Subgroups — #16858917

Order 3968.

1660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758566	C₂	1	1	yes	maximal, characteristic
2	#14521531	C₂	1	2	yes	maximal, complemented
3	#14521530	C₂	1	3	yes	maximal, complemented
4	#14521523	C₂	1	3	yes	maximal, complemented
5	#14521521	C₂	1	6	yes	maximal, complemented
6	#41×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#92499	C₂	1	2	yes	maximal, complemented
8	#14521517	C₂	1	3	yes	maximal, complemented
9	C₂×#11758470	C₂	1	3	yes	maximal, complemented
10	C₄×D₄×D₃₁	C₂	1	6	yes	maximal, complemented
11	C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	#11758566	C₄	1	16	yes	complemented
14	C₂×#92499	C₂²	1	2	yes	—
15	C₂×D₄×D₃₁	C₂²	1	6	yes	—
16	C₂×#92495	C₂²	1	3	yes	—
17	C₂²×D₁₂₄	C₂²	1	3	yes	—
18	C₂²×C₄×D₃₁	C₂²	1	6	yes	complemented
19	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
20	C₄×D₄×C₃₁	C₂²	1	6	yes	complemented
21	C₃₁×#117	C₂²	1	3	yes	complemented
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	3	yes	complemented
23	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
24	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
25	D₄×Dic₃₁	C₂²	1	3	yes	complemented
26	D₄×Dic₃₁	C₂²	1	3	yes	complemented
27	C₄×C₂²⋊D₃₁	C₂²	1	12	yes	complemented
28	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	6	yes	complemented
29	#11758503	C₂²	1	3	yes	complemented
30	C₂×C₄×Dic₃₁	C₂²	1	3	yes	complemented
31	#11758470	C₂²	1	6	yes	complemented
32	#11758470	C₂²	1	3	yes	complemented
33	#11758470	C₂²	1	6	yes	complemented
34	#11758470	C₂²	1	6	yes	complemented
35	#11758470	C₂²	1	3	yes	complemented
36	#11758469	C₂²	1	6	yes	complemented
37	C₄⋊C₄×D₃₁	C₂²	1	6	yes	complemented
38	#11758464	C₂²	1	1	yes	characteristic, complemented
39	#11758464	C₂²	1	3	yes	complemented
40	(C₂²⋊D₃₁)⋊C₄	C₂²	1	6	yes	complemented
41	(C₂²⋊D₃₁)⋊C₄	C₂²	1	6	yes	complemented
42	C₂²⋊C₄×D₃₁	C₂²	1	12	yes	complemented
43	C₄×D₁₂₄	C₂²	1	6	yes	complemented
44	C₄×D₁₂₄	C₂²	1	6	yes	complemented
45	#11758449	C₂²	1	6	yes	complemented
46	C₄²×D₃₁	C₂²	1	6	yes	complemented
47	C₂³×D₃₁	C₂³	1	6	yes	—
48	#41×C₃₁	C₂×C₄	1	8	yes	complemented
49	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₃₁	C₂³	1	3	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
52	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
53	C₂×C₂²⋊D₃₁	C₂³	1	6	yes	—
54	C₂²×Dic₃₁	C₂³	1	3	yes	—
55	C₂²×Dic₃₁	C₂³	1	3	yes	—
56	#92499	C₂×C₄	1	16	yes	complemented
57	D₄×D₃₁	C₂×C₄	1	48	yes	complemented
58	#92495	C₂×C₄	1	24	yes	complemented
59	C₂×D₁₂₄	C₂×C₄	1	24	yes	complemented
60	C₂×D₁₂₄	C₂³	1	6	yes	—
61	C₂×D₁₂₄	C₂³	1	3	yes	—
62	C₂×C₄×D₃₁	—	4	12	no	—
63	C₂×C₄×D₃₁	C₂³	1	6	yes	—
64	C₂×C₄×D₃₁	C₂³	1	6	yes	—
65	C₂×C₄×D₃₁	C₂³	1	12	yes	—
66	C₂×C₄×D₃₁	C₂³	1	6	yes	—
67	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
