Subgroups — C₂³×C₃₁×#100 #21516830

Order 7936.

7134 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#100	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶×C₃₁	C₄	1	2	yes	—
5	C₂³×D₄×C₃₁	—	2	14	no	—
6	C₂³×D₄×C₃₁	—	2	2	no	—
7	C₂³×D₄×C₃₁	C₄	1	14	yes	—
8	C₂³×D₄×C₃₁	C₂²	1	14	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	complemented
12	C₂³×C₈×C₃₁	—	2	2	no	—
13	C₂×C₃₁×#100	C₂²	1	112	yes	direct product factor
14	C₂⁵×C₃₁	—	2	14	no	—
15	C₂⁵×C₃₁	—	2	2	no	—
16	C₂⁵×C₃₁	—	4	16	no	—
17	C₂⁵×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂⁵×C₃₁	C₂×C₄	1	28	yes	—
19	C₂²×D₄×C₃₁	—	2	84	no	—
20	C₂²×D₄×C₃₁	—	2	28	no	—
21	C₂²×D₄×C₃₁	—	2	28	no	—
22	C₂²×D₄×C₃₁	—	2	84	no	—
23	C₂²×D₄×C₃₁	C₂×C₄	1	84	yes	—
24	C₂²×D₄×C₃₁	C₂³	1	28	yes	—
25	C₂³×C₄×C₃₁	C₂×C₄	1	7	yes	—
26	C₂³×C₄×C₃₁	D₄	1	14	yes	—
27	C₂³×C₄×C₃₁	D₄	1	2	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	7	yes	—
29	C₂×M₁₆×C₃₁	—	2	84	no	—
30	C₂×M₁₆×C₃₁	C₂³	1	56	yes	complemented
31	C₂²×C₈×C₃₁	—	2	28	no	—
32	C₃₁×#100	C₂³	1	64	yes	direct product factor
33	C₂⁴×C₃₁	—	2	8	no	—
34	C₂⁴×C₃₁	—	2	84	no	—
35	C₂⁴×C₃₁	—	2	28	no	—
36	C₂⁴×C₃₁	—	4	112	no	—
37	C₂⁴×C₃₁	—	4	112	no	—
38	C₂⁴×C₃₁	—	4	16	no	—
39	C₂⁴×C₃₁	C₂²⋊C₄	1	7	yes	—
40	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂⁴×C₃₁	C₂²×C₄	1	7	yes	—
42	C₂⁴×C₃₁	C₂²×C₄	1	56	yes	—
43	C₂×D₄×C₃₁	—	2	168	no	—
44	C₂×D₄×C₃₁	—	2	168	no	—
45	C₂×D₄×C₃₁	—	2	56	no	—
46	C₂×D₄×C₃₁	—	2	56	no	—
47	C₂×D₄×C₃₁	C₂²×C₄	1	56	yes	—
48	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	42	yes	—
50	C₂²×C₄×C₃₁	C₂²⋊C₄	1	14	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	21	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	14	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	42	yes	—
54	C₂²×C₄×C₃₁	C₂⁴	1	7	yes	—
55	M₁₆×C₃₁	—	2	56	no	—
56	M₁₆×C₃₁	C₂⁴	1	16	yes	complemented
57	C₂×C₈×C₃₁	—	2	56	no	—
58	C₂³×#100	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	—	2	28	no	—
60	C₂³×C₃₁	—	2	28	no	—
61	C₂³×C₃₁	—	2	56	no	—
62	C₂³×C₃₁	—	2	56	no	—
63	C₂³×C₃₁	—	4	112	no	—
64	C₂³×C₃₁	—	4	336	no	—
65	C₂³×C₃₁	—	4	112	no	—
66	C₂³×C₃₁	#100	1	8	yes	direct product factor
67	C₂³×C₃₁	C₂×C₂²⋊C₄	1	21	yes	—
68	C₂³×C₃₁	C₂×C₂²⋊C₄	1	7	yes	—
69	C₂³×C₃₁	C₂³×C₄	1	7	yes	—
70	C₂³×C₃₁	C₂³×C₄	1	16	yes	—
71	D₄×C₃₁	—	2	112	no	—
72	D₄×C₃₁	—	2	16	no	—
73	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	42	yes	—
74	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	42	yes	—
