Subgroups — C₂×C₃₁×#540 #16843385

Order 3968.

1260 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#540	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#121	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#121	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
10	C₂²×D₄×C₃₁	C₂²	1	4	yes	—
11	C₂²×D₄×C₃₁	C₂²	1	4	yes	—
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	C₂²	1	4	yes	—
14	C₃₁×#127	C₂²	1	8	yes	complemented
15	C₃₁×#125	C₂²	1	8	yes	complemented
16	C₃₁×#123	C₂²	1	32	yes	complemented
17	C₃₁×#121	C₂²	1	32	yes	complemented
18	C₃₁×#121	C₂²	1	16	yes	complemented
19	C₄×D₄×C₃₁	C₂²	1	16	yes	complemented
20	C₃₁×#117	C₂²	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
30	C₂⁴×C₃₁	C₂³	1	4	yes	—
31	C₂×D₄×C₃₁	—	2	8	no	—
32	C₂×D₄×C₃₁	—	4	8	no	—
33	C₂×D₄×C₃₁	—	4	8	no	—
34	C₂×D₄×C₃₁	C₂³	1	8	yes	—
35	C₂×D₄×C₃₁	C₂³	1	16	yes	—
36	C₂×D₄×C₃₁	C₂³	1	16	yes	—
37	C₂²×C₄×C₃₁	—	2	4	no	—
38	C₂²×C₄×C₃₁	—	4	8	no	—
39	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	16	yes	—
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
47	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
48	C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
49	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
50	C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
51	C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
52	C₄²×C₃₁	C₂³	1	4	yes	—
53	C₄²×C₃₁	C₂³	1	4	yes	—
54	C₂³×C₃₁	—	2	2	no	—
55	C₂³×C₃₁	—	2	8	no	—
56	C₂³×C₃₁	—	2	8	no	—
57	C₂³×C₃₁	—	2	4	no	—
58	C₂³×C₃₁	—	2	8	no	—
59	C₂³×C₃₁	C₂⁴	1	2	yes	—
60	C₂³×C₃₁	C₂⁴	1	8	yes	—
61	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
62	D₄×C₃₁	—	2	16	no	—
63	D₄×C₃₁	—	4	16	no	—
64	D₄×C₃₁	—	4	16	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	4	no	—
67	C₂×C₄×C₃₁	—	2	8	no	—
68	C₂×C₄×C₃₁	—	2	4	no	—
69	C₂×C₄×C₃₁	—	2	8	no	—
70	C₂×C₄×C₃₁	—	4	16	no	—
71	C₂×C₄×C₃₁	#41	1	4	yes	—
72	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
73	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
74	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
75	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
76	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
77	C₂×#540	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₃₁	—	2	2	no	—
79	C₂²×C₃₁	—	2	8	no	—
80	C₂²×C₃₁	—	2	8	no	—
81	C₂²×C₃₁	—	2	4	no	—
82	C₂²×C₃₁	—	2	8	no	—
83	C₂²×C₃₁	—	4	4	no	—
84	C₂²×C₃₁	—	4	16	no	—
85	C₂²×C₃₁	C₂×#41	1	2	yes	—
86	C₂²×C₃₁	H(5,𝔽₂)	1	4	yes	—
87	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
88	C₄×C₃₁	—	2	2	no	—
89	C₄×C₃₁	—	2	4	no	—
90	C₄×C₃₁	—	2	8	no	—
91	C₄×C₃₁	—	2	4	no	—
92	C₄×C₃₁	C₂×#41	1	4	yes	—
93	#540	C₂×C₃₁	1	16	yes	direct product factor
94	C₂×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
95	C₂×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
96	C₂×#123	C₂×C₃₁	1	4	yes	complemented
97	C₂×#121	C₂×C₃₁	1	4	yes	complemented
98	C₂×#121	C₂×C₃₁	1	2	yes	complemented
99	C₂×C₄×D₄	C₂×C₃₁	1	2	yes	complemented
100	C₂×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
101	C₂×C₃₁	—	4	2	no	—
102	C₂×C₃₁	—	4	8	no	—
103	C₂×C₃₁	#540	1	4	yes	direct product factor
104	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
105	C₂×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
106	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
107	C₂²×D₄	C₂²×C₃₁	1	4	yes	—
108	C₂²×D₄	C₂²×C₃₁	1	4	yes	—
109	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
110	C₂³×C₄	C₂²×C₃₁	1	4	yes	—
111	#127	C₂²×C₃₁	1	8	yes	complemented
112	#125	C₂²×C₃₁	1	8	yes	complemented
113	#123	C₂²×C₃₁	1	32	yes	complemented
114	#121	C₂²×C₃₁	1	32	yes	complemented
115	#121	C₂²×C₃₁	1	16	yes	complemented
116	C₄×D₄	C₂²×C₃₁	1	16	yes	complemented
117	#117	C₂²×C₃₁	1	8	yes	complemented
118	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
