Subgroups — #16854133

Order 3968.

572 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758484	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758478	C₂	1	1	yes	maximal, characteristic
3	#14521517	C₂	1	1	yes	maximal, characteristic
4	#14521006	C₂	1	2	yes	maximal, complemented
5	#14521005	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521003	C₂	1	2	yes	maximal, complemented
7	#14521001	C₂	1	2	yes	maximal, complemented
8	M₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₃₁	C₂	1	2	yes	maximal, complemented
11	C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#92495	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	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×#92474	C₂²	1	1	yes	characteristic
18	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	#11758484	C₄	1	8	yes	complemented
20	#11758478	C₄	1	8	yes	complemented
21	C₂×#92469	C₂²	1	2	yes	—
22	C₂×#92469	C₂²	1	2	yes	—
23	C₂×C₈×D₃₁	C₂²	1	2	yes	—
24	C₂×C₈×D₃₁	C₂²	1	2	yes	—
25	#11758470	C₄	1	2	yes	—
26	#11758464	C₄	1	2	yes	—
27	(C₂²⋊D₃₁)⋊C₄	C₄	1	4	yes	—
28	#11758449	C₂²	1	2	yes	—
29	C₄²×D₃₁	C₂²	1	2	yes	—
30	C₃₁×#97	C₂²	1	2	yes	—
31	C₄×C₈×C₃₁	C₂²	1	2	yes	—
32	#11758403	C₂²	1	2	yes	—
33	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
34	C₈×Dic₃₁	C₂²	1	2	yes	—
35	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
36	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
37	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
38	#11758391	C₂²	1	2	yes	—
39	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
42	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
43	#92495	C₂×C₄	1	4	yes	—
44	#92495	C₂×C₄	1	4	yes	—
45	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₃₁	C₂³	1	2	yes	—
48	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
49	M₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₄²×C₃₁	C₂³	1	2	yes	—
57	(C₂×D₃₁)⋊C₄	C₂×C₄	1	4	yes	—
58	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
60	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
61	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
62	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
63	#92474	C₂×C₄	1	4	yes	complemented
64	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	complemented
65	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
67	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
68	#92469	C₂×C₄	1	8	yes	complemented
69	#92469	C₂×C₄	1	8	yes	complemented
70	C₈×D₃₁	C₂×C₄	1	8	yes	complemented
71	C₈×D₃₁	C₂×C₄	1	8	yes	complemented
72	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
79	C₂²⋊D₃₁	C₄²	1	8	yes	—
80	C₂×Dic₃₁	—	2	4	no	—
81	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
83	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
84	D₁₂₄	C₄²	1	4	yes	—
85	C₄×D₃₁	—	2	8	no	—
86	C₄×D₃₁	C₂²×C₄	1	4	yes	—
87	C₄×D₃₁	C₂²×C₄	1	4	yes	—
88	Dic₆₂	C₄²	1	4	yes	—
89	C₈×C₃₁	C₂²×C₄	1	2	yes	—
90	C₈×C₃₁	C₂²×C₄	1	4	yes	—
91	C₈×C₃₁	C₂²×C₄	1	2	yes	—
92	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
93	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
94	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
95	C₄×#131	—	31	1	no	maximal, Sylow
96	C₂²×C₃₁	—	2	2	no	—
97	C₂²×C₃₁	C₂×C₄²	1	2	yes	—
98	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₃₁	—	2	4	no	—
100	C₂×D₃₁	C₂×C₄²	1	4	yes	—
101	C₄×C₃₁	—	2	4	no	—
102	C₄×C₃₁	C₂×C₄²	1	2	yes	—
103	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
104	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
105	Dic₃₁	—	2	2	no	—
106	Dic₃₁	C₂×C₄²	1	4	yes	—
107	Dic₃₁	#131	1	4	yes	complemented
108	C₂×#131	—	31	1	no	—
109	C₂×#131	—	31	1	no	—
110	C₄×#41	—	31	1	no	—
111	#405	—	31	1	no	—
112	#405	—	31	2	no	—
113	#405	—	31	2	no	—
114	#405	D₃₁	1	1	yes	characteristic, complemented
115	C₄×M₁₆	—	31	1	no	—
116	C₄×M₁₆	—	31	2	no	—
117	C₂×C₄×C₈	—	31	1	no	—
118	C₂×C₄×C₈	—	31	2	no	—
119	C₂×C₃₁	—	2	2	no	—
120	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
