Subgroups — C₃₁×#3341 #16843274

Order 3968.

364 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#555	C₂	1	2	yes	maximal, complemented
2	C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#482	C₂	1	4	yes	maximal, complemented
4	C₃₁×#476	C₂	1	2	yes	maximal, complemented
5	C₃₁×#466	C₂	1	2	yes	maximal, complemented
6	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#427	C₂	1	2	yes	maximal, complemented
9	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃₁×#124	—	2	4	no	—
13	C₃₁×#121	—	2	2	no	—
14	C₃₁×#121	C₂²	1	4	yes	—
15	C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#117	—	2	2	no	—
18	C₃₁×#117	C₂²	1	4	yes	—
19	C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₃₁×#107	C₂²	1	2	yes	—
21	C₃₁×#107	C₂²	1	2	yes	—
22	C₃₁×#106	C₂²	1	4	yes	—
23	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₃₁×#102	C₂²	1	2	yes	—
25	C₃₁×#102	C₂²	1	4	yes	—
26	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₃₁	—	2	2	no	—
29	C₂×D₄×C₃₁	—	2	4	no	—
30	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
31	C₂×D₄×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₃₁	—	2	2	no	—
33	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
34	D₈×C₃₁	—	2	2	no	—
35	C₂×C₈×C₃₁	—	2	4	no	—
36	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₃₁	—	2	4	no	—
44	C₂²⋊C₄×C₃₁	—	2	4	no	—
45	C₂²⋊C₄×C₃₁	—	2	2	no	—
46	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
47	C₄²×C₃₁	—	2	4	no	—
48	C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₃₁	—	2	2	no	—
50	C₂³×C₃₁	C₂×D₄	1	2	yes	—
51	Q₈×C₃₁	—	8	2	no	—
52	D₄×C₃₁	—	2	2	no	—
53	D₄×C₃₁	—	2	2	no	—
54	D₄×C₃₁	—	2	2	no	—
55	D₄×C₃₁	—	8	4	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	4	no	—
60	C₂×C₄×C₃₁	—	4	4	no	—
61	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
65	C₈×C₃₁	—	2	2	no	—
66	C₈×C₃₁	—	2	1	no	—
67	C₈×C₃₁	#41	1	2	yes	—
68	#3341	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₃₁	—	2	2	no	—
70	C₂²×C₃₁	—	2	2	no	—
71	C₂²×C₃₁	—	2	2	no	—
72	C₂²×C₃₁	—	4	2	no	—
73	C₂²×C₃₁	—	4	2	no	—
74	C₂²×C₃₁	—	4	2	no	—
75	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	—	2	1	no	—
78	C₄×C₃₁	—	4	2	no	—
79	C₄×C₃₁	—	4	4	no	—
80	C₄×C₃₁	#135	1	2	yes	—
81	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
82	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
83	#555	C₂×C₃₁	1	2	yes	complemented
84	#500	C₂×C₃₁	1	1	yes	characteristic, complemented
85	#482	C₂×C₃₁	1	4	yes	complemented
86	#476	C₂×C₃₁	1	2	yes	complemented
87	#466	C₂×C₃₁	1	2	yes	complemented
88	C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
89	C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
90	#427	C₂×C₃₁	1	2	yes	complemented
91	C₂×C₃₁	—	4	2	no	—
92	C₂×C₃₁	—	8	2	no	—
93	C₂×C₃₁	#547	1	1	yes	characteristic
94	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
95	C₂×C₃₁	#576	1	1	yes	characteristic
96	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₃₁	1	2	yes	—
98	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
99	#124	—	2	4	no	—
100	#121	—	2	2	no	—
101	#121	C₂²×C₃₁	1	4	yes	—
102	C₄×D₄	C₂²×C₃₁	1	2	yes	—
103	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	#117	C₂²×C₃₁	1	4	yes	—
106	#107	C₂²×C₃₁	1	1	yes	characteristic
107	#107	C₂²×C₃₁	1	2	yes	—
108	#107	C₂²×C₃₁	1	2	yes	—
109	#106	C₂²×C₃₁	1	4	yes	—
110	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
111	#102	C₂²×C₃₁	1	2	yes	—
112	#102	C₂²×C₃₁	1	4	yes	—
113	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
114	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
115	C₃₁	#3341	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	2	4	no	—
118	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
119	C₂×D₄	C₂³×C₃₁	1	2	yes	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
122	D₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₃₁	1	2	yes	—
126	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
129	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
130	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	C₂³×C₃₁	1	1	yes	characteristic
137	C₂³	—	2	2	no	—
138	C₂³	C₂×D₄×C₃₁	1	2	yes	—
139	Q₈	—	8	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	8	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	#41×C₃₁	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	4	no	—
167	C₄	C₃₁×#135	1	2	yes	—
168	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
169	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
170	C₂	—	4	2	no	—
171	C₂	—	8	2	no	—
172	C₂	C₃₁×#547	1	1	yes	characteristic
173	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
174	C₂	C₃₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#555	C₂	semidirect
2. C₃₁×#500	C₂	semidirect
3. C₃₁×#482	C₂	semidirect
4. C₃₁×#476	C₂	semidirect
5. C₃₁×#466	C₂	semidirect
6. C₄×D₈×C₃₁	C₂	semidirect
7. C₈×D₄×C₃₁	C₂	semidirect
8. C₃₁×#427	C₂	semidirect
68. #3341	C₃₁	direct (N × H)
83. #555	C₂×C₃₁	semidirect
84. #500	C₂×C₃₁	semidirect
85. #482	C₂×C₃₁	semidirect
86. #476	C₂×C₃₁	semidirect
87. #466	C₂×C₃₁	semidirect
88. C₄×D₈	C₂×C₃₁	semidirect
89. C₈×D₄	C₂×C₃₁	semidirect
90. #427	C₂×C₃₁	semidirect
115. C₃₁	#3341	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#555	2
C₃₁×#500	2
C₃₁×#482	2
C₃₁×#476	2
C₃₁×#466	2
C₄×D₈×C₃₁	2
C₈×D₄×C₃₁	2
C₃₁×#427	2
#3341	31

Subgroups — C31×#3341 #16843274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3341 #16843274