Subgroups — #117×Dic₃₁ #16848054

Order 3968.

752 conjugacy classes of subgroups.

The subgroup lattice has 174 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	C₂²×C₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₃₁	C₂	1	4	yes	maximal, complemented
4	C₂²⋊C₄×Dic₃₁	C₂	1	4	yes	maximal
5	#14520895	C₂	1	2	yes	maximal, complemented
6	C₄²×Dic₃₁	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×Dic₃₁	C₂²	1	2	yes	—
10	C₃₁×#117	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₃₁	C₂²	1	2	yes	—
15	#11758503	C₄	1	8	yes	complemented
16	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃₁	C₄	1	8	yes	complemented
18	C₂×C₄×Dic₃₁	C₂²	1	8	yes	—
19	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
23	C₃₁⋊#95	C₂²	1	8	yes	—
24	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₁	—	2	8	no	—
32	C₂²×Dic₃₁	C₂×C₄	1	8	yes	—
33	C₂²×Dic₃₁	C₂³	1	2	yes	—
34	C₂²×Dic₃₁	C₂³	1	4	yes	—
35	C₂²×Dic₃₁	C₂³	1	2	yes	—
36	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
37	C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
38	C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
39	C₂²⋊Dic₃₁	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	complemented
41	C₄×Dic₃₁	—	2	16	no	—
42	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
43	C₄×Dic₃₁	C₂×C₄	1	8	yes	—
44	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
45	C₄×Dic₃₁	C₂×C₄	1	8	yes	complemented
46	C₂³×C₃₁	—	2	4	no	—
47	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
48	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
50	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₃₁	—	2	8	no	—
52	C₂×C₄×C₃₁	C₄²	1	16	yes	complemented
53	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×Dic₃₁	—	2	8	no	—
61	C₂×Dic₃₁	—	2	4	no	—
62	C₂×Dic₃₁	—	2	16	no	—
63	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
64	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
65	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
66	C₂×Dic₃₁	#41	1	8	yes	—
67	C₄×#117	—	31	1	no	maximal, Sylow
68	C₂²×C₃₁	—	2	4	no	—
69	C₂²×C₃₁	—	2	4	no	—
70	C₂²×C₃₁	—	2	4	no	—
71	C₂²×C₃₁	C₂×C₄²	1	4	yes	—
72	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₃₁	C₂×#41	1	4	yes	—
76	C₄×C₃₁	—	2	8	no	—
77	C₄×C₃₁	C₂×C₄²	1	4	yes	—
78	C₄×C₃₁	C₂×C₄²	1	4	yes	—
79	Dic₃₁	—	2	4	no	—
80	Dic₃₁	—	2	8	no	—
81	Dic₃₁	#117	1	8	yes	direct product factor
82	C₂×#117	—	31	1	no	—
83	C₂×#117	D₃₁	1	1	yes	characteristic
84	C₂²×C₄²	—	31	1	no	—
85	C₄×C₄⋊C₄	—	31	4	no	—
86	C₄×C₂²⋊C₄	—	31	4	no	—
87	#376	—	31	2	no	—
88	C₄³	—	31	2	no	—
89	C₂×C₃₁	—	2	4	no	—
90	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
91	C₂×C₃₁	C₂×#117	1	2	yes	—
92	C₂×C₃₁	C₄×#41	1	2	yes	—
93	C₂×C₃₁	C₄×#41	1	2	yes	—
94	C₂³×C₄	—	31	2	no	—
95	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
96	#117	—	31	8	no	—
97	#117	Dic₃₁	1	8	yes	direct product factor
98	C₂×C₄⋊C₄	—	31	2	no	—
99	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
100	C₂×C₂²⋊C₄	—	31	2	no	—
101	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
102	C₂×C₄²	—	31	8	no	—
103	C₂×C₄²	—	31	2	no	—
104	C₂×C₄²	—	31	2	no	—
105	C₂×C₄²	—	31	1	no	—
