Subgroups — #16856281

Order 3968.

380 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521583	C₂	1	2	yes	maximal, complemented
2	#14521451	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521433	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521404	C₂	1	2	yes	maximal, complemented
5	#14521290	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521285	C₂	1	1	yes	maximal, characteristic
7	#14521278	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#92470	C₂	1	2	yes	maximal, complemented
10	C₄×D₂₄₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	#11758507	C₂²	1	2	yes	—
16	#11758506	—	2	2	no	—
17	C₄×C₂²⋊D₃₁	—	2	2	no	—
18	#11758501	C₂²	1	2	yes	—
19	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
20	C₂×D₂₄₈	C₂²	1	1	yes	characteristic
21	C₂×#92470	C₂²	1	2	yes	—
22	#11758453	—	2	2	no	—
23	#11758452	C₂²	1	2	yes	—
24	C₄×D₁₂₄	C₂²	1	2	yes	—
25	#11758449	—	2	2	no	—
26	#11758447	C₂²	1	2	yes	—
27	C₄×Dic₆₂	C₂²	1	2	yes	—
28	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	#11758409	C₂²	1	2	yes	—
33	#11758409	C₂²	1	2	yes	—
34	#11758406	C₂²	1	1	yes	characteristic
35	#11758406	C₂²	1	1	yes	characteristic
36	#11758405	C₂²	1	2	yes	—
37	#11758404	C₂²	1	2	yes	—
38	#11758404	C₂²	1	2	yes	—
39	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₃₁	—	2	2	no	—
43	C₂×D₁₂₄	C₂³	1	2	yes	—
44	C₂×C₄×D₃₁	—	2	2	no	—
45	C₂×Dic₆₂	C₂³	1	2	yes	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄²×C₃₁	D₄	1	1	yes	characteristic
53	C₄²×C₃₁	D₄	1	1	yes	characteristic
54	C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₃₁	—	2	2	no	—
57	(C₂×D₃₁)⋊C₄	—	2	2	no	—
58	(C₂×D₃₁)⋊C₄	—	2	2	no	—
59	(C₂×D₃₁)⋊C₄	—	2	2	no	—
60	(C₂×D₃₁)⋊C₄	—	2	2	no	—
61	C₄⋊Dic₃₁	C₂³	1	2	yes	—
62	C₄⋊Dic₃₁	C₂³	1	2	yes	—
63	Dic₃₁⋊C₄	—	2	2	no	—
64	Dic₃₁⋊C₄	—	2	2	no	—
65	Dic₃₁⋊C₄	—	2	2	no	—
66	Dic₃₁⋊C₄	—	2	2	no	—
67	C₄×Dic₃₁	—	2	2	no	—
68	Dic₁₂₄	—	2	2	no	—
69	D₂₄₈	—	2	2	no	—
70	#92470	—	2	4	no	—
71	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	1	no	—
76	C₂×C₄×C₃₁	—	2	1	no	—
77	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₃₁	—	4	4	no	—
84	C₂×Dic₃₁	—	2	2	no	—
85	C₂×Dic₃₁	—	2	2	no	—
86	C₂×Dic₃₁	—	2	2	no	—
87	D₁₂₄	—	2	2	no	—
88	D₁₂₄	—	2	2	no	—
89	C₄×D₃₁	—	4	4	no	—
90	Dic₆₂	—	2	2	no	—
91	Dic₆₂	—	2	2	no	—
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	#41	1	2	yes	—
95	C₈×C₃₁	#41	1	2	yes	—
96	#3168	—	31	1	no	maximal, Sylow
97	C₂²×C₃₁	—	2	1	no	—
98	C₂²×C₃₁	—	2	1	no	—
99	C₂²×C₃₁	#135	1	2	yes	—
100	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×D₃₁	—	4	2	no	—
102	C₂×D₃₁	—	4	2	no	—
103	C₂×D₃₁	—	4	2	no	—
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	2	1	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	#135	1	2	yes	—
109	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
110	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
111	Dic₃₁	—	4	2	no	—
112	Dic₃₁	—	4	2	no	—
113	Dic₃₁	—	4	2	no	—
114	#529	—	31	2	no	—
115	#499	—	31	1	no	—
116	#495	—	31	1	no	—
117	#488	—	31	2	no	—
118	#467	—	31	1	no	—
119	#466	—	31	1	no	—
120	#465	—	31	2	no	—
121	C₄×Q₁₆	—	31	1	no	—
122	C₄×SD₁₆	—	31	2	no	—
123	C₄×D₈	—	31	1	no	—
124	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic, complemented
125	C₂×C₃₁	—	2	2	no	—
126	C₂×C₃₁	#532	1	1	yes	characteristic
127	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
128	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
129	D₃₁	—	8	2	no	—
130	C₂×Q₁₆	—	31	1	no	—
131	C₂×SD₁₆	—	31	2	no	—
