1	1	yes	characteristic
39	C₃₁×#104	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
41	#11758412	C₂²	1	2	yes	complemented
42	#11758411	C₂²	1	2	yes	complemented
43	#11758407	C₂²	1	1	yes	characteristic, complemented
44	#11758393	C₂²	1	2	yes	complemented
45	#41×C₃₁	C₂³	1	1	yes	characteristic
46	#41×C₃₁	C₂³	1	1	yes	characteristic
47	#92499	—	2	2	no	—
48	Q₈×D₃₁	—	2	2	no	—
49	#92497	—	2	2	no	—
50	#92497	—	4	2	no	—
51	D₄×D₃₁	—	2	2	no	—
52	D₄×D₃₁	—	4	2	no	—
53	#92495	—	2	2	no	—
54	#92495	—	2	2	no	—
55	#92495	C₂³	1	2	yes	—
56	C₂×D₁₂₄	—	2	2	no	—
57	C₂×D₁₂₄	C₂³	1	2	yes	—
58	C₂×Dic₆₂	—	2	2	no	—
59	C₂×Dic₆₂	C₂³	1	2	yes	—
60	Q₁₆×C₃₁	D₄	1	1	yes	characteristic
61	SD₁₆×C₃₁	D₄	1	1	yes	characteristic
62	SD₁₆×C₃₁	D₄	1	1	yes	characteristic
63	D₈×C₃₁	D₄	1	1	yes	characteristic
64	M₁₆×C₃₁	—	2	1	no	—
65	M₁₆×C₃₁	—	2	1	no	—
66	M₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₃₁	—	2	1	no	—
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	#92482	—	2	2	no	—
74	#92481	—	2	2	no	—
75	#92480	—	2	2	no	—
76	#92479	—	2	2	no	—
77	(C₂×D₃₁)⋊C₄	—	4	2	no	—
78	#92474	C₂³	1	2	yes	—
79	Dic₁₂₄	—	2	2	no	—
80	Dic₁₂₄	—	2	1	no	—
81	Dic₁₂₄	—	2	1	no	—
82	Dic₁₂₄	D₄	1	1	yes	characteristic
83	D₂₄₈	—	2	2	no	—
84	D₂₄₈	—	2	1	no	—
85	D₂₄₈	—	2	1	no	—
86	D₂₄₈	D₄	1	1	yes	characteristic
87	#92470	—	2	2	no	—
88	#92470	—	2	2	no	—
89	#92470	—	2	2	no	—
90	#92470	—	2	2	no	—
91	#92470	D₄	1	2	yes	—
92	#92469	—	2	2	no	—
93	C₈×D₃₁	—	2	2	no	—
94	C₂²×D₃₁	—	2	2	no	—
95	C₂²×D₃₁	—	4	2	no	—
96	Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	D₄×C₃₁	—	2	1	no	—
99	D₄×C₃₁	—	2	1	no	—
100	D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₃₁	—	2	2	no	—
107	C₂²⋊D₃₁	—	4	2	no	—
108	C₂²⋊D₃₁	—	4	2	no	—
109	C₂×Dic₃₁	—	2	2	no	—
110	C₂×Dic₃₁	—	4	2	no	—
111	D₁₂₄	—	2	2	no	—
112	D₁₂₄	—	2	2	no	—
113	D₁₂₄	—	4	2	no	—
114	D₁₂₄	C₂×D₄	1	2	yes	—
115	C₄×D₃₁	—	2	2	no	—
116	C₄×D₃₁	—	4	2	no	—
117	C₄×D₃₁	—	4	2	no	—
118	Dic₆₂	—	2	2	no	—
119	Dic₆₂	—	2	2	no	—
120	Dic₆₂	—	4	2	no	—
121	Dic₆₂	C₂×D₄	1	2	yes	—
122	C₈×C₃₁	—	2	1	no	—
123	C₈×C₃₁	—	2	1	no	—
124	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₃₁⋊C₈	—	2	2	no	—
129	#3290	—	31	1	no	maximal, Sylow
130	C₂²×C₃₁	—	2	1	no	—
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×D₃₁	—	2	2	no	—
134	C₂×D₃₁	—	4	2	no	—
135	C₂×D₃₁	—	8	2	no	—
136	C₂×D₃₁	—	16	2	no	—
137	C₄×C₃₁	—	2	1	no	—
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	4	1	no	—
140	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	Dic₃₁	—	2	2	no	—
143	Dic₃₁	—	4	2	no	—
144	#578	—	31	1	no	—
