Subgroups — #16845422

Order 3968.

404 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
3	#14520810	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₃₁	—	2	2	no	—
7	C₂×M₁₆×C₃₁	—	2	1	no	—
8	C₂×M₁₆×C₃₁	C₄	1	4	yes	complemented
9	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂²×C₈×C₃₁	—	2	1	no	—
11	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
12	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
13	#11758503	—	2	1	no	—
14	#11758503	—	2	1	no	—
15	#11758503	—	2	2	no	—
16	#11758503	C₄	1	2	yes	complemented
17	#11758503	C₄	1	4	yes	complemented
18	#11758503	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₃₁	—	2	1	no	—
20	C₂×C₄⋊Dic₃₁	C₄	1	2	yes	complemented
21	C₂×C₄×Dic₃₁	—	2	1	no	—
22	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
23	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	D₄	1	2	yes	—
32	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃₁	—	2	2	no	—
35	C₂²×Dic₃₁	—	2	2	no	—
36	M₁₆×C₃₁	—	2	4	no	—
37	M₁₆×C₃₁	—	2	2	no	—
38	M₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
39	C₂×C₈×C₃₁	—	2	4	no	—
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂²⋊Dic₃₁	—	2	2	no	—
42	C₂²⋊Dic₃₁	—	2	4	no	—
43	C₂²⋊Dic₃₁	—	2	2	no	—
44	C₄⋊Dic₃₁	—	2	4	no	—
45	C₄⋊Dic₃₁	—	2	2	no	—
46	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
47	C₄×Dic₃₁	—	2	2	no	—
48	C₄×Dic₃₁	—	2	2	no	—
49	C₄×Dic₃₁	—	2	2	no	—
50	C₄×Dic₃₁	C₂×C₄	1	4	yes	complemented
51	C₂³×C₃₁	—	4	2	no	—
52	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
55	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
56	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₄²	1	2	yes	—
58	C₂×C₄×C₃₁	C₄²	1	2	yes	—
59	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
75	C₂×Dic₃₁	—	2	4	no	—
76	C₂×Dic₃₁	—	2	4	no	—
77	C₂×Dic₃₁	—	4	4	no	—
78	C₂×Dic₃₁	—	4	4	no	—
79	C₈×C₃₁	—	2	4	no	—
80	#1733	—	31	1	no	maximal, Sylow
81	C₂²×C₃₁	—	2	2	no	—
82	C₂²×C₃₁	—	2	2	no	—
83	C₂²×C₃₁	—	2	2	no	—
84	C₂²×C₃₁	—	4	2	no	—
85	C₂²×C₃₁	—	4	2	no	—
86	C₂²×C₃₁	#95	1	2	yes	—
87	C₂²×C₃₁	#95	1	2	yes	—
88	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₃₁	#95	1	2	yes	—
92	C₄×C₃₁	#95	1	2	yes	—
93	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
94	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₃₁	—	4	4	no	—
98	Dic₃₁	—	4	4	no	—
99	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
100	C₂×#117	—	31	1	no	—
101	C₂×#117	—	31	1	no	—
102	#339	—	31	4	no	—
103	C₂×C₃₁	—	2	1	no	—
104	C₂×C₃₁	—	2	2	no	—
105	C₂×C₃₁	—	2	2	no	—
106	C₂×C₃₁	—	2	1	no	—
107	C₂×C₃₁	C₂×#95	1	1	yes	characteristic
108	C₂×C₃₁	#421	1	2	yes	—
109	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	2	1	no	—
112	C₂×M₁₆	Dic₃₁	1	4	yes	complemented
113	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
114	C₂²×C₈	—	2	1	no	—
115	#117	—	31	4	no	—
116	#117	—	31	2	no	—
117	#117	—	31	2	no	—
118	#117	—	62	1	no	—
119	#117	—	62	2	no	—
120	#117	—	62	1	no	—
121	C₂×C₄⋊C₄	—	31	2	no	—
122	C₂×C₄⋊C₄	—	62	1	no	—
123	C₂×C₂²⋊C₄	—	62	1	no	—
124	C₂×C₂²⋊C₄	—	62	1	no	—
125	C₂×C₄²	—	31	2	no	—
126	C₂×C₄²	—	62	1	no	—
127	C₃₁	#1733	1	1	yes	characteristic, complemented, Sylow
128	C₂⁴	D₁₂₄	1	1	yes	characteristic
129	C₂²×C₄	—	62	2	no	—
130	C₂²×C₄	—	62	2	no	—
131	C₂²×C₄	Dic₆₂	1	2	yes	—
132	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
134	C₂²×C₄	C₄×D₃₁	1	2	yes	—
135	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
137	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
138	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	—	2	2	no	—
143	M₁₆	C₂×Dic₃₁	1	4	yes	complemented
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	2	no	—
146	C₄⋊C₄	—	31	4	no	—
147	C₄⋊C₄	—	62	2	no	—
148	C₄⋊C₄	—	62	4	no	—
149	C₂²⋊C₄	—	62	4	no	—
150	C₂²⋊C₄	—	62	2	no	—
151	C₂²⋊C₄	—	62	2	no	—
152	C₄²	—	31	4	no	—
153	C₄²	—	62	2	no	—
154	C₄²	—	62	2	no	—
155	C₄²	—	62	2	no	—
156	C₂³	—	4	2	no	—
157	C₂³	C₄⋊Dic₃₁	1	2	yes	—
158	C₂³	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
159	C₂³	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
160	C₂³	(C₂×D₃₁)⋊C₄	1	2	yes	—
161	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
162	C₂×C₄	—	62	4	no	—
163	C₂×C₄	—	62	4	no	—
164	C₂×C₄	—	124	4	no	—
165	C₂×C₄	—	124	4	no	—
166	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
167	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
168	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
169	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
170	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
171	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
172	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
173	C₂×C₄	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
175	C₂×C₄	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
177	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
178	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
182	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	C₃₁⋊#95	1	2	yes	—
191	C₂²	C₃₁⋊#95	1	2	yes	—
192	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
193	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
194	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
195	C₄	—	124	4	no	—
196	C₄	—	124	4	no	—
197	C₄	C₃₁⋊#95	1	2	yes	—
198	C₄	C₃₁⋊#95	1	2	yes	—
199	C₄	C₂×C₄×Dic₃₁	1	1	yes	characteristic
200	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
201	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
202	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	2	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	1	no	—
207	C₂	C₂×C₃₁⋊#95	1	1	yes	characteristic
208	C₂	#14521065	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	#1733
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14520810	C₂	semidirect
8. C₂×M₁₆×C₃₁	C₄	semidirect
16. #11758503	C₄	semidirect
17. #11758503	C₄	semidirect
20. C₂×C₄⋊Dic₃₁	C₄	semidirect
38. M₁₆×C₃₁	C₂×C₄	semidirect
50. C₄×Dic₃₁	C₂×C₄	semidirect
112. C₂×M₁₆	Dic₃₁	semidirect
127. C₃₁	#1733	semidirect
143. M₁₆	C₂×Dic₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758503	2
#14520810	2
C₂²×M₁₆×C₃₁	2
#1733	31