132	C₂×D₈	—	31	1	no	—
133	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
135	#126	—	62	2	no	—
136	#125	—	31	2	no	—
137	#124	—	31	2	no	—
138	#123	—	62	2	no	—
139	C₂²⋊Q₈	—	31	2	no	—
140	#121	—	31	2	no	—
141	C₄×Q₈	—	31	2	no	—
142	C₄×D₄	—	31	2	no	—
143	C₄×D₄	—	62	2	no	—
144	#117	—	62	2	no	—
145	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
146	#107	—	31	1	no	—
147	#107	—	31	1	no	—
148	#106	—	31	2	no	—
149	#103	—	31	2	no	—
150	#103	—	31	2	no	—
151	#102	—	31	2	no	—
152	#102	—	31	2	no	—
153	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
154	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
155	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
156	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
157	C₃₁	#3168	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	31	2	no	—
159	C₂×D₄	—	31	2	no	—
160	C₂×D₄	—	62	2	no	—
161	C₂²×C₄	—	62	2	no	—
162	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
163	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
165	Q₁₆	—	62	2	no	—
166	SD₁₆	—	62	4	no	—
167	D₈	—	62	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
174	C₄⋊C₄	—	31	2	no	—
175	C₄⋊C₄	—	31	2	no	—
176	C₄⋊C₄	—	62	2	no	—
177	C₄⋊C₄	—	62	2	no	—
178	C₄⋊C₄	—	62	2	no	—
179	C₄⋊C₄	—	62	2	no	—
180	C₂²⋊C₄	—	62	2	no	—
181	C₂²⋊C₄	—	62	2	no	—
182	C₂²⋊C₄	—	62	2	no	—
183	C₂²⋊C₄	—	62	2	no	—
184	C₂²⋊C₄	—	62	2	no	—
185	C₄²	—	62	2	no	—
186	C₄²	D₁₂₄	1	1	yes	characteristic
187	C₄²	D₁₂₄	1	1	yes	characteristic
188	C₄²	C₂²×D₃₁	1	1	yes	characteristic
189	C₄²	C₂²×D₃₁	1	1	yes	characteristic
190	C₂³	—	62	2	no	—
191	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
192	Q₈	—	62	2	no	—
193	Q₈	—	62	2	no	—
194	D₄	—	62	2	no	—
195	D₄	—	62	2	no	—
196	D₄	—	124	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	62	2	no	—
202	C₂×C₄	—	62	2	no	—
203	C₂×C₄	—	62	2	no	—
204	C₂×C₄	—	124	4	no	—
205	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
206	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
207	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
208	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
209	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	#92495	1	2	yes	—
214	C₈	#92495	1	2	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	124	2	no	—
218	C₂²	—	124	2	no	—
219	C₂²	—	124	2	no	—
220	C₂²	#11758481	1	2	yes	—
221	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	124	2	no	—
227	C₄	—	124	2	no	—
228	C₄	—	124	2	no	—
229	C₄	#11758481	1	2	yes	—
230	C₄	C₂×#92495	1	1	yes	characteristic
231	C₄	C₂×#92495	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	248	2	no	—
234	C₂	#14521599	1	1	yes	characteristic
235	C₂	C₂×#11758481	1	1	yes	characteristic
236	C₂	C₂×#11758481	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	#14521285	C₂
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	#3168
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #14521583	C₂	semidirect
2. #14521451	C₂	semidirect
3. #14521433	C₂	semidirect
4. #14521404	C₂	semidirect
5. #14521290	C₂	semidirect
7. #14521278	C₂	semidirect
8. C₄×Dic₁₂₄	C₂	semidirect
9. C₄×#92470	C₂	semidirect
10. C₄×D₂₄₈	C₂	semidirect
11. C₂×C₄×C₈×C₃₁	C₂	semidirect
124. C₂×C₄×C₈	D₃₁	semidirect
157. C₃₁	#3168	semidirect

Maximal subgroups

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

subgroup	index
#14521583	2
#14521451	2
#14521433	2
#14521404	2
#14521290	2
#14521285	2
#14521278	2
C₄×Dic₁₂₄	2
C₄×#92470	2
C₄×D₂₄₈	2
C₂×C₄×C₈×C₃₁	2
#3168	31