Subgroups — #16854127

Order 3968.

564 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92469	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₃₁	C₂	1	1	yes	maximal, characteristic
3	#117×D₃₁	C₂	1	1	yes	maximal, characteristic
4	#14521007	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521006	C₂	1	2	yes	maximal, complemented
6	#14521005	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521004	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521003	C₂	1	2	yes	maximal, complemented
9	#14521001	C₂	1	2	yes	maximal, complemented
10	C₄×#92469	C₂	1	2	yes	maximal, complemented
11	C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#117	C₂²	1	1	yes	characteristic
16	#11758503	C₂²	1	1	yes	characteristic
17	C₂×#92474	C₂²	1	1	yes	characteristic
18	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	M₁₆×D₃₁	C₄	1	8	yes	complemented
20	C₂×#92469	C₄	1	8	yes	complemented
21	C₂×#92469	C₂²	1	1	yes	characteristic
22	C₂×#92469	C₂²	1	2	yes	—
23	C₂×#92469	C₂²	1	1	yes	characteristic
24	C₂×#92469	C₂²	1	1	yes	characteristic
25	C₂×#92469	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₃₁	C₂²	1	2	yes	—
27	#11758469	C₄	1	2	yes	—
28	C₄⋊C₄×D₃₁	C₄	1	2	yes	—
29	C₂²⋊C₄×D₃₁	C₄	1	2	yes	—
30	#11758454	C₄	1	2	yes	—
31	#11758449	C₂²	1	2	yes	—
32	C₄²×D₃₁	C₂²	1	2	yes	—
33	C₃₁×#97	C₂²	1	2	yes	complemented
34	C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
35	#11758403	C₂²	1	2	yes	complemented
36	#11758403	C₂²	1	2	yes	complemented
37	C₈×Dic₃₁	C₂²	1	2	yes	complemented
38	C₈×Dic₃₁	C₂²	1	2	yes	complemented
39	#11758391	C₂²	1	2	yes	complemented
40	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
41	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
49	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
51	M₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₄²×C₃₁	C₂³	1	2	yes	—
59	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
60	(C₂×D₃₁)⋊C₄	C₂×C₄	1	4	yes	—
61	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
62	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
63	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
64	C₄×Dic₃₁	C₂³	1	2	yes	—
65	#92474	C₂×C₄	1	4	yes	complemented
66	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	complemented
67	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
69	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	#92469	C₂×C₄	1	4	yes	—
71	#92469	C₂×C₄	1	4	yes	—
72	#92469	C₂×C₄	1	8	yes	—
73	#92469	C₂×C₄	1	4	yes	complemented
74	#92469	C₂×C₄	1	4	yes	complemented
75	C₈×D₃₁	C₂×C₄	1	8	yes	complemented
76	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₃₁	—	4	2	no	—
78	C₂²×D₃₁	C₄²	1	4	yes	—
79	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₃₁	C₄²	1	4	yes	—
87	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