69	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
71	C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
72	C₄²×C₃₁	C₂³	1	6	yes	—
73	C₂²⋊Dic₃₁	C₂³	1	6	yes	—
74	(C₂×D₃₁)⋊C₄	C₂³	1	6	yes	—
75	(C₂×D₃₁)⋊C₄	C₂³	1	6	yes	—
76	(C₂×D₃₁)⋊C₄	C₂³	1	12	yes	—
77	C₄⋊Dic₃₁	C₂³	1	3	yes	—
78	C₄⋊Dic₃₁	C₂³	1	3	yes	—
79	Dic₃₁⋊C₄	C₂³	1	2	yes	—
80	Dic₃₁⋊C₄	C₂³	1	6	yes	—
81	C₄×Dic₃₁	C₂³	1	2	yes	—
82	C₄×Dic₃₁	C₂³	1	3	yes	—
83	C₄×Dic₃₁	C₂³	1	6	yes	—
84	C₄×Dic₃₁	C₂³	1	3	yes	—
85	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
86	C₄×Dic₃₁	C₂³	1	3	yes	—
87	C₂³×C₃₁	C₂⁴	1	3	yes	—
88	C₂²×D₃₁	—	4	6	no	—
89	C₂²×D₃₁	—	4	6	no	—
90	C₂²×D₃₁	C₂²×C₄	1	24	yes	complemented
91	C₂²×D₃₁	C₂⁴	1	6	yes	—
92	Q₈×C₃₁	C₂²×C₄	1	4	yes	complemented
93	D₄×C₃₁	C₂²×C₄	1	12	yes	complemented
94	C₂×C₄×C₃₁	—	4	6	no	—
95	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	complemented
96	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
97	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
98	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂⁴	1	3	yes	—
100	C₂²⋊D₃₁	C₂²×C₄	1	24	yes	complemented
101	C₂×Dic₃₁	—	2	6	no	—
102	C₂×Dic₃₁	—	4	3	no	—
103	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₃₁	C₂⁴	1	3	yes	—
105	C₂×Dic₃₁	C₂⁴	1	3	yes	—
106	C₂×Dic₃₁	C₂⁴	1	2	yes	—
107	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
108	D₁₂₄	C₂²×C₄	1	12	yes	complemented
109	D₁₂₄	C₂²×C₄	1	24	yes	complemented
110	C₄×D₃₁	—	2	12	no	—
111	C₄×D₃₁	—	4	12	no	—
112	C₄×D₃₁	—	4	6	no	—
113	C₄×D₃₁	C₂²×C₄	1	24	yes	complemented
114	Dic₆₂	C₂²×C₄	1	4	yes	complemented
115	C₄×H(5,𝔽₂)	—	31	1	no	maximal, Sylow
116	C₂²×C₃₁	—	2	3	no	—
117	C₂²×C₃₁	—	2	3	no	—
118	C₂²×C₃₁	C₂³×C₄	1	6	yes	—
119	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
120	C₂×D₃₁	—	2	6	no	—
121	C₂×D₃₁	—	2	6	no	—
122	C₂×D₃₁	—	4	24	no	—
123	C₂×D₃₁	C₂³×C₄	1	12	yes	—
124	C₄×C₃₁	—	2	6	no	—
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	C₂³×C₄	1	2	yes	—
127	C₄×C₃₁	C₂³×C₄	1	6	yes	—
128	Dic₃₁	—	2	3	no	—
129	Dic₃₁	—	2	3	no	—
130	Dic₃₁	—	2	1	no	—
131	Dic₃₁	C₂³×C₄	1	4	yes	—
132	Dic₃₁	H(5,𝔽₂)	1	2	yes	complemented
133	C₂×H(5,𝔽₂)	—	31	1	no	—
134	#520	—	31	2	no	—
135	#520	—	31	3	no	—
136	#520	D₃₁	1	1	yes	characteristic, complemented
137	#518	—	31	6	no	—
138	#518	—	31	3	no	—
139	C₄×#41	—	31	1	no	—
140	C₄×#41	—	31	2	no	—
141	C₄×#41	—	31	3	no	—
142	C₂×C₄×D₄	—	31	6	no	—
143	C₂×C₄×D₄	—	31	3	no	—
144	C₂×C₃₁	—	2	6	no	—
145	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
146	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₃₁	#585	1	1	yes	characteristic
148	D₃₁	—	2	12	no	—
149	H(5,𝔽₂)	—	31	16	no	—
150	C₂×#41	—	31	2	no	—
151	C₂×#41	—	31	3	no	—
152	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
153	C₂²×D₄	—	31	6	no	—
154	C₂²×D₄	—	31	3	no	—