75	C₂×C₄×C₃₁	C₂³×C₄	1	7	yes	—
76	C₂×C₄×C₃₁	C₂²×D₄	1	14	yes	—
77	C₂×C₄×C₃₁	C₂²×D₄	1	14	yes	—
78	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
79	C₈×C₃₁	—	2	16	no	—
80	C₂⁴×D₄	C₂×C₃₁	1	1	yes	characteristic
81	C₂³×M₁₆	C₂×C₃₁	1	2	yes	complemented
82	C₂²×#100	C₂×C₃₁	1	28	yes	direct product factor
83	C₂²×C₃₁	—	2	42	no	—
84	C₂²×C₃₁	—	2	14	no	—
85	C₂²×C₃₁	—	2	16	no	—
86	C₂²×C₃₁	—	4	16	no	—
87	C₂²×C₃₁	—	4	112	no	—
88	C₂²×C₃₁	—	4	112	no	—
89	C₂²×C₃₁	C₂×#100	1	28	yes	direct product factor
90	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
91	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
92	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
93	C₄×C₃₁	C₂²×C₂²⋊C₄	1	14	yes	—
94	C₄×C₃₁	C₂³×D₄	1	2	yes	—
95	C₂⁶	C₄×C₃₁	1	2	yes	—
96	C₂³×D₄	—	2	2	no	—
97	C₂³×D₄	—	2	14	no	—
98	C₂³×D₄	C₄×C₃₁	1	14	yes	—
99	C₂³×D₄	C₂²×C₃₁	1	14	yes	—
100	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
101	C₂²×M₁₆	—	2	14	no	—
102	C₂²×M₁₆	C₂²×C₃₁	1	28	yes	complemented
103	C₂³×C₈	—	2	2	no	—
104	C₂×#100	C₂²×C₃₁	1	112	yes	direct product factor
105	C₂×C₃₁	—	2	7	no	—
106	C₂×C₃₁	—	2	1	no	—
107	C₂×C₃₁	—	4	16	no	—
108	C₂×C₃₁	C₂²×#100	1	14	yes	direct product factor
109	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
110	C₂⁵	—	2	14	no	—
111	C₂⁵	—	2	2	no	—
112	C₂⁵	—	4	16	no	—
113	C₂⁵	C₂×C₄×C₃₁	1	1	yes	characteristic
114	C₂⁵	C₂×C₄×C₃₁	1	28	yes	—
115	C₂²×D₄	—	2	84	no	—
116	C₂²×D₄	—	2	28	no	—
117	C₂²×D₄	—	2	28	no	—
118	C₂²×D₄	—	2	84	no	—
119	C₂²×D₄	C₂×C₄×C₃₁	1	84	yes	—
120	C₂²×D₄	C₂³×C₃₁	1	28	yes	—
121	C₂³×C₄	C₂×C₄×C₃₁	1	7	yes	—
122	C₂³×C₄	D₄×C₃₁	1	2	yes	—
123	C₂³×C₄	D₄×C₃₁	1	14	yes	—
124	C₂³×C₄	C₂³×C₃₁	1	7	yes	—
125	C₂×M₁₆	—	2	84	no	—
126	C₂×M₁₆	C₂³×C₃₁	1	56	yes	complemented
127	C₂²×C₈	—	2	28	no	—
128	#100	C₂³×C₃₁	1	64	yes	direct product factor
129	C₃₁	C₂³×#100	1	1	yes	characteristic, direct product factor, Sylow
130	C₂⁴	—	2	8	no	—
131	C₂⁴	—	2	84	no	—
132	C₂⁴	—	2	28	no	—
133	C₂⁴	—	4	112	no	—
134	C₂⁴	—	4	112	no	—
135	C₂⁴	—	4	16	no	—
136	C₂⁴	C₂²⋊C₄×C₃₁	1	7	yes	—
137	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
138	C₂⁴	C₂²×C₄×C₃₁	1	7	yes	—
139	C₂⁴	C₂²×C₄×C₃₁	1	56	yes	—
140	C₂×D₄	—	2	168	no	—
141	C₂×D₄	—	2	168	no	—
142	C₂×D₄	—	2	56	no	—
143	C₂×D₄	—	2	56	no	—
144	C₂×D₄	C₂²×C₄×C₃₁	1	56	yes	—
145	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₃₁	1	42	yes	—
147	C₂²×C₄	C₂²⋊C₄×C₃₁	1	14	yes	—
148	C₂²×C₄	C₂²×C₄×C₃₁	1	21	yes	—