121	C₂×C₃₁	C₂×#131	1	2	yes	—
122	D₃₁	—	2	4	no	—
123	C₂×#41	—	31	1	no	—
124	#131	—	31	8	no	—
125	#131	—	31	8	no	—
126	C₂×M₁₆	—	31	1	no	—
127	C₂×M₁₆	—	31	2	no	—
128	C₂×M₁₆	—	31	2	no	—
129	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
130	C₂²×C₈	—	31	1	no	—
131	C₂²×C₈	—	31	2	no	—
132	C₂²×C₈	—	31	2	no	—
133	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
134	C₄×Q₈	—	31	2	no	—
135	C₄×D₄	—	31	2	no	—
136	C₄×D₄	—	31	4	no	—
137	#117	—	31	2	no	—
138	#117	C₂×D₃₁	1	1	yes	characteristic
139	C₂×C₄²	—	31	1	no	—
140	C₂×C₄²	—	31	2	no	—
141	#97	—	31	2	no	—
142	#97	—	31	2	no	—
143	#97	C₂×D₃₁	1	2	yes	—
144	C₄×C₈	—	31	1	no	—
145	C₄×C₈	—	31	2	no	—
146	C₄×C₈	—	31	1	no	—
147	C₄×C₈	—	31	1	no	—
148	C₄×C₈	—	31	1	no	—
149	C₄×C₈	—	31	2	no	—
150	C₄×C₈	C₂×D₃₁	1	2	yes	—
151	C₃₁	C₄×#131	1	1	yes	characteristic, complemented, Sylow
152	#41	—	31	4	no	—
153	#41	—	31	4	no	—
154	C₂×Q₈	—	31	1	no	—
155	C₂×D₄	—	31	2	no	—
156	C₂×D₄	—	31	1	no	—
157	C₂²×C₄	—	31	2	no	—
158	C₂²×C₄	—	31	2	no	—
159	C₂²×C₄	—	31	4	no	—
160	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
161	M₁₆	—	31	4	no	—
162	M₁₆	—	31	8	no	—
163	M₁₆	—	31	8	no	—
164	M₁₆	C₄×D₃₁	1	4	yes	complemented
165	C₂×C₈	—	31	4	no	—
166	C₂×C₈	—	31	1	no	—
167	C₂×C₈	—	31	2	no	—
168	C₂×C₈	—	31	1	no	—
169	C₂×C₈	—	31	8	no	—
170	C₂×C₈	—	31	8	no	—
171	C₂×C₈	C₄×D₃₁	1	4	yes	complemented
172	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₃₁	1	2	yes	—
174	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
175	C₄⋊C₄	—	31	4	no	—
176	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
177	C₂²⋊C₄	—	31	4	no	—
178	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
179	C₄²	—	31	4	no	—
180	C₄²	—	31	2	no	—
181	C₄²	—	31	1	no	—
182	C₄²	—	31	1	no	—
183	C₄²	C₂²×D₃₁	1	2	yes	—
184	C₂³	—	31	2	no	—
185	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
186	Q₈	—	31	4	no	—
187	D₄	—	31	8	no	—
188	D₄	—	31	4	no	—
189	C₂×C₄	—	31	2	no	—
190	C₂×C₄	—	31	2	no	—
191	C₂×C₄	—	31	4	no	—
192	C₂×C₄	—	31	4	no	—
193	C₂×C₄	—	31	2	no	—
194	C₂×C₄	—	62	8	no	—
195	C₂×C₄	—	62	4	no	—
196	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
197	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
200	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
201	C₈	—	31	2	no	—
202	C₈	—	31	4	no	—
203	C₈	—	31	2	no	—
204	C₈	C₂×C₄×D₃₁	1	2	yes	—
205	C₈	C₂×C₄×D₃₁	1	4	yes	—
206	C₈	C₂×C₄×D₃₁	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	31	4	no	—
209	C₂²	—	62	4	no	—
210	C₂²	C₄²×D₃₁	1	2	yes	—
211	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	31	4	no	—
214	C₄	—	31	4	no	—
215	C₄	—	62	2	no	—
216	C₄	C₄²×D₃₁	1	2	yes	—
217	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
218	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	62	4	no	—
221	C₂	C₂×C₄²×D₃₁	1	1	yes	characteristic
222	C₂	#131×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#405	C₂
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂×#92495	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#405	D₃₁
hypocenter	C₃₁	C₄×#131
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #14521006	C₂	semidirect
5. #14521005	C₂	semidirect
6. #14521003	C₂	semidirect
7. #14521001	C₂	semidirect
8. M₁₆×Dic₃₁	C₂	semidirect
9. C₂×C₈×Dic₃₁	C₂	semidirect
10. C₄×C₈×D₃₁	C₂	semidirect
11. C₃₁×#405	C₂	semidirect
19. #11758484	C₄	semidirect
20. #11758478	C₄	semidirect
49. M₁₆×C₃₁	C₂×C₄	semidirect
50. C₂×C₈×C₃₁	C₂×C₄	semidirect
63. #92474	C₂×C₄	semidirect
64. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
68. #92469	C₂×C₄	semidirect
69. #92469	C₂×C₄	semidirect
70. C₈×D₃₁	C₂×C₄	semidirect
71. C₈×D₃₁	C₂×C₄	semidirect
107. Dic₃₁	#131	semidirect
114. #405	D₃₁	semidirect
151. C₃₁	C₄×#131	semidirect
164. M₁₆	C₄×D₃₁	semidirect
171. 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
C₂×#11758484	2
C₂×#11758478	2
#14521517	2
#14521006	2
#14521005	2
#14521003	2
#14521001	2
M₁₆×Dic₃₁	2
C₂×C₈×Dic₃₁	2
C₄×C₈×D₃₁	2
C₃₁×#405	2
C₄×#131	31