Subgroups — C₂×C₃₁×#556 #16843407

Order 3968.

1112 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#556	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#126	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#124	C₂	1	2	yes	maximal
5	C₂×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Q₈×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×#117	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
12	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
14	C₃₁×C₄⋊Q₈	C₂²	1	8	yes	complemented
15	C₃₁×#126	C₂²	1	32	yes	complemented
16	C₃₁×#124	C₂²	1	16	yes	—
17	C₃₁×C₂²⋊Q₈	C₂²	1	16	yes	complemented
18	C₄×Q₈×C₃₁	C₂²	1	16	yes	complemented
19	C₄×Q₈×C₃₁	C₂²	1	8	yes	complemented
20	C₄×D₄×C₃₁	C₂²	1	8	yes	—
21	C₃₁×#117	C₂²	1	16	yes	complemented
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
29	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
31	C₂×C₄²×C₃₁	C₂²	1	4	yes	—
32	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
34	C₂⁴×C₃₁	C₂³	1	2	yes	—
35	C₂×Q₈×C₃₁	—	2	4	no	—
36	C₂×Q₈×C₃₁	—	2	8	no	—
37	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
38	C₂×Q₈×C₃₁	C₂³	1	8	yes	—
39	C₂×D₄×C₃₁	—	2	4	no	—
40	C₂×D₄×C₃₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₃₁	—	2	8	no	—
42	C₂²×C₄×C₃₁	C₂³	1	8	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
47	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
51	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
52	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
53	C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
54	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
55	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
56	C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
57	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
58	C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
59	C₄²×C₃₁	C₂³	1	16	yes	—
60	C₄²×C₃₁	C₂³	1	4	yes	—
61	C₄²×C₃₁	C₂³	1	8	yes	—
62	C₂³×C₃₁	—	2	4	no	—
63	C₂³×C₃₁	—	2	4	no	—
64	C₂³×C₃₁	—	2	4	no	—
65	C₂³×C₃₁	C₂⁴	1	4	yes	—
66	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
67	Q₈×C₃₁	—	2	8	no	—
68	Q₈×C₃₁	—	2	16	no	—
69	D₄×C₃₁	—	2	8	no	—
70	C₂×C₄×C₃₁	—	2	4	no	—
71	C₂×C₄×C₃₁	—	2	4	no	—
72	C₂×C₄×C₃₁	—	2	8	no	—
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	16	no	—
76	C₂×C₄×C₃₁	#41	1	4	yes	—
77	C₂×C₄×C₃₁	#41	1	4	yes	—
78	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
79	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
80	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
81	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
82	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
83	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
84	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
85	C₂×#556	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₃₁	—	2	4	no	—
87	C₂²×C₃₁	—	2	4	no	—
88	C₂²×C₃₁	—	2	4	no	—
89	C₂²×C₃₁	—	4	8	no	—
90	C₂²×C₃₁	C₂×#41	1	2	yes	—
91	C₂²×C₃₁	C₂×#41	1	2	yes	—
92	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
93	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
94	C₄×C₃₁	—	2	4	no	—
95	C₄×C₃₁	—	2	4	no	—
96	C₄×C₃₁	—	2	8	no	—
97	C₄×C₃₁	—	2	4	no	—
98	C₄×C₃₁	—	2	2	no	—
99	C₄×C₃₁	C₂×#41	1	4	yes	—
100	C₄×C₃₁	C₂×#41	1	4	yes	—
101	#556	C₂×C₃₁	1	16	yes	direct product factor
102	C₂×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
103	C₂×#126	C₂×C₃₁	1	4	yes	complemented
104	C₂×#124	C₂×C₃₁	1	2	yes	—
105	C₂×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
106	C₂×C₄×Q₈	C₂×C₃₁	1	2	yes	complemented
107	C₂×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
109	C₂×#117	C₂×C₃₁	1	2	yes	complemented
110	C₂×C₃₁	—	4	4	no	—
111	C₂×C₃₁	#556	1	4	yes	direct product factor
112	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
113	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
114	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
115	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
116	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
117	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
118	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
119	C₄⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
120	#126	C₂²×C₃₁	1	32	yes	complemented
