Subgroups — C₂³×C₃₁×#102 #21516836

Order 7936.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 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	C₂⁴×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#102	C₂	1	28	yes	maximal, direct product factor
5	C₂⁶×C₃₁	—	2	1	no	—
6	C₂³×D₄×C₃₁	—	2	7	no	—
7	C₂³×D₄×C₃₁	—	2	8	no	—
8	C₂³×D₄×C₃₁	C₄	1	16	yes	complemented
9	C₂³×D₄×C₃₁	C₂²	1	14	yes	—
10	C₂⁴×C₄×C₃₁	—	2	1	no	—
11	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	—	2	8	no	—
13	C₂³×C₈×C₃₁	C₂²	1	14	yes	complemented
14	C₂²×C₄⋊C₄×C₃₁	—	2	7	no	—
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	14	yes	complemented
16	C₂×C₃₁×#102	C₂²	1	112	yes	direct product factor
17	C₂⁵×C₃₁	—	2	16	no	—
18	C₂⁵×C₃₁	—	2	14	no	—
19	C₂⁵×C₃₁	—	4	8	no	—
20	C₂⁵×C₃₁	—	4	8	no	—
21	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×D₄×C₃₁	—	2	112	no	—
23	C₂²×D₄×C₃₁	—	2	42	no	—
24	C₂²×D₄×C₃₁	—	2	56	no	—
25	C₂²×D₄×C₃₁	C₂×C₄	1	112	yes	complemented
26	C₂²×D₄×C₃₁	C₂³	1	28	yes	—
27	C₂³×C₄×C₃₁	—	2	14	no	—
28	C₂³×C₄×C₃₁	—	4	8	no	—
29	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₃₁	D₄	1	7	yes	—
31	C₂³×C₄×C₃₁	D₄	1	8	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	7	yes	—
33	C₂²×C₈×C₃₁	—	2	56	no	—
34	C₂²×C₈×C₃₁	C₂³	1	28	yes	complemented
35	C₂×C₄⋊C₄×C₃₁	—	2	42	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	28	yes	complemented
37	C₃₁×#102	C₂³	1	64	yes	direct product factor
38	C₂⁴×C₃₁	—	2	112	no	—
39	C₂⁴×C₃₁	—	2	28	no	—
40	C₂⁴×C₃₁	—	4	128	no	—
41	C₂⁴×C₃₁	—	4	56	no	—
42	C₂⁴×C₃₁	—	4	56	no	—
43	C₂⁴×C₃₁	C₂²⋊C₄	1	8	yes	—
44	C₂⁴×C₃₁	D₈	1	8	yes	—
45	C₂⁴×C₃₁	SD₁₆	1	8	yes	—
46	C₂⁴×C₃₁	C₂×D₄	1	7	yes	—
47	C₂×D₄×C₃₁	—	2	336	no	—
48	C₂×D₄×C₃₁	—	2	28	no	—
49	C₂×D₄×C₃₁	—	2	56	no	—
50	C₂×D₄×C₃₁	C₂²×C₄	1	112	yes	complemented
51	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
52	C₂²×C₄×C₃₁	—	2	28	no	—
53	C₂²×C₄×C₃₁	—	4	56	no	—
54	C₂²×C₄×C₃₁	C₂²⋊C₄	1	56	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	28	yes	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	21	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	28	yes	—
58	C₂²×C₄×C₃₁	C₂⁴	1	7	yes	—
59	C₂×C₈×C₃₁	—	2	56	no	—
60	C₂×C₈×C₃₁	C₂⁴	1	8	yes	complemented
61	C₄⋊C₄×C₃₁	—	2	28	no	—
62	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	complemented
63	C₂³×#102	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₃₁	—	2	112	no	—
65	C₂³×C₃₁	—	2	8	no	—
66	C₂³×C₃₁	—	4	56	no	—
67	C₂³×C₃₁	—	4	56	no	—
68	C₂³×C₃₁	—	4	448	no	—
69	C₂³×C₃₁	#102	1	64	yes	direct product factor
70	C₂³×C₃₁	C₂×C₂²⋊C₄	1	28	yes	—
71	C₂³×C₃₁	C₂×D₈	1	28	yes	—
72	C₂³×C₃₁	C₂×SD₁₆	1	28	yes	—
73	C₂³×C₃₁	C₂²×D₄	1	7	yes	—
74	D₄×C₃₁	—	2	112	no	—
75	D₄×C₃₁	—	2	8	no	—
76	D₄×C₃₁	C₂³×C₄	1	16	yes	complemented
77	C₂×C₄×C₃₁	—	2	8	no	—
78	C₂×C₄×C₃₁	—	4	56	no	—
79	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	84	yes	—
80	C₂×C₄×C₃₁	C₂³×C₄	1	14	yes	—
81	C₂×C₄×C₃₁	C₂²×D₄	1	7	yes	—
82	C₂×C₄×C₃₁	C₂²×D₄	1	14	yes	—
83	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
84	C₈×C₃₁	—	2	8	no	—
85	C₂⁴×D₄	C₂×C₃₁	1	1	yes	characteristic
86	C₂⁴×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
87	C₂³×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