155	C₂³×C₄	—	31	6	no	—
156	C₄×Q₈	—	31	1	no	—
157	C₄×Q₈	—	31	1	no	—
158	C₄×Q₈	—	31	1	no	—
159	C₄×Q₈	—	31	3	no	—
160	C₄×Q₈	C₂×D₃₁	1	2	yes	complemented
161	C₄×D₄	—	31	3	no	—
162	C₄×D₄	—	31	3	no	—
163	C₄×D₄	—	31	12	no	—
164	C₄×D₄	—	31	6	no	—
165	C₄×D₄	—	31	3	no	—
166	C₄×D₄	—	31	6	no	—
167	C₄×D₄	—	31	6	no	—
168	C₄×D₄	—	31	3	no	—
169	C₄×D₄	—	31	6	no	—
170	C₄×D₄	—	31	6	no	—
171	C₄×D₄	—	31	6	no	—
172	C₄×D₄	—	31	6	no	—
173	C₄×D₄	C₂×D₃₁	1	6	yes	complemented
174	#117	—	31	3	no	—
175	#117	—	31	6	no	—
176	#117	—	31	6	no	—
177	#117	C₂×D₃₁	1	3	yes	complemented
178	C₂×C₄⋊C₄	—	31	6	no	—
179	C₂×C₄⋊C₄	C₂×D₃₁	1	3	yes	complemented
180	C₂×C₂²⋊C₄	—	31	6	no	—
181	C₂×C₂²⋊C₄	—	31	12	no	—
182	C₂×C₄²	—	31	3	no	—
183	C₂×C₄²	—	31	6	no	—
184	C₃₁	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
185	C₂⁴	—	31	6	no	—
186	#41	—	31	16	no	—
187	#41	—	31	24	no	—
188	#41	C₄×D₃₁	1	8	yes	complemented
189	C₂×Q₈	—	31	1	no	—
190	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
191	C₂×D₄	—	31	6	no	—
192	C₂×D₄	—	31	48	no	—
193	C₂×D₄	—	31	24	no	—
194	C₂×D₄	—	31	3	no	—
195	C₂×D₄	—	31	6	no	—
196	C₂×D₄	C₂²×D₃₁	1	3	yes	—
197	C₂²×C₄	—	31	6	no	—
198	C₂²×C₄	—	31	3	no	—
199	C₂²×C₄	—	31	3	no	—
200	C₂²×C₄	—	31	6	no	—
201	C₂²×C₄	—	31	6	no	—
202	C₂²×C₄	—	31	12	no	—
203	C₂²×C₄	—	124	12	no	—
204	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
205	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
206	C₄⋊C₄	—	31	6	no	—
207	C₄⋊C₄	—	31	2	no	—
208	C₄⋊C₄	—	31	3	no	—
209	C₄⋊C₄	—	31	3	no	—
210	C₄⋊C₄	C₂²×D₃₁	1	6	yes	—
211	C₄⋊C₄	C₂²×D₃₁	1	3	yes	—
212	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
213	C₂²⋊C₄	—	31	6	no	—
214	C₂²⋊C₄	—	31	6	no	—
215	C₂²⋊C₄	—	31	12	no	—
216	C₂²⋊C₄	—	31	6	no	—
217	C₂²⋊C₄	C₂²×D₃₁	1	6	yes	—
218	C₄²	—	31	2	no	—
219	C₄²	—	31	1	no	—
220	C₄²	—	31	3	no	—
221	C₄²	—	31	6	no	—
222	C₄²	—	31	3	no	—
223	C₄²	—	31	3	no	—
224	C₄²	C₂²×D₃₁	1	6	yes	—
225	C₂³	—	31	6	no	—
226	C₂³	—	31	24	no	—
227	C₂³	—	124	6	no	—
228	C₂³	—	124	6	no	—
229	C₂³	C₂³×D₃₁	1	3	yes	—
230	Q₈	—	31	4	no	—
231	Q₈	C₂×C₄×D₃₁	1	4	yes	complemented
232	D₄	—	31	24	no	—
233	D₄	—	31	24	no	—
234	D₄	—	31	12	no	—
235	D₄	C₂×C₄×D₃₁	1	12	yes	complemented
236	C₂×C₄	—	4	6	no	—
237	C₂×C₄	—	31	1	no	—
238	C₂×C₄	—	31	2	no	—
239	C₂×C₄	—	31	24	no	—
240	C₂×C₄	—	31	1	no	—
241	C₂×C₄	—	31	3	no	—
242	C₂×C₄	—	31	3	no	—
243	C₂×C₄	—	62	12	no	—
244	C₂×C₄	—	62	6	no	—
245	C₂×C₄	—	124	6	no	—
246	C₂×C₄	—	124	12	no	—
247	C₂×C₄	—	124	3	no	—
248	C₂×C₄	C₂×C₄×D₃₁	1	12	yes	complemented
249	C₂×C₄	C₂³×D₃₁	1	2	yes	—
250	C₂×C₄	C₂³×D₃₁	1	6	yes	—
251	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₃₁	1	3	yes	—
253	C₂²	—	2	3	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	31	12	no	—
256	C₂²	—	62	6	no	—
257	C₂²	—	62	6	no	—
258	C₂²	—	124	24	no	—
259	C₂²	C₂²×C₄×D₃₁	1	6	yes	—