145	#577	—	31	2	no	—
146	#576	—	31	1	no	—
147	#495	—	31	1	no	—
148	#473	—	31	1	no	—
149	#472	—	31	1	no	—
150	#471	—	31	2	no	—
151	#455	—	31	2	no	—
152	#454	—	31	2	no	—
153	#443	—	31	1	no	—
154	#443	D₃₁	1	1	yes	characteristic, complemented
155	C₂×C₃₁	—	2	1	no	—
156	C₂×C₃₁	—	4	1	no	—
157	C₂×C₃₁	—	4	1	no	—
158	C₂×C₃₁	D₄²	1	1	yes	characteristic
159	D₃₁	—	4	2	no	—
160	D₃₁	—	8	2	no	—
161	2₋¹⁺⁴	—	31	2	no	—
162	H(5,𝔽₂)	—	31	2	no	—
163	#137	—	31	2	no	—
164	#137	—	31	2	no	—
165	#137	—	62	2	no	—
166	#137	—	62	2	no	—
167	AGL(1,ℤ/8)	—	31	2	no	—
168	AGL(1,ℤ/8)	—	31	2	no	—
169	AGL(1,ℤ/8)	—	62	2	no	—
170	AGL(1,ℤ/8)	—	62	2	no	—
171	#135	—	31	1	no	—
172	#135	—	62	1	no	—
173	#135	—	62	1	no	—
174	#135	—	62	2	no	—
175	#135	C₂×D₃₁	1	1	yes	characteristic
176	C₂×Q₁₆	—	31	1	no	—
177	C₂×Q₁₆	—	62	1	no	—
178	C₂×SD₁₆	—	31	2	no	—
179	C₂×SD₁₆	—	62	2	no	—
180	C₂×D₈	—	31	1	no	—
181	C₂×D₈	—	62	1	no	—
182	#131	—	31	2	no	—
183	#131	C₂×D₃₁	1	1	yes	characteristic
184	#131	C₂×D₃₁	1	1	yes	characteristic, complemented
185	#124	—	31	2	no	—
186	#108	—	31	1	no	—
187	#108	C₂×D₃₁	1	1	yes	characteristic, complemented
188	#104	—	31	2	no	—
189	#104	C₂×D₃₁	1	1	yes	characteristic
190	#104	C₂×D₃₁	1	1	yes	characteristic, complemented
191	#101	—	31	2	no	—
192	#100	—	31	2	no	—
193	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
194	C₃₁	#3290	1	1	yes	characteristic, complemented, Sylow
195	#41	—	31	2	no	—
196	#41	—	62	2	no	—
197	#41	—	62	2	no	—
198	#41	—	62	2	no	—
199	#41	—	62	2	no	—
200	#41	—	124	2	no	—
201	#41	C₂²×D₃₁	1	1	yes	characteristic
202	#41	C₂²×D₃₁	1	1	yes	characteristic
203	C₂×Q₈	—	31	2	no	—
204	C₂×Q₈	—	62	2	no	—
205	C₂×Q₈	—	62	2	no	—
206	C₂×D₄	—	31	2	no	—
207	C₂×D₄	—	62	2	no	—
208	C₂×D₄	—	62	2	no	—
209	C₂×D₄	—	124	2	no	—
210	Q₁₆	—	31	1	no	—
211	Q₁₆	—	62	2	no	—
212	Q₁₆	—	62	2	no	—
213	Q₁₆	—	62	1	no	—
214	Q₁₆	—	62	1	no	—
215	Q₁₆	D₁₂₄	1	1	yes	characteristic
216	SD₁₆	—	31	2	no	—
217	SD₁₆	—	62	2	no	—
218	SD₁₆	—	62	2	no	—
219	SD₁₆	—	62	2	no	—
220	SD₁₆	—	62	2	no	—
221	SD₁₆	—	62	2	no	—
222	SD₁₆	—	62	2	no	—
223	SD₁₆	D₁₂₄	1	1	yes	characteristic
224	SD₁₆	D₁₂₄	1	1	yes	characteristic
225	D₈	—	31	1	no	—
226	D₈	—	62	2	no	—
227	D₈	—	62	2	no	—
228	D₈	—	62	1	no	—
229	D₈	—	62	1	no	—
230	D₈	D₁₂₄	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	31	2	no	—
234	M₁₆	—	62	2	no	—
235	M₁₆	C₂²×D₃₁	1	1	yes	characteristic
236	M₁₆	C₂²×D₃₁	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	62	2	no	—
240	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
242	C₂²⋊C₄	—	124	2	no	—
