Subgroups — #16854558

Order 3968.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521521	C₂	1	2	yes	maximal, complemented
3	#14521159	C₂	1	2	yes	maximal, complemented
4	#14521158	C₂	1	2	yes	maximal, complemented
5	C₄×D₂₄₈	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521131	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521086	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521052	C₂	1	2	yes	maximal, complemented
10	#14521003	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₃₁	C₂²	1	2	yes	—
13	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
15	D₄×Dic₃₁	C₂²	1	2	yes	complemented
16	C₂×#92479	C₂²	1	2	yes	—
17	C₄×C₂²⋊D₃₁	—	2	2	no	—
18	C₂×(C₂×D₃₁)⋊C₄	—	2	2	no	—
19	D₈×D₃₁	C₄	1	8	yes	complemented
20	C₂×D₂₄₈	C₂²	1	1	yes	characteristic
21	C₂×#92469	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
23	#11758470	C₂²	1	2	yes	complemented
24	#11758469	C₂²	1	2	yes	—
25	(C₂²⋊D₃₁)⋊C₄	—	2	2	no	—
26	C₂²⋊C₄×D₃₁	—	2	2	no	—
27	C₄×D₁₂₄	C₂²	1	2	yes	complemented
28	#11758449	C₂²	1	1	yes	characteristic
29	C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#102	C₂²	1	2	yes	complemented
31	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
32	#11758420	C₂²	1	2	yes	complemented
33	#11758409	C₂²	1	2	yes	complemented
34	#11758406	C₂²	1	1	yes	characteristic, complemented
35	#11758403	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
37	#11758397	C₂²	1	2	yes	complemented
38	#11758396	C₂²	1	2	yes	complemented
39	#11758391	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₃₁	—	2	2	no	—
41	C₂×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	—	2	2	no	—
43	C₂×C₂²⋊D₃₁	—	2	2	no	—
44	C₂²×Dic₃₁	—	2	2	no	—
45	D₄×D₃₁	—	2	4	no	—
46	D₄×D₃₁	C₂×C₄	1	8	yes	complemented
47	C₂×D₁₂₄	C₂³	1	2	yes	—
48	C₂×C₄×D₃₁	—	2	2	no	—
49	C₂×C₄×D₃₁	—	2	2	no	—
50	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
51	D₈×C₃₁	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₃₁	—	2	2	no	—
56	C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₃₁	—	2	2	no	—
58	#92479	C₂×C₄	1	8	yes	complemented
59	(C₂×D₃₁)⋊C₄	—	2	2	no	—
60	(C₂×D₃₁)⋊C₄	—	2	2	no	—
61	(C₂×D₃₁)⋊C₄	—	2	2	no	—
62	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
63	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
64	C₄⋊Dic₃₁	C₂³	1	2	yes	—
65	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
66	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
67	C₄×Dic₃₁	—	2	2	no	—
68	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	D₂₄₈	C₂×C₄	1	4	yes	complemented
72	#92469	—	2	2	no	—
73	C₈×D₃₁	C₂×C₄	1	4	yes	complemented
74	C₂³×C₃₁	—	2	2	no	—
75	C₂²×D₃₁	—	2	2	no	—
76	C₂²×D₃₁	—	2	8	no	—
77	C₂²×D₃₁	—	8	2	no	—
78	C₂²×D₃₁	—	8	2	no	—
79	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
80	D₄×C₃₁	—	2	2	no	—
81	D₄×C₃₁	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	8	2	no	—