88	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₄×D₃₁	—	4	4	no	—
91	C₄×D₃₁	C₄²	1	4	yes	—
92	C₄×D₃₁	C₄²	1	4	yes	—
93	C₄×D₃₁	C₂²×C₄	1	2	yes	—
94	C₄×D₃₁	C₂²×C₄	1	2	yes	—
95	C₄×D₃₁	C₂²×C₄	1	2	yes	—
96	C₄×D₃₁	C₂²×C₄	1	2	yes	—
97	C₈×C₃₁	C₂²×C₄	1	4	yes	—
98	C₈×C₃₁	C₂²×C₄	1	2	yes	—
99	C₈×C₃₁	C₂²×C₄	1	2	yes	—
100	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
101	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
102	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
103	#2873	—	31	1	no	maximal, Sylow
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	C₂×C₄²	1	2	yes	—
106	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₃₁	—	2	2	no	—
108	C₂×D₃₁	—	2	2	no	—
109	C₂×D₃₁	—	4	4	no	—
110	C₂×D₃₁	C₂×C₄²	1	2	yes	—
111	C₂×D₃₁	C₂×C₄²	1	2	yes	—
112	C₄×C₃₁	—	2	4	no	—
113	C₄×C₃₁	C₂×C₄²	1	2	yes	—
114	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
116	Dic₃₁	—	2	4	no	—
117	Dic₃₁	C₂×C₄²	1	2	yes	—
118	Dic₃₁	C₂×C₄²	1	2	yes	—
119	C₂²×M₁₆	—	31	1	no	—
120	C₂²×M₁₆	—	31	1	no	—
121	C₂×#117	—	31	1	no	—
122	#405	—	31	1	no	—
123	#405	—	31	1	no	—
124	#405	—	31	1	no	—
125	#405	—	31	2	no	—
126	#405	—	31	2	no	—
127	#405	D₃₁	1	1	yes	characteristic, complemented
128	C₄×M₁₆	—	31	2	no	—
129	C₄×M₁₆	—	31	2	no	—
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
132	C₂×C₃₁	#568	1	2	yes	—
133	D₃₁	—	2	2	no	—
134	D₃₁	—	2	2	no	—
135	C₂³×C₄	—	31	1	no	—
136	C₂×M₁₆	—	31	1	no	—
137	C₂×M₁₆	—	31	8	no	—
138	C₂×M₁₆	—	31	8	no	—
139	C₂×M₁₆	—	31	1	no	—
140	C₂×M₁₆	—	31	2	no	—
141	C₂×M₁₆	—	31	1	no	—
142	C₂×M₁₆	—	31	1	no	—
143	C₂×M₁₆	—	31	1	no	—
144	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
145	C₂²×C₈	—	31	1	no	—
146	C₂²×C₈	—	31	2	no	—
147	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
148	#117	—	31	1	no	—
149	#117	—	31	2	no	—
150	#117	—	31	2	no	—
151	#117	—	31	2	no	—
152	#117	C₂×D₃₁	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	31	2	no	—
154	C₂×C₂²⋊C₄	—	31	2	no	—
155	C₂×C₄²	—	31	2	no	—
156	#97	—	31	2	no	—
157	#97	—	31	2	no	—
158	#97	—	31	2	no	—
159	#97	C₂×D₃₁	1	2	yes	complemented
160	C₄×C₈	—	31	2	no	—
161	C₄×C₈	—	31	2	no	—
162	C₄×C₈	—	31	2	no	—
163	C₄×C₈	C₂×D₃₁	1	2	yes	complemented
164	C₃₁	#2873	1	1	yes	characteristic, complemented, Sylow
165	C₂⁴	—	31	1	no	—
166	C₂²×C₄	—	31	1	no	—
167	C₂²×C₄	—	31	1	no	—
168	C₂²×C₄	—	31	1	no	—
169	C₂²×C₄	—	31	4	no	—
170	C₂²×C₄	—	31	4	no	—
171	C₂²×C₄	—	31	1	no	—
172	C₂²×C₄	—	31	1	no	—
173	C₂²×C₄	—	31	4	no	—
174	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
175	M₁₆	—	31	4	no	—
176	M₁₆	—	31	8	no	—
177	M₁₆	—	31	4	no	—
178	M₁₆	—	31	4	no	—
179	M₁₆	—	31	4	no	—
180	M₁₆	—	31	4	no	—
181	M₁₆	C₄×D₃₁	1	4	yes	complemented
182	C₂×C₈	—	31	4	no	—
183	C₂×C₈	—	31	1	no	—
184	C₂×C₈	—	31	2	no	—
185	C₂×C₈	—	31	1	no	—
186	C₂×C₈	—	31	8	no	—
187	C₂×C₈	C₄×D₃₁	1	4	yes	complemented
188	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₃₁	1	2	yes	—