243	C₄²	C₂²×D₃₁	1	1	yes	characteristic
244	C₂³	—	62	2	no	—
245	C₂³	—	124	2	no	—
246	Q₈	—	31	2	no	—
247	Q₈	—	62	2	no	—
248	Q₈	—	62	2	no	—
249	Q₈	—	124	2	no	—
250	Q₈	C₂×D₁₂₄	1	1	yes	characteristic
251	Q₈	C₂×D₁₂₄	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	31	2	no	—
255	D₄	—	62	2	no	—
256	D₄	—	62	2	no	—
257	D₄	—	62	2	no	—
258	D₄	—	124	2	no	—
259	D₄	—	124	2	no	—
260	D₄	—	124	2	no	—
261	D₄	C₂×D₁₂₄	1	1	yes	characteristic
262	D₄	C₂×D₁₂₄	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	62	2	no	—
267	C₂×C₄	—	62	2	no	—
268	C₂×C₄	—	124	2	no	—
269	C₂×C₄	—	124	2	no	—
270	C₂×C₄	—	124	2	no	—
271	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	62	2	no	—
275	C₈	C₂×D₁₂₄	1	1	yes	characteristic
276	C₈	C₂×D₁₂₄	1	1	yes	characteristic
277	C₈	D₄×D₃₁	1	1	yes	characteristic
278	C₈	D₄×D₃₁	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	62	2	no	—
282	C₂²	—	124	2	no	—
283	C₂²	—	496	2	no	—
284	C₂²	—	248	2	no	—
285	C₂²	#11758566	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	62	2	no	—
290	C₄	—	124	2	no	—
291	C₄	C₂²×D₁₂₄	1	1	yes	characteristic
292	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	124	2	no	—
297	C₂	—	248	2	no	—
298	C₂	D₄×D₁₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#443	C₂
socle	C₂×C₃₁	D₄²
hypercenter	#443	D₃₁
hypocenter	C₃₁	#3290
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #14521866	C₂	semidirect
2. #14521856	C₂	semidirect
3. #14521854	C₂	semidirect
4. #14521433	C₂	semidirect
5. #14521324	C₂	semidirect
6. #14521319	C₂	semidirect
7. #14521313	C₂	semidirect
8. #14521228	C₂	semidirect
9. #14521223	C₂	semidirect
10. #14521167	C₂	semidirect
11. C₃₁×#443	C₂	semidirect
12. #11758567	C₂²	semidirect
16. C₃₁×#131	C₂²	semidirect
18. #11758526	C₂²	semidirect
24. #11758486	C₂²	semidirect
28. C₂×Dic₁₂₄	C₂²	semidirect
34. C₂×#92470	C₂²	semidirect
35. #11758478	C₂²	semidirect
36. #11758452	C₂²	semidirect
37. C₃₁×#108	C₂²	semidirect
39. C₃₁×#104	C₂²	semidirect
40. C₄×C₈×C₃₁	C₂²	semidirect
41. #11758412	C₂²	semidirect
42. #11758411	C₂²	semidirect
43. #11758407	C₂²	semidirect
44. #11758393	C₂²	semidirect
154. #443	D₃₁	semidirect
184. #131	C₂×D₃₁	semidirect
187. #108	C₂×D₃₁	semidirect
190. #104	C₂×D₃₁	semidirect
193. C₄×C₈	C₂×D₃₁	semidirect
194. C₃₁	#3290	semidirect

Maximal subgroups

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

subgroup	index
#14521866	2
#14521856	2
#14521854	2
#14521433	2
#14521324	2
#14521319	2
#14521313	2
#14521228	2
#14521223	2
#14521167	2
C₃₁×#443	2
#3290	31