121	#124	C₂²×C₃₁	1	16	yes	—
122	C₂²⋊Q₈	C₂²×C₃₁	1	16	yes	complemented
123	C₄×Q₈	C₂²×C₃₁	1	16	yes	complemented
124	C₄×Q₈	C₂²×C₃₁	1	8	yes	complemented
125	C₄×D₄	C₂²×C₃₁	1	8	yes	—
126	#117	C₂²×C₃₁	1	16	yes	complemented
127	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
131	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
136	C₂×C₄²	C₂²×C₃₁	1	4	yes	—
137	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
138	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
139	C₃₁	C₂×#556	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁴	C₂³×C₃₁	1	2	yes	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	2	8	no	—
143	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
144	C₂×Q₈	C₂³×C₃₁	1	8	yes	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	C₂³×C₃₁	1	4	yes	—
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
149	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
150	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
152	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
153	C₂²×C₄	C₂³×C₃₁	1	8	yes	—
154	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
156	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
157	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
158	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
159	C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
160	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
161	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
162	C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
163	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
164	C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
165	C₄²	C₂³×C₃₁	1	16	yes	—
166	C₄²	C₂³×C₃₁	1	4	yes	—
167	C₄²	C₂³×C₃₁	1	8	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	4	no	—
171	C₂³	C₂⁴×C₃₁	1	4	yes	—
172	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	Q₈	—	2	16	no	—
175	D₄	—	2	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	#41×C₃₁	1	4	yes	—
183	C₂×C₄	#41×C₃₁	1	4	yes	—
184	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
185	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
186	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
187	C₂×C₄	C₂⁴×C₃₁	1	8	yes	—
188	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
189	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
190	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	8	no	—
195	C₂²	C₂×#41×C₃₁	1	2	yes	—
196	C₂²	C₂×#41×C₃₁	1	2	yes	—
197	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
198	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	8	no	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	C₂×#41×C₃₁	1	4	yes	—
205	C₄	C₂×#41×C₃₁	1	4	yes	—
206	C₂	—	4	4	no	—
207	C₂	C₃₁×#556	1	4	yes	direct product factor
208	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
209	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
210	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#556	C₂	direct (N × H)
2. C₂×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₂×C₃₁×#126	C₂	semidirect
5. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₄×Q₈×C₃₁	C₂	semidirect
7. C₂×C₄×Q₈×C₃₁	C₂	semidirect
9. C₂×C₃₁×#117	C₂	semidirect
14. C₃₁×C₄⋊Q₈	C₂²	semidirect
15. C₃₁×#126	C₂²	semidirect
17. C₃₁×C₂²⋊Q₈	C₂²	semidirect
18. C₄×Q₈×C₃₁	C₂²	semidirect
19. C₄×Q₈×C₃₁	C₂²	semidirect
21. C₃₁×#117	C₂²	semidirect
85. C₂×#556	C₃₁	direct (N × H)
101. #556	C₂×C₃₁	direct (N × H)
102. C₂×C₄⋊Q₈	C₂×C₃₁	semidirect
103. C₂×#126	C₂×C₃₁	semidirect
105. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
106. C₂×C₄×Q₈	C₂×C₃₁	semidirect
107. C₂×C₄×Q₈	C₂×C₃₁	semidirect
109. C₂×#117	C₂×C₃₁	semidirect
111. C₂×C₃₁	#556	direct (N × H)
119. C₄⋊Q₈	C₂²×C₃₁	semidirect
120. #126	C₂²×C₃₁	semidirect
122. C₂²⋊Q₈	C₂²×C₃₁	semidirect
123. C₄×Q₈	C₂²×C₃₁	semidirect
124. C₄×Q₈	C₂²×C₃₁	semidirect
126. #117	C₂²×C₃₁	semidirect
139. C₃₁	C₂×#556	direct (N × H)
207. C₂	C₃₁×#556	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×#556	2
C₂×C₃₁×C₄⋊Q₈	2
C₂×C₃₁×#126	2
C₂×C₃₁×#124	2
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₄×Q₈×C₃₁	2
C₂×C₄×D₄×C₃₁	2
C₂×C₃₁×#117	2
C₂×#556	31

Subgroups — C2×C31×#556 #16843407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#556 #16843407