84	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₃₁	—	2	8	no	—
88	C₂×Dic₃₁	—	2	2	no	—
89	C₂×Dic₃₁	—	8	2	no	—
90	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
93	D₁₂₄	—	2	2	no	—
94	D₁₂₄	C₂²×C₄	1	4	yes	—
95	C₄×D₃₁	—	8	2	no	—
96	C₄×D₃₁	—	8	2	no	—
97	C₄×D₃₁	C₂²×C₄	1	2	yes	—
98	C₄×D₃₁	#41	1	2	yes	—
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	C₂²×C₄	1	2	yes	—
101	C₃₁⋊C₈	—	2	1	no	—
102	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
103	#2948	—	31	1	no	maximal, Sylow
104	C₂²×C₃₁	—	2	4	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×D₃₁	—	2	1	no	—
109	C₂×D₃₁	—	2	1	no	—
110	C₂×D₃₁	—	2	4	no	—
111	C₂×D₃₁	—	4	2	no	—
112	C₂×D₃₁	—	4	2	no	—
113	C₂×D₃₁	—	8	8	no	—
114	C₂×D₃₁	C₄×D₄	1	2	yes	—
115	C₄×C₃₁	—	2	1	no	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
120	Dic₃₁	—	2	1	no	—
121	Dic₃₁	—	2	1	no	—
122	Dic₃₁	—	4	2	no	—
123	Dic₃₁	C₄×D₄	1	2	yes	—
124	C₂²×D₈	—	31	1	no	—
125	#518	—	31	2	no	—
126	#442	—	31	2	no	—
127	#442	—	31	2	no	—
128	C₄×D₈	—	31	1	no	—
129	C₄×D₈	—	31	1	no	—
130	C₄×D₈	—	31	1	no	—
131	C₄×D₈	D₃₁	1	1	yes	characteristic, complemented
132	#427	—	31	1	no	—
133	#418	—	31	2	no	—
134	#405	—	31	1	no	—
135	C₂×C₃₁	—	4	4	no	—
136	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₃₁	#576	1	1	yes	characteristic
138	C₂×C₃₁	#576	1	1	yes	characteristic
139	D₃₁	—	2	2	no	—
140	D₃₁	—	4	4	no	—
141	C₂²×D₄	—	31	2	no	—
142	C₂×D₈	—	31	2	no	—
143	C₂×D₈	—	31	8	no	—
144	C₂×D₈	—	31	1	no	—
145	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
146	C₂×M₁₆	—	31	1	no	—
147	C₂²×C₈	—	31	1	no	—
148	C₄×D₄	—	31	2	no	—
149	C₄×D₄	—	31	2	no	—
150	C₄×D₄	—	31	2	no	—
151	C₄×D₄	—	62	2	no	—
152	C₄×D₄	—	62	2	no	—
153	C₄×D₄	C₂×D₃₁	1	2	yes	complemented
154	#117	—	31	1	no	—
155	#117	—	31	2	no	—
156	C₂×C₂²⋊C₄	—	62	2	no	—
157	C₂×C₂²⋊C₄	—	62	2	no	—
158	#107	—	31	1	no	—
159	#107	C₂×D₃₁	1	1	yes	characteristic, complemented
160	#106	—	31	2	no	—
161	#102	—	31	2	no	—
162	#102	—	31	2	no	—
163	#102	—	31	2	no	—
164	#102	C₂×D₃₁	1	2	yes	complemented
165	#97	—	31	1	no	—
166	#97	—	31	1	no	—
167	C₄×C₈	—	31	1	no	—
168	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
169	C₃₁	#2948	1	1	yes	characteristic, complemented, Sylow
170	C₂⁴	—	62	2	no	—
171	C₂×D₄	—	31	8	no	—
172	C₂×D₄	—	31	2	no	—
173	C₂×D₄	—	62	4	no	—
174	C₂×D₄	—	62	2	no	—
175	C₂×D₄	C₂²×D₃₁	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	31	1	no	—
178	C₂²×C₄	—	62	2	no	—
179	C₂²×C₄	—	62	2	no	—
180	C₂²×C₄	—	62	2	no	—
181	D₈	—	31	8	no	—
182	D₈	—	31	4	no	—
183	D₈	C₄×D₃₁	1	4	yes	complemented
184	M₁₆	—	62	2	no	—
185	C₂×C₈	—	31	1	no	—
186	C₂×C₈	—	31	4	no	—
187	C₂×C₈	—	31	1	no	—
188	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
190	C₄⋊C₄	—	31	1	no	—
191	C₄⋊C₄	—	31	1	no	—
192	C₄⋊C₄	—	31	2	no	—
193	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	31	1	no	—
196	C₂²⋊C₄	—	31	1	no	—
197	C₂²⋊C₄	—	62	2	no	—
198	C₂²⋊C₄	—	62	2	no	—
199	C₂²⋊C₄	—	62	2	no	—