190	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
191	C₄⋊C₄	—	31	4	no	—
192	C₄⋊C₄	—	31	2	no	—
193	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
194	C₂²⋊C₄	—	31	2	no	—
195	C₂²⋊C₄	—	31	4	no	—
196	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
197	C₄²	—	31	4	no	—
198	C₄²	—	31	2	no	—
199	C₄²	C₂²×D₃₁	1	2	yes	—
200	C₂³	—	31	1	no	—
201	C₂³	—	31	1	no	—
202	C₂³	—	31	4	no	—
203	C₂³	—	124	2	no	—
204	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
205	C₂×C₄	—	31	4	no	—
206	C₂×C₄	—	31	1	no	—
207	C₂×C₄	—	31	1	no	—
208	C₂×C₄	—	31	4	no	—
209	C₂×C₄	—	31	4	no	—
210	C₂×C₄	—	31	2	no	—
211	C₂×C₄	—	31	2	no	—
212	C₂×C₄	—	31	2	no	—
213	C₂×C₄	—	31	2	no	—
214	C₂×C₄	—	31	4	no	—
215	C₂×C₄	—	124	4	no	—
216	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
220	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
221	C₈	—	31	4	no	—
222	C₈	—	31	2	no	—
223	C₈	—	31	2	no	—
224	C₈	C₂×C₄×D₃₁	1	4	yes	—
225	C₈	C₂×C₄×D₃₁	1	2	yes	—
226	C₈	C₂×C₄×D₃₁	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	31	2	no	—
229	C₂²	—	31	2	no	—
230	C₂²	—	62	2	no	—
231	C₂²	—	62	2	no	—
232	C₂²	—	124	4	no	—
233	C₂²	C₄²×D₃₁	1	2	yes	—
234	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	31	2	no	—
237	C₄	—	31	2	no	—
238	C₄	—	62	4	no	—
239	C₄	C₄²×D₃₁	1	2	yes	—
240	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
241	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	62	2	no	—
244	C₂	—	62	2	no	—
245	C₂	C₂×C₄²×D₃₁	1	1	yes	characteristic
246	C₂	#14521814	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Fitting	C₃₁×#405	C₂
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#405	D₃₁
hypocenter	C₃₁	#2873
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #14521007	C₂	semidirect
5. #14521006	C₂	semidirect
6. #14521005	C₂	semidirect
7. #14521004	C₂	semidirect
8. #14521003	C₂	semidirect
9. #14521001	C₂	semidirect
10. C₄×#92469	C₂	semidirect
11. C₃₁×#405	C₂	semidirect
19. M₁₆×D₃₁	C₄	semidirect
20. C₂×#92469	C₄	semidirect
33. C₃₁×#97	C₂²	semidirect
34. C₄×C₈×C₃₁	C₂²	semidirect
35. #11758403	C₂²	semidirect
36. #11758403	C₂²	semidirect
37. C₈×Dic₃₁	C₂²	semidirect
38. C₈×Dic₃₁	C₂²	semidirect
39. #11758391	C₂²	semidirect
40. C₄×C₃₁⋊C₈	C₂²	semidirect
51. M₁₆×C₃₁	C₂×C₄	semidirect
52. C₂×C₈×C₃₁	C₂×C₄	semidirect
65. #92474	C₂×C₄	semidirect
66. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
73. #92469	C₂×C₄	semidirect
74. #92469	C₂×C₄	semidirect
75. C₈×D₃₁	C₂×C₄	semidirect
127. #405	D₃₁	semidirect
159. #97	C₂×D₃₁	semidirect
163. C₄×C₈	C₂×D₃₁	semidirect
164. C₃₁	#2873	semidirect
181. M₁₆	C₄×D₃₁	semidirect
187. C₂×C₈	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#92469	2
C₂×M₁₆×D₃₁	2
#117×D₃₁	2
#14521007	2
#14521006	2
#14521005	2
#14521004	2
#14521003	2
#14521001	2
C₄×#92469	2
C₃₁×#405	2
#2873	31