260	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
261	C₄	—	2	6	no	—
262	C₄	—	2	2	no	—
263	C₄	—	31	4	no	—
264	C₄	—	31	2	no	—
265	C₄	—	62	3	no	—
266	C₄	—	62	3	no	—
267	C₄	—	62	1	no	—
268	C₄	C₂²×C₄×D₃₁	1	6	yes	—
269	C₄	C₂²×C₄×D₃₁	1	2	yes	—
270	C₂	—	2	6	no	—
271	C₂	—	62	12	no	—
272	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
273	C₂	H(5,𝔽₂)×D₃₁	1	1	yes	characteristic
274	C₂	#14521886	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Fitting	C₃₁×#520	C₂
socle, cosocle	C₂²×C₃₁	C₂⁵
prime radical	C₂×#11758566	C₂
hypercenter	#520	D₃₁
hypocenter	C₃₁	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #14521531	C₂	semidirect
3. #14521530	C₂	semidirect
4. #14521523	C₂	semidirect
5. #14521521	C₂	semidirect
6. #41×Dic₃₁	C₂	semidirect
7. C₄×#92499	C₂	semidirect
8. #14521517	C₂	semidirect
9. C₂×#11758470	C₂	semidirect
10. C₄×D₄×D₃₁	C₂	semidirect
11. C₃₁×#520	C₂	semidirect
13. #11758566	C₄	semidirect
18. C₂²×C₄×D₃₁	C₂²	semidirect
19. C₄×Q₈×C₃₁	C₂²	semidirect
20. C₄×D₄×C₃₁	C₂²	semidirect
21. C₃₁×#117	C₂²	semidirect
22. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
23. Q₈×Dic₃₁	C₂²	semidirect
24. Q₈×Dic₃₁	C₂²	semidirect
25. D₄×Dic₃₁	C₂²	semidirect
26. D₄×Dic₃₁	C₂²	semidirect
27. C₄×C₂²⋊D₃₁	C₂²	semidirect
28. C₂×(C₂×D₃₁)⋊C₄	C₂²	semidirect
29. #11758503	C₂²	semidirect
30. C₂×C₄×Dic₃₁	C₂²	semidirect
31. #11758470	C₂²	semidirect
32. #11758470	C₂²	semidirect
33. #11758470	C₂²	semidirect
34. #11758470	C₂²	semidirect
35. #11758470	C₂²	semidirect
36. #11758469	C₂²	semidirect
37. C₄⋊C₄×D₃₁	C₂²	semidirect
38. #11758464	C₂²	semidirect
39. #11758464	C₂²	semidirect
40. (C₂²⋊D₃₁)⋊C₄	C₂²	semidirect
41. (C₂²⋊D₃₁)⋊C₄	C₂²	semidirect
42. C₂²⋊C₄×D₃₁	C₂²	semidirect
43. C₄×D₁₂₄	C₂²	semidirect
44. C₄×D₁₂₄	C₂²	semidirect
45. #11758449	C₂²	semidirect
46. C₄²×D₃₁	C₂²	semidirect
48. #41×C₃₁	C₂×C₄	semidirect
56. #92499	C₂×C₄	semidirect
57. D₄×D₃₁	C₂×C₄	semidirect
58. #92495	C₂×C₄	semidirect
59. C₂×D₁₂₄	C₂×C₄	semidirect
90. C₂²×D₃₁	C₂²×C₄	semidirect
92. Q₈×C₃₁	C₂²×C₄	semidirect
93. D₄×C₃₁	C₂²×C₄	semidirect
95. C₂×C₄×C₃₁	C₂²×C₄	semidirect
100. C₂²⋊D₃₁	C₂²×C₄	semidirect
108. D₁₂₄	C₂²×C₄	semidirect
109. D₁₂₄	C₂²×C₄	semidirect
113. C₄×D₃₁	C₂²×C₄	semidirect
114. Dic₆₂	C₂²×C₄	semidirect
132. Dic₃₁	H(5,𝔽₂)	semidirect
136. #520	D₃₁	semidirect
160. C₄×Q₈	C₂×D₃₁	semidirect
173. C₄×D₄	C₂×D₃₁	semidirect
177. #117	C₂×D₃₁	semidirect
179. C₂×C₄⋊C₄	C₂×D₃₁	semidirect
184. C₃₁	C₄×H(5,𝔽₂)	semidirect
188. #41	C₄×D₃₁	semidirect
231. Q₈	C₂×C₄×D₃₁	semidirect
235. D₄	C₂×C₄×D₃₁	semidirect
248. C₂×C₄	C₂×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×#11758566	2
#14521531	2
#14521530	2
#14521523	2
#14521521	2
#41×Dic₃₁	2
C₄×#92499	2
#14521517	2
C₂×#11758470	2
C₄×D₄×D₃₁	2
C₃₁×#520	2
C₄×H(5,𝔽₂)	31