200	C₂²⋊C₄	—	62	2	no	—
201	C₄²	—	31	1	no	—
202	C₄²	—	62	2	no	—
203	C₄²	C₂²×D₃₁	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	31	1	no	—
206	C₂³	—	62	8	no	—
207	C₂³	—	62	2	no	—
208	C₂³	—	248	2	no	—
209	C₂³	—	248	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	31	4	no	—
212	D₄	—	62	8	no	—
213	D₄	—	62	2	no	—
214	D₄	C₂×C₄×D₃₁	1	4	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	31	1	no	—
218	C₂×C₄	—	31	2	no	—
219	C₂×C₄	—	31	2	no	—
220	C₂×C₄	—	31	1	no	—
221	C₂×C₄	—	31	1	no	—
222	C₂×C₄	—	62	2	no	—
223	C₂×C₄	—	248	2	no	—
224	C₂×C₄	—	248	2	no	—
225	C₂×C₄	—	248	2	no	—
226	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
227	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	31	2	no	—
231	C₈	—	62	1	no	—
232	C₈	C₂×C₄×D₃₁	1	2	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	31	2	no	—
237	C₂²	—	62	1	no	—
238	C₂²	—	62	1	no	—
239	C₂²	—	62	4	no	—
240	C₂²	—	124	2	no	—
241	C₂²	—	124	2	no	—
242	C₂²	—	248	8	no	—
243	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	31	2	no	—
248	C₄	—	62	1	no	—
249	C₄	—	62	1	no	—
250	C₄	—	124	2	no	—
251	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
252	C₄	#41×D₃₁	1	1	yes	characteristic
253	C₂	—	4	4	no	—
254	C₂	—	62	2	no	—
255	C₂	—	124	4	no	—
256	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic
257	C₂	#14521850	1	1	yes	characteristic
258	C₂	#14521851	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×D₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₈×D₃₁	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×D₈	D₃₁
hypocenter	C₃₁	#2948
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #14521521	C₂	semidirect
3. #14521159	C₂	semidirect
4. #14521158	C₂	semidirect
5. C₄×D₂₄₈	C₂	semidirect
6. D₈×Dic₃₁	C₂	semidirect
7. #14521131	C₂	semidirect
8. #14521086	C₂	semidirect
9. #14521052	C₂	semidirect
10. #14521003	C₂	semidirect
11. C₄×D₈×C₃₁	C₂	semidirect
14. C₄×D₄×C₃₁	C₂²	semidirect
15. D₄×Dic₃₁	C₂²	semidirect
19. D₈×D₃₁	C₄	semidirect
23. #11758470	C₂²	semidirect
27. C₄×D₁₂₄	C₂²	semidirect
29. C₃₁×#107	C₂²	semidirect
30. C₃₁×#102	C₂²	semidirect
31. C₄×C₈×C₃₁	C₂²	semidirect
32. #11758420	C₂²	semidirect
33. #11758409	C₂²	semidirect
34. #11758406	C₂²	semidirect
35. #11758403	C₂²	semidirect
36. C₈×Dic₃₁	C₂²	semidirect
37. #11758397	C₂²	semidirect
38. #11758396	C₂²	semidirect
39. #11758391	C₂²	semidirect
46. D₄×D₃₁	C₂×C₄	semidirect
51. D₈×C₃₁	C₂×C₄	semidirect
58. #92479	C₂×C₄	semidirect
71. D₂₄₈	C₂×C₄	semidirect
73. C₈×D₃₁	C₂×C₄	semidirect
131. C₄×D₈	D₃₁	semidirect
153. C₄×D₄	C₂×D₃₁	semidirect
159. #107	C₂×D₃₁	semidirect
164. #102	C₂×D₃₁	semidirect
168. C₄×C₈	C₂×D₃₁	semidirect
169. C₃₁	#2948	semidirect
183. D₈	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×D₃₁	2
#14521521	2
#14521159	2
#14521158	2
C₄×D₂₄₈	2
D₈×Dic₃₁	2
#14521131	2
#14521086	2
#14521052	2
#14521003	2
C₄×D₈×C₃₁	2
#2948	31