149	C₂²×C₄	C₂×D₄×C₃₁	1	14	yes	—
150	C₂²×C₄	C₂×D₄×C₃₁	1	42	yes	—
151	C₂²×C₄	C₂⁴×C₃₁	1	7	yes	—
152	M₁₆	—	2	56	no	—
153	M₁₆	C₂⁴×C₃₁	1	16	yes	complemented
154	C₂×C₈	—	2	56	no	—
155	C₂³	—	2	28	no	—
156	C₂³	—	2	28	no	—
157	C₂³	—	2	56	no	—
158	C₂³	—	2	56	no	—
159	C₂³	—	4	112	no	—
160	C₂³	—	4	336	no	—
161	C₂³	—	4	112	no	—
162	C₂³	C₃₁×#100	1	8	yes	direct product factor
163	C₂³	C₂×C₂²⋊C₄×C₃₁	1	21	yes	—
164	C₂³	C₂×C₂²⋊C₄×C₃₁	1	7	yes	—
165	C₂³	C₂³×C₄×C₃₁	1	7	yes	—
166	C₂³	C₂³×C₄×C₃₁	1	16	yes	—
167	D₄	—	2	112	no	—
168	D₄	—	2	16	no	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	42	yes	—
170	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	42	yes	—
171	C₂×C₄	C₂³×C₄×C₃₁	1	7	yes	—
172	C₂×C₄	C₂²×D₄×C₃₁	1	14	yes	—
173	C₂×C₄	C₂²×D₄×C₃₁	1	14	yes	—
174	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
175	C₈	—	2	16	no	—
176	C₂²	—	2	42	no	—
177	C₂²	—	2	14	no	—
178	C₂²	—	2	16	no	—
179	C₂²	—	4	16	no	—
180	C₂²	—	4	112	no	—
181	C₂²	—	4	112	no	—
182	C₂²	C₂×C₃₁×#100	1	28	yes	direct product factor
183	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
184	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
185	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
186	C₄	C₂²×C₂²⋊C₄×C₃₁	1	14	yes	—
187	C₄	C₂³×D₄×C₃₁	1	2	yes	—
188	C₂	—	2	7	no	—
189	C₂	—	2	1	no	—
190	C₂	—	4	16	no	—
191	C₂	C₂²×C₃₁×#100	1	14	yes	direct product factor
192	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	C₂⁴×D₄×C₃₁	C₂
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
2. C₂³×M₁₆×C₃₁	C₂	semidirect
3. C₂²×C₃₁×#100	C₂	direct (N × H)
11. C₂²×M₁₆×C₃₁	C₂²	semidirect
13. C₂×C₃₁×#100	C₂²	direct (N × H)
30. C₂×M₁₆×C₃₁	C₂³	semidirect
32. C₃₁×#100	C₂³	direct (N × H)
56. M₁₆×C₃₁	C₂⁴	semidirect
58. C₂³×#100	C₃₁	direct (N × H)
66. C₂³×C₃₁	#100	direct (N × H)
81. C₂³×M₁₆	C₂×C₃₁	semidirect
82. C₂²×#100	C₂×C₃₁	direct (N × H)
89. C₂²×C₃₁	C₂×#100	direct (N × H)
102. C₂²×M₁₆	C₂²×C₃₁	semidirect
104. C₂×#100	C₂²×C₃₁	direct (N × H)
108. C₂×C₃₁	C₂²×#100	direct (N × H)
126. C₂×M₁₆	C₂³×C₃₁	semidirect
128. #100	C₂³×C₃₁	direct (N × H)
129. C₃₁	C₂³×#100	direct (N × H)
153. M₁₆	C₂⁴×C₃₁	semidirect
162. C₂³	C₃₁×#100	direct (N × H)
182. C₂²	C₂×C₃₁×#100	direct (N × H)
191. C₂	C₂²×C₃₁×#100	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×D₄×C₃₁	2
C₂³×M₁₆×C₃₁	2
C₂²×C₃₁×#100	2
C₂³×#100	31

Subgroups — C23×C31×#100 #21516830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃₁×#100 #21516830