122	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
124	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
125	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
126	C₃₁	C₂×#540	1	1	yes	characteristic, direct product factor, Sylow
127	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
128	C₂⁴	C₂³×C₃₁	1	4	yes	—
129	C₂×D₄	—	2	8	no	—
130	C₂×D₄	—	4	8	no	—
131	C₂×D₄	—	4	8	no	—
132	C₂×D₄	C₂³×C₃₁	1	8	yes	—
133	C₂×D₄	C₂³×C₃₁	1	16	yes	—
134	C₂×D₄	C₂³×C₃₁	1	16	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	4	8	no	—
137	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
138	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
139	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
141	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
142	C₂²×C₄	C₂³×C₃₁	1	16	yes	—
143	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
144	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
145	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
146	C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
147	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
148	C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
149	C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
150	C₄²	C₂³×C₃₁	1	4	yes	—
151	C₄²	C₂³×C₃₁	1	4	yes	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	8	no	—
154	C₂³	—	2	8	no	—
155	C₂³	—	2	4	no	—
156	C₂³	—	2	8	no	—
157	C₂³	C₂⁴×C₃₁	1	2	yes	—
158	C₂³	C₂⁴×C₃₁	1	8	yes	—
159	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
160	D₄	—	2	16	no	—
161	D₄	—	4	16	no	—
162	D₄	—	4	16	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	4	16	no	—
169	C₂×C₄	#41×C₃₁	1	4	yes	—
170	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
171	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
172	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
173	C₂×C₄	C₂⁴×C₃₁	1	8	yes	—
174	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	8	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	8	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	16	no	—
182	C₂²	C₂×#41×C₃₁	1	2	yes	—
183	C₂²	C₃₁×H(5,𝔽₂)	1	4	yes	—
184	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	2	8	no	—
188	C₄	—	2	4	no	—
189	C₄	C₂×#41×C₃₁	1	4	yes	—
190	C₂	—	4	2	no	—
191	C₂	—	4	8	no	—
192	C₂	C₃₁×#540	1	4	yes	direct product factor
193	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
194	C₂	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
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×#540	C₂	direct (N × H)
2. C₂×C₃₁×#127	C₂	semidirect
3. C₂×C₃₁×#125	C₂	semidirect
4. C₂×C₃₁×#123	C₂	semidirect
5. C₂×C₃₁×#121	C₂	semidirect
6. C₂×C₃₁×#121	C₂	semidirect
7. C₂×C₄×D₄×C₃₁	C₂	semidirect
8. C₂×C₃₁×#117	C₂	semidirect
14. C₃₁×#127	C₂²	semidirect
15. C₃₁×#125	C₂²	semidirect
16. C₃₁×#123	C₂²	semidirect
17. C₃₁×#121	C₂²	semidirect
18. C₃₁×#121	C₂²	semidirect
19. C₄×D₄×C₃₁	C₂²	semidirect
20. C₃₁×#117	C₂²	semidirect
77. C₂×#540	C₃₁	direct (N × H)
93. #540	C₂×C₃₁	direct (N × H)
94. C₂×#127	C₂×C₃₁	semidirect
95. C₂×#125	C₂×C₃₁	semidirect
96. C₂×#123	C₂×C₃₁	semidirect
97. C₂×#121	C₂×C₃₁	semidirect
98. C₂×#121	C₂×C₃₁	semidirect
99. C₂×C₄×D₄	C₂×C₃₁	semidirect
100. C₂×#117	C₂×C₃₁	semidirect
103. C₂×C₃₁	#540	direct (N × H)
111. #127	C₂²×C₃₁	semidirect
112. #125	C₂²×C₃₁	semidirect
113. #123	C₂²×C₃₁	semidirect
114. #121	C₂²×C₃₁	semidirect
115. #121	C₂²×C₃₁	semidirect
116. C₄×D₄	C₂²×C₃₁	semidirect
117. #117	C₂²×C₃₁	semidirect
126. C₃₁	C₂×#540	direct (N × H)
192. C₂	C₃₁×#540	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×#540	2
C₂×C₃₁×#127	2
C₂×C₃₁×#125	2
C₂×C₃₁×#123	2
C₂×C₃₁×#121	2
C₂×C₄×D₄×C₃₁	2
C₂×C₃₁×#117	2
C₂×#540	31

Subgroups — C2×C31×#540 #16843385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#540 #16843385