106	C₂×C₄²	—	31	1	no	—
107	C₂×C₄²	—	31	8	no	—
108	C₂×C₄²	C₂×D₃₁	1	2	yes	—
109	#95	—	31	8	no	—
110	C₃₁	C₄×#117	1	1	yes	characteristic, complemented, Sylow
111	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
112	C₂²×C₄	—	31	2	no	—
113	C₂²×C₄	—	31	8	no	—
114	C₂²×C₄	—	31	2	no	—
115	C₂²×C₄	—	31	4	no	—
116	C₂²×C₄	—	62	8	no	—
117	C₂²×C₄	C₄×D₃₁	1	4	yes	—
118	C₂²×C₄	C₄×D₃₁	1	4	yes	—
119	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
120	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
121	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
123	C₄⋊C₄	—	31	8	no	—
124	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	complemented
125	C₂²⋊C₄	—	31	8	no	—
126	C₂²⋊C₄	C₂×Dic₃₁	1	8	yes	—
127	C₄²	—	31	4	no	—
128	C₄²	—	31	8	no	—
129	C₄²	—	31	4	no	—
130	C₄²	—	31	8	no	—
131	C₄²	—	62	16	no	—
132	C₄²	C₂×Dic₃₁	1	8	yes	complemented
133	C₂³	—	2	4	no	—
134	C₂³	C₂×C₄×D₃₁	1	2	yes	—
135	C₂³	C₂×C₄×D₃₁	1	2	yes	—
136	C₂³	C₂²×Dic₃₁	1	2	yes	—
137	C₂³	C₂³×D₃₁	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	31	4	no	—
140	C₂×C₄	—	31	8	no	—
141	C₂×C₄	—	31	4	no	—
142	C₂×C₄	—	31	8	no	—
143	C₂×C₄	—	62	16	no	—
144	C₂×C₄	—	62	8	no	—
145	C₂×C₄	—	62	4	no	—
146	C₂×C₄	C₄×Dic₃₁	1	16	yes	complemented
147	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
148	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
149	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
150	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
151	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
152	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
153	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	4	no	—
157	C₂²	C₂×C₄×Dic₃₁	1	4	yes	—
158	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
159	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
160	C₂²	#41×D₃₁	1	4	yes	—
161	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
162	C₄	—	2	8	no	—
163	C₄	—	31	8	no	—
164	C₄	—	62	4	no	—
165	C₄	—	62	8	no	—
166	C₄	C₂×C₄×Dic₃₁	1	4	yes	—
167	C₄	C₂×C₄×Dic₃₁	1	4	yes	—
168	C₂	—	2	4	no	—
169	C₂	C₂²×C₄×Dic₃₁	1	1	yes	characteristic
170	C₂	#117×D₃₁	1	2	yes	—
171	C₂	#14521517	1	2	yes	—
172	C₂	#41×Dic₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
prime radical	C₂⁴×C₃₁	C₂³
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	C₄×#117
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×Dic₃₁	C₂	semidirect
5. #14520895	C₂	semidirect
6. C₄²×Dic₃₁	C₂	semidirect
10. C₃₁×#117	C₄	semidirect
15. #11758503	C₄	semidirect
17. C₂×C₄×Dic₃₁	C₄	semidirect
36. C₄⋊C₄×C₃₁	C₂×C₄	semidirect
38. C₄²×C₃₁	C₂×C₄	semidirect
40. C₄⋊Dic₃₁	C₂×C₄	semidirect
45. C₄×Dic₃₁	C₂×C₄	semidirect
52. C₂×C₄×C₃₁	C₄²	semidirect
81. Dic₃₁	#117	direct (N × H)
97. #117	Dic₃₁	direct (N × H)
110. C₃₁	C₄×#117	semidirect
124. C₄⋊C₄	C₂×Dic₃₁	semidirect
132. C₄²	C₂×Dic₃₁	semidirect
146. C₂×C₄	C₄×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
C₂²×C₄×Dic₃₁	2
C₄⋊C₄×Dic₃₁	2
C₂²⋊C₄×Dic₃₁	2
#14520895	2
C₄²×Dic₃₁	2
C₂×C₃₁×#117	2
C₄×#117	31

Subgroups — #117×Dic31 #16848054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #117×Dic₃₁ #16848054