Subgroups — #16854574

Order 3968.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521525	C₂	1	2	yes	maximal
3	#14521154	C₂	1	2	yes	maximal, complemented
4	#14521153	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521143	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521086	C₂	1	1	yes	maximal, characteristic
9	#14521056	C₂	1	2	yes	maximal
10	#14521003	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₃₁	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	Q₈×Dic₃₁	C₂²	1	2	yes	—
16	C₂×#92482	C₂²	1	2	yes	—
17	Q₁₆×D₃₁	C₄	1	8	yes	complemented
18	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
19	C₂×#92469	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
21	#11758469	—	2	2	no	—
22	#11758469	C₂²	1	2	yes	—
23	#11758464	—	2	2	no	—
24	#11758464	C₂²	1	2	yes	—
25	#11758449	—	2	2	no	—
26	#11758449	C₂²	1	1	yes	characteristic
27	C₄×Dic₆₂	—	2	2	no	—
28	C₄×Dic₆₂	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#103	C₂²	1	2	yes	—
31	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	#11758423	C₂²	1	2	yes	—
33	#11758406	C₂²	1	1	yes	characteristic
34	#11758404	C₂²	1	2	yes	—
35	#11758403	C₂²	1	1	yes	characteristic
36	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
37	#11758398	C₂²	1	2	yes	—
38	#11758396	C₂²	1	2	yes	—
39	#11758391	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
41	Q₈×D₃₁	—	2	4	no	—
42	Q₈×D₃₁	C₂×C₄	1	8	yes	—
43	C₂×C₄×D₃₁	—	2	2	no	—
44	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₆₂	—	2	2	no	—
46	C₂×Dic₆₂	C₂³	1	2	yes	—
47	Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
48	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
52	C₄²×C₃₁	—	2	2	no	—
53	C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	#92482	C₂×C₄	1	8	yes	—
55	(C₂×D₃₁)⋊C₄	—	2	2	no	—
56	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
57	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
58	C₄⋊Dic₃₁	—	2	2	no	—
59	C₄⋊Dic₃₁	C₂³	1	2	yes	—
60	Dic₃₁⋊C₄	—	2	2	no	—
61	Dic₃₁⋊C₄	—	2	2	no	—
62	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
63	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
64	C₄×Dic₃₁	—	2	2	no	—
65	C₄×Dic₃₁	—	2	2	no	—
66	C₄×Dic₃₁	—	2	2	no	—
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	Dic₁₂₄	C₂×C₄	1	4	yes	—
72	#92469	—	2	2	no	—
73	C₈×D₃₁	C₂×C₄	1	4	yes	—
74	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₃₁	—	2	2	no	—
76	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₃₁	—	2	2	no	—
83	C₂×Dic₃₁	—	2	2	no	—
84	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄×D₃₁	—	2	8	no	—
88	C₄×D₃₁	C₂²×C₄	1	2	yes	—
89	C₄×D₃₁	#41	1	2	yes	—
90	Dic₆₂	—	2	2	no	—
91	Dic₆₂	—	2	8	no	—
92	Dic₆₂	C₂²×C₄	1	4	yes	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	C₂²×C₄	1	2	yes	—
95	C₃₁⋊C₈	—	2	1	no	—
96	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
97	#2950	—	31	1	no	maximal, Sylow
98	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×D₃₁	—	2	1	no	—
100	C₂×D₃₁	—	2	1	no	—
101	C₂×D₃₁	C₄×D₄	1	2	yes	—
102	C₄×C₃₁	—	2	4	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	4	2	no	—
106	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
108	Dic₃₁	—	2	4	no	—
109	Dic₃₁	—	2	1	no	—
110	Dic₃₁	—	2	1	no	—
111	Dic₃₁	—	4	2	no	—
112	Dic₃₁	C₄×D₄	1	2	yes	—
113	C₂²×Q₁₆	—	31	1	no	—
114	#519	—	31	2	no	—
115	#441	—	31	2	no	—
116	#441	—	31	2	no	—
117	C₄×Q₁₆	—	31	1	no	—
118	C₄×Q₁₆	—	31	1	no	—
119	C₄×Q₁₆	—	31	1	no	—
120	C₄×Q₁₆	D₃₁	1	1	yes	characteristic, complemented
121	#427	—	31	1	no	—
122	#419	—	31	2	no	—
