Subgroups — C₂³×C₃₁×#101 #21516831

Order 7936.

4990 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₂²×C₃₁×#101	C₂	1	28	yes	maximal, direct product factor
4	C₂³×Q₈×C₃₁	—	2	14	no	—
5	C₂³×Q₈×C₃₁	—	2	2	no	—
6	C₂³×Q₈×C₃₁	C₄	1	14	yes	—
7	C₂³×Q₈×C₃₁	C₂²	1	14	yes	—
8	C₂⁴×C₄×C₃₁	C₄	1	2	yes	—
9	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	—	2	14	no	—
11	C₂²×M₁₆×C₃₁	C₂²	1	28	yes	—
12	C₂³×C₈×C₃₁	—	2	2	no	—
13	C₂×C₃₁×#101	C₂²	1	112	yes	direct product factor
14	C₂⁵×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Q₈×C₃₁	—	2	84	no	—
16	C₂²×Q₈×C₃₁	—	2	28	no	—
17	C₂²×Q₈×C₃₁	—	2	28	no	—
18	C₂²×Q₈×C₃₁	—	2	84	no	—
19	C₂²×Q₈×C₃₁	C₂×C₄	1	84	yes	—
20	C₂²×Q₈×C₃₁	C₂³	1	28	yes	—
21	C₂³×C₄×C₃₁	—	2	14	no	—
22	C₂³×C₄×C₃₁	—	2	2	no	—
23	C₂³×C₄×C₃₁	C₂×C₄	1	28	yes	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	7	yes	—
25	C₂³×C₄×C₃₁	D₄	1	14	yes	—
26	C₂³×C₄×C₃₁	D₄	1	2	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	7	yes	—
28	C₂×M₁₆×C₃₁	—	2	84	no	—
29	C₂×M₁₆×C₃₁	C₂³	1	56	yes	—
30	C₂²×C₈×C₃₁	—	2	28	no	—
31	C₃₁×#101	C₂³	1	64	yes	direct product factor
32	C₂⁴×C₃₁	—	2	8	no	—
33	C₂⁴×C₃₁	C₂²⋊C₄	1	7	yes	—
34	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂⁴×C₃₁	C₂²×C₄	1	7	yes	—
36	C₂×Q₈×C₃₁	—	2	168	no	—
37	C₂×Q₈×C₃₁	—	2	168	no	—
38	C₂×Q₈×C₃₁	—	2	56	no	—
39	C₂×Q₈×C₃₁	—	2	56	no	—
40	C₂×Q₈×C₃₁	C₂²×C₄	1	56	yes	—
41	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
42	C₂²×C₄×C₃₁	—	2	84	no	—
43	C₂²×C₄×C₃₁	—	2	28	no	—
44	C₂²×C₄×C₃₁	C₂²⋊C₄	1	42	yes	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	14	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	56	yes	—
47	C₂²×C₄×C₃₁	C₂²×C₄	1	21	yes	—
48	C₂²×C₄×C₃₁	C₂×D₄	1	14	yes	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	42	yes	—
50	C₂²×C₄×C₃₁	C₂⁴	1	7	yes	—
51	M₁₆×C₃₁	—	2	56	no	—
52	M₁₆×C₃₁	C₂⁴	1	16	yes	—
53	C₂×C₈×C₃₁	—	2	56	no	—
54	C₂³×#101	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₃₁	—	2	28	no	—
56	C₂³×C₃₁	—	2	28	no	—
57	C₂³×C₃₁	#101	1	8	yes	direct product factor
58	C₂³×C₃₁	C₂×C₂²⋊C₄	1	21	yes	—
59	C₂³×C₃₁	C₂×C₂²⋊C₄	1	7	yes	—
60	C₂³×C₃₁	C₂³×C₄	1	7	yes	—
61	Q₈×C₃₁	—	2	112	no	—
62	Q₈×C₃₁	—	2	16	no	—
63	C₂×C₄×C₃₁	—	2	56	no	—
64	C₂×C₄×C₃₁	—	2	56	no	—
65	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	42	yes	—
66	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	42	yes	—
67	C₂×C₄×C₃₁	C₂³×C₄	1	16	yes	—
68	C₂×C₄×C₃₁	C₂³×C₄	1	7	yes	—
69	C₂×C₄×C₃₁	C₂²×D₄	1	14	yes	—
70	C₂×C₄×C₃₁	C₂²×D₄	1	14	yes	—
71	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
72	C₈×C₃₁	—	2	16	no	—
73	C₂⁴×Q₈	C₂×C₃₁	1	1	yes	characteristic
74	C₂³×M₁₆	C₂×C₃₁	1	2	yes	—
75	C₂²×#101	C₂×C₃₁	1	28	yes	direct product factor
76	C₂²×C₃₁	—	2	42	no	—
77	C₂²×C₃₁	—	2	14	no	—
78	C₂²×C₃₁	C₂×#101	1	28	yes	direct product factor
79	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
80	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
81	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
82	C₄×C₃₁	—	2	16	no	—
83	C₄×C₃₁	C₂²×C₂²⋊C₄	1	14	yes	—
84	C₄×C₃₁	C₂³×D₄	1	2	yes	—
85	C₂³×Q₈	—	2	2	no	—
86	C₂³×Q₈	—	2	14	no	—
87	C₂³×Q₈	C₄×C₃₁	1	14	yes	—
88	C₂³×Q₈	C₂²×C₃₁	1	14	yes	—
89	C₂⁴×C₄	C₄×C₃₁	1	2	yes	—
90	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
91	C₂²×M₁₆	—	2	14	no	—
92	C₂²×M₁₆	C₂²×C₃₁	1	28	yes	—
93	C₂³×C₈	—	2	2	no	—
94	C₂×#101	C₂²×C₃₁	1	112	yes	direct product factor
