Subgroups — #16845427

Order 3968.

424 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	C₂³×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#117	—	2	2	no	—
7	C₃₁×#117	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
10	C₂×C₄²×C₃₁	—	2	1	no	—
11	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
12	#11758503	—	2	2	no	—
13	#11758503	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
15	C₂×C₄×Dic₃₁	—	2	1	no	—
16	C₂²×C₃₁⋊C₈	—	2	2	no	—
17	C₂²×C₃₁⋊C₈	—	2	2	no	—
18	C₂²×C₃₁⋊C₈	C₄	1	4	yes	complemented
19	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
20	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	—	2	2	no	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	D₄	1	2	yes	—
27	C₂²×C₄×C₃₁	D₄	1	2	yes	—
28	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×Dic₃₁	—	2	2	no	—
31	C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₃₁	—	2	4	no	—
34	C₄²×C₃₁	—	2	4	no	—
35	C₂²⋊Dic₃₁	—	2	4	no	—
36	C₄⋊Dic₃₁	—	2	2	no	—
37	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
38	C₄×Dic₃₁	—	2	4	no	—
39	C₂×C₃₁⋊C₈	—	2	8	no	—
40	C₂×C₃₁⋊C₈	—	2	2	no	—
41	C₂×C₃₁⋊C₈	—	2	2	no	—
42	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
43	C₂³×C₃₁	—	2	4	no	—
44	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
47	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	—	2	4	no	—
49	C₂×C₄×C₃₁	—	4	4	no	—
50	C₂×C₄×C₃₁	C₄²	1	4	yes	—
51	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×Dic₃₁	—	2	4	no	—
64	C₂×Dic₃₁	—	4	4	no	—
65	C₃₁⋊C₈	—	2	4	no	—
66	#1735	—	31	1	no	maximal, Sylow
67	C₂²×C₃₁	—	2	4	no	—
68	C₂²×C₃₁	—	2	4	no	—
69	C₂²×C₃₁	—	2	4	no	—
70	C₂²×C₃₁	#95	1	4	yes	—
71	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₃₁	#135	1	4	yes	—
75	C₄×C₃₁	—	4	4	no	—
76	C₄×C₃₁	#95	1	4	yes	—
77	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
78	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	Dic₃₁	—	4	4	no	—
82	C₂³×C₈	—	31	1	no	—
83	C₂×#117	—	31	1	no	—
84	C₂×#117	D₃₁	1	1	yes	characteristic
85	#340	—	31	4	no	—
86	C₂×C₃₁	—	2	4	no	—
87	C₂×C₃₁	C₂×#95	1	1	yes	characteristic
88	C₂×C₃₁	#416	1	2	yes	—
89	C₂×C₃₁	#416	1	2	yes	—
90	C₂×C₃₁	#427	1	2	yes	—
91	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
92	C₂²×C₈	—	31	4	no	—
93	C₂²×C₈	—	31	2	no	—
94	C₂²×C₈	—	62	2	no	—
95	C₂²×C₈	—	62	2	no	—
96	#117	—	2	2	no	—
97	#117	—	31	4	no	—
98	#117	—	62	2	no	—
99	#117	Dic₃₁	1	4	yes	complemented
100	C₂×C₄⋊C₄	—	31	2	no	—
101	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
102	C₂×C₂²⋊C₄	—	2	1	no	—
103	C₂×C₂²⋊C₄	—	62	1	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	—	62	1	no	—
106	C₃₁	#1735	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴	C₂²⋊D₃₁	1	1	yes	characteristic
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	62	2	no	—
110	C₂²×C₄	Dic₆₂	1	2	yes	—
111	C₂²×C₄	C₄×D₃₁	1	2	yes	—
112	C₂²×C₄	C₄×D₃₁	1	2	yes	—
113	C₂²×C₄	D₁₂₄	1	2	yes	—
114	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
116	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
118	C₂×C₈	—	31	4	no	—
119	C₂×C₈	—	62	8	no	—
120	C₂×C₈	—	62	2	no	—
121	C₂×C₈	—	62	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	31	4	no	—
124	C₄⋊C₄	—	62	2	no	—
125	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	62	4	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	62	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	Dic₃₁⋊C₄	1	2	yes	—
132	C₂³	(C₂×D₃₁)⋊C₄	1	2	yes	—
133	C₂³	C₂²⋊Dic₃₁	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	62	4	no	—
138	C₂×C₄	—	124	4	no	—
139	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
140	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
141	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
142	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
143	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
144	C₂×C₄	(C₂×D₃₁)⋊C₄	1	4	yes	—
145	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
146	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
149	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
150	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
152	C₈	—	62	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	C₃₁⋊#95	1	4	yes	—
157	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
158	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
159	C₂²	#11758527	1	4	yes	—
160	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
161	C₄	—	4	4	no	—
162	C₄	—	124	4	no	—
163	C₄	C₃₁⋊#95	1	4	yes	—
164	C₄	C₂×C₄×Dic₃₁	1	1	yes	characteristic
165	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
166	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
167	C₄	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
168	C₂	—	2	4	no	—
169	C₂	C₂×C₃₁⋊#95	1	1	yes	characteristic
170	C₂	#14521043	1	2	yes	—
171	C₂	#14521045	1	2	yes	—
172	C₂	#14521085	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	#1735
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #14520811	C₂	semidirect
7. C₃₁×#117	C₄	semidirect
13. #11758503	C₄	semidirect
18. C₂²×C₃₁⋊C₈	C₄	semidirect
99. #117	Dic₃₁	semidirect
106. C₃₁	#1735	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃₁⋊C₈	2
C₂×#11758503	2
#14520811	2
C₂×C₃₁×#117	2
#1735	31