123	#405	—	31	1	no	—
124	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂×C₃₁	#578	1	1	yes	characteristic
126	C₂×C₃₁	#578	1	1	yes	characteristic
127	D₃₁	—	2	2	no	—
128	C₂²×Q₈	—	31	2	no	—
129	C₂×Q₁₆	—	31	2	no	—
130	C₂×Q₁₆	—	31	8	no	—
131	C₂×Q₁₆	—	31	1	no	—
132	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
133	C₂×M₁₆	—	31	1	no	—
134	C₂²×C₈	—	31	1	no	—
135	C₄×Q₈	—	31	2	no	—
136	C₄×Q₈	—	31	2	no	—
137	C₄×Q₈	—	31	2	no	—
138	C₄×Q₈	—	62	2	no	—
139	C₄×Q₈	—	62	2	no	—
140	C₄×Q₈	C₂×D₃₁	1	2	yes	—
141	#117	—	31	1	no	—
142	#117	—	31	2	no	—
143	#117	—	62	2	no	—
144	#117	—	62	2	no	—
145	#107	—	31	1	no	—
146	#107	C₂×D₃₁	1	1	yes	characteristic
147	#106	—	31	2	no	—
148	#103	—	31	2	no	—
149	#103	—	31	2	no	—
150	#103	—	31	2	no	—
151	#103	C₂×D₃₁	1	2	yes	—
152	#97	—	31	1	no	—
153	#97	—	31	1	no	—
154	C₄×C₈	—	31	1	no	—
155	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
156	C₃₁	#2950	1	1	yes	characteristic, complemented, Sylow
157	C₂×Q₈	—	31	8	no	—
158	C₂×Q₈	—	31	2	no	—
159	C₂×Q₈	—	62	2	no	—
160	C₂×Q₈	—	62	4	no	—
161	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
162	C₂²×C₄	—	31	1	no	—
163	C₂²×C₄	—	62	2	no	—
164	Q₁₆	—	31	8	no	—
165	Q₁₆	—	31	4	no	—
166	Q₁₆	C₄×D₃₁	1	4	yes	—
167	M₁₆	—	62	2	no	—
168	C₂×C₈	—	31	1	no	—
169	C₂×C₈	—	31	4	no	—
170	C₂×C₈	—	31	1	no	—
171	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	31	1	no	—
175	C₄⋊C₄	—	31	1	no	—
176	C₄⋊C₄	—	31	2	no	—
177	C₄⋊C₄	—	62	2	no	—
178	C₄⋊C₄	—	62	2	no	—
179	C₄⋊C₄	—	62	2	no	—
180	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
181	C₂²⋊C₄	—	31	1	no	—
182	C₂²⋊C₄	—	31	1	no	—
183	C₂²⋊C₄	—	62	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	31	1	no	—
186	C₄²	—	62	2	no	—
187	C₄²	—	62	2	no	—
188	C₄²	—	62	2	no	—
189	C₄²	—	62	2	no	—
190	C₄²	C₂²×D₃₁	1	1	yes	characteristic
191	C₂³	—	31	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	31	4	no	—
194	Q₈	—	62	8	no	—
195	Q₈	—	62	2	no	—
196	Q₈	C₂×C₄×D₃₁	1	4	yes	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	31	1	no	—
200	C₂×C₄	—	31	2	no	—
201	C₂×C₄	—	31	2	no	—
202	C₂×C₄	—	31	1	no	—
203	C₂×C₄	—	31	1	no	—
204	C₂×C₄	—	62	8	no	—
205	C₂×C₄	—	62	2	no	—
206	C₂×C₄	—	62	2	no	—
207	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
208	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	31	2	no	—
212	C₈	—	62	1	no	—
213	C₈	C₂×C₄×D₃₁	1	2	yes	—
214	C₂²	—	31	2	no	—
215	C₂²	—	62	1	no	—
216	C₂²	—	62	1	no	—
217	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	—	31	2	no	—
223	C₄	—	62	1	no	—
224	C₄	—	62	1	no	—
225	C₄	—	62	4	no	—
226	C₄	—	124	2	no	—
227	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
228	C₄	#41×D₃₁	1	1	yes	characteristic
229	C₂	—	62	2	no	—
230	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic
231	C₂	#14521862	1	1	yes	characteristic
232	C₂	#14521863	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×Q₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂²×D₃₁	C₂×D₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×Q₁₆	D₃₁
hypocenter	C₃₁	#2950
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #14521154	C₂	semidirect
4. #14521153	C₂	semidirect
5. C₄×Dic₁₂₄	C₂	semidirect
6. Q₁₆×Dic₃₁	C₂	semidirect
7. #14521143	C₂	semidirect
11. C₄×Q₁₆×C₃₁	C₂	semidirect
17. Q₁₆×D₃₁	C₄	semidirect
120. C₄×Q₁₆	D₃₁	semidirect
156. C₃₁	#2950	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₃₁	2
#14521525	2
#14521154	2
#14521153	2
C₄×Dic₁₂₄	2
Q₁₆×Dic₃₁	2
#14521143	2
#14521086	2
#14521056	2
#14521003	2
C₄×Q₁₆×C₃₁	2
#2950	31