95	C₂×C₃₁	—	2	7	no	—
96	C₂×C₃₁	—	2	1	no	—
97	C₂×C₃₁	C₂²×#101	1	14	yes	direct product factor
98	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
99	C₂⁵	C₂×C₄×C₃₁	1	1	yes	characteristic
100	C₂²×Q₈	—	2	84	no	—
101	C₂²×Q₈	—	2	28	no	—
102	C₂²×Q₈	—	2	28	no	—
103	C₂²×Q₈	—	2	84	no	—
104	C₂²×Q₈	C₂×C₄×C₃₁	1	84	yes	—
105	C₂²×Q₈	C₂³×C₃₁	1	28	yes	—
106	C₂³×C₄	—	2	14	no	—
107	C₂³×C₄	—	2	2	no	—
108	C₂³×C₄	C₂×C₄×C₃₁	1	28	yes	—
109	C₂³×C₄	C₂×C₄×C₃₁	1	7	yes	—
110	C₂³×C₄	D₄×C₃₁	1	2	yes	—
111	C₂³×C₄	D₄×C₃₁	1	14	yes	—
112	C₂³×C₄	C₂³×C₃₁	1	7	yes	—
113	C₂×M₁₆	—	2	84	no	—
114	C₂×M₁₆	C₂³×C₃₁	1	56	yes	—
115	C₂²×C₈	—	2	28	no	—
116	#101	C₂³×C₃₁	1	64	yes	direct product factor
117	C₃₁	C₂³×#101	1	1	yes	characteristic, direct product factor, Sylow
118	C₂⁴	—	2	8	no	—
119	C₂⁴	C₂²⋊C₄×C₃₁	1	7	yes	—
120	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
121	C₂⁴	C₂²×C₄×C₃₁	1	7	yes	—
122	C₂×Q₈	—	2	168	no	—
123	C₂×Q₈	—	2	168	no	—
124	C₂×Q₈	—	2	56	no	—
125	C₂×Q₈	—	2	56	no	—
126	C₂×Q₈	C₂²×C₄×C₃₁	1	56	yes	—
127	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
128	C₂²×C₄	—	2	84	no	—
129	C₂²×C₄	—	2	28	no	—
130	C₂²×C₄	C₂²⋊C₄×C₃₁	1	42	yes	—
131	C₂²×C₄	C₂²⋊C₄×C₃₁	1	14	yes	—
132	C₂²×C₄	C₂²×C₄×C₃₁	1	56	yes	—
133	C₂²×C₄	C₂²×C₄×C₃₁	1	21	yes	—
134	C₂²×C₄	C₂×D₄×C₃₁	1	14	yes	—
135	C₂²×C₄	C₂×D₄×C₃₁	1	42	yes	—
136	C₂²×C₄	C₂⁴×C₃₁	1	7	yes	—
137	M₁₆	—	2	56	no	—
138	M₁₆	C₂⁴×C₃₁	1	16	yes	—
139	C₂×C₈	—	2	56	no	—
140	C₂³	—	2	28	no	—
141	C₂³	—	2	28	no	—
142	C₂³	C₃₁×#101	1	8	yes	direct product factor
143	C₂³	C₂×C₂²⋊C₄×C₃₁	1	21	yes	—
144	C₂³	C₂×C₂²⋊C₄×C₃₁	1	7	yes	—
145	C₂³	C₂³×C₄×C₃₁	1	7	yes	—
146	Q₈	—	2	112	no	—
147	Q₈	—	2	16	no	—
148	C₂×C₄	—	2	56	no	—
149	C₂×C₄	—	2	56	no	—
150	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	42	yes	—
151	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	42	yes	—
152	C₂×C₄	C₂³×C₄×C₃₁	1	16	yes	—
153	C₂×C₄	C₂³×C₄×C₃₁	1	7	yes	—
154	C₂×C₄	C₂²×D₄×C₃₁	1	14	yes	—
155	C₂×C₄	C₂²×D₄×C₃₁	1	14	yes	—
156	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
157	C₈	—	2	16	no	—
158	C₂²	—	2	42	no	—
159	C₂²	—	2	14	no	—
160	C₂²	C₂×C₃₁×#101	1	28	yes	direct product factor
161	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
162	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
163	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
164	C₄	—	2	16	no	—
165	C₄	C₂²×C₂²⋊C₄×C₃₁	1	14	yes	—
166	C₄	C₂³×D₄×C₃₁	1	2	yes	—
167	C₂	—	2	7	no	—
168	C₂	—	2	1	no	—
169	C₂	C₂²×C₃₁×#101	1	14	yes	direct product factor
170	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂²⋊C₄
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂⁵×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₃₁×#101	C₂	direct (N × H)
13. C₂×C₃₁×#101	C₂²	direct (N × H)
31. C₃₁×#101	C₂³	direct (N × H)
54. C₂³×#101	C₃₁	direct (N × H)
57. C₂³×C₃₁	#101	direct (N × H)
75. C₂²×#101	C₂×C₃₁	direct (N × H)
78. C₂²×C₃₁	C₂×#101	direct (N × H)
94. C₂×#101	C₂²×C₃₁	direct (N × H)
97. C₂×C₃₁	C₂²×#101	direct (N × H)
116. #101	C₂³×C₃₁	direct (N × H)
117. C₃₁	C₂³×#101	direct (N × H)
142. C₂³	C₃₁×#101	direct (N × H)
160. C₂²	C₂×C₃₁×#101	direct (N × H)
169. C₂	C₂²×C₃₁×#101	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₃₁	2
C₂³×M₁₆×C₃₁	2
C₂²×C₃₁×#101	2
C₂³×#101	31

Subgroups — C23×C31×#101 #21516831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃₁×#101 #21516831