88	C₂²×#102	C₂×C₃₁	1	28	yes	direct product factor
89	C₂²×C₃₁	—	2	16	no	—
90	C₂²×C₃₁	—	4	224	no	—
91	C₂²×C₃₁	—	4	8	no	—
92	C₂²×C₃₁	—	4	8	no	—
93	C₂²×C₃₁	C₂×#102	1	112	yes	direct product factor
94	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	14	yes	—
95	C₂²×C₃₁	C₂²×D₈	1	14	yes	—
96	C₂²×C₃₁	C₂²×SD₁₆	1	14	yes	—
97	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
98	C₄×C₃₁	—	4	8	no	—
99	C₄×C₃₁	C₂²×C₂²⋊C₄	1	14	yes	—
100	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
102	C₂⁶	—	2	1	no	—
103	C₂³×D₄	—	2	7	no	—
104	C₂³×D₄	—	2	8	no	—
105	C₂³×D₄	C₄×C₃₁	1	16	yes	complemented
106	C₂³×D₄	C₂²×C₃₁	1	14	yes	—
107	C₂⁴×C₄	—	2	1	no	—
108	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
109	C₂³×C₈	—	2	8	no	—
110	C₂³×C₈	C₂²×C₃₁	1	14	yes	complemented
111	C₂²×C₄⋊C₄	—	2	7	no	—
112	C₂²×C₄⋊C₄	C₂²×C₃₁	1	14	yes	complemented
113	C₂×#102	C₂²×C₃₁	1	112	yes	direct product factor
114	C₂×C₃₁	—	4	16	no	—
115	C₂×C₃₁	C₂²×#102	1	28	yes	direct product factor
116	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
118	C₂×C₃₁	C₂³×SD₁₆	1	1	yes	characteristic
119	C₂⁵	—	2	16	no	—
120	C₂⁵	—	2	14	no	—
121	C₂⁵	—	4	8	no	—
122	C₂⁵	—	4	8	no	—
123	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
124	C₂²×D₄	—	2	112	no	—
125	C₂²×D₄	—	2	42	no	—
126	C₂²×D₄	—	2	56	no	—
127	C₂²×D₄	C₂×C₄×C₃₁	1	112	yes	complemented
128	C₂²×D₄	C₂³×C₃₁	1	28	yes	—
129	C₂³×C₄	—	2	14	no	—
130	C₂³×C₄	—	4	8	no	—
131	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	—
132	C₂³×C₄	D₄×C₃₁	1	7	yes	—
133	C₂³×C₄	D₄×C₃₁	1	8	yes	—
134	C₂³×C₄	C₂³×C₃₁	1	7	yes	—
135	C₂²×C₈	—	2	56	no	—
136	C₂²×C₈	C₂³×C₃₁	1	28	yes	complemented
137	C₂×C₄⋊C₄	—	2	42	no	—
138	C₂×C₄⋊C₄	C₂³×C₃₁	1	28	yes	complemented
139	#102	C₂³×C₃₁	1	64	yes	direct product factor
140	C₃₁	C₂³×#102	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	—	2	112	no	—
142	C₂⁴	—	2	28	no	—
143	C₂⁴	—	4	128	no	—
144	C₂⁴	—	4	56	no	—
145	C₂⁴	—	4	56	no	—
146	C₂⁴	C₂²⋊C₄×C₃₁	1	8	yes	—
147	C₂⁴	D₈×C₃₁	1	8	yes	—
148	C₂⁴	SD₁₆×C₃₁	1	8	yes	—
149	C₂⁴	C₂×D₄×C₃₁	1	7	yes	—
150	C₂×D₄	—	2	336	no	—
151	C₂×D₄	—	2	28	no	—
152	C₂×D₄	—	2	56	no	—
153	C₂×D₄	C₂²×C₄×C₃₁	1	112	yes	complemented
154	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
155	C₂²×C₄	—	2	28	no	—
156	C₂²×C₄	—	4	56	no	—
157	C₂²×C₄	C₂²⋊C₄×C₃₁	1	56	yes	—
158	C₂²×C₄	C₂²×C₄×C₃₁	1	28	yes	—
159	C₂²×C₄	C₂×D₄×C₃₁	1	21	yes	—
160	C₂²×C₄	C₂×D₄×C₃₁	1	28	yes	—
161	C₂²×C₄	C₂⁴×C₃₁	1	7	yes	—
162	C₂×C₈	—	2	56	no	—
163	C₂×C₈	C₂⁴×C₃₁	1	8	yes	complemented
164	C₄⋊C₄	—	2	28	no	—
165	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	complemented
166	C₂³	—	2	112	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	4	448	no	—
169	C₂³	—	4	56	no	—
170	C₂³	—	4	56	no	—
171	C₂³	C₃₁×#102	1	64	yes	direct product factor
172	C₂³	C₂×C₂²⋊C₄×C₃₁	1	28	yes	—
173	C₂³	C₂×D₈×C₃₁	1	28	yes	—
174	C₂³	C₂×SD₁₆×C₃₁	1	28	yes	—
175	C₂³	C₂²×D₄×C₃₁	1	7	yes	—
176	D₄	—	2	112	no	—
177	D₄	—	2	8	no	—
178	D₄	C₂³×C₄×C₃₁	1	16	yes	complemented
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	4	56	no	—
181	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	84	yes	—
182	C₂×C₄	C₂³×C₄×C₃₁	1	14	yes	—
183	C₂×C₄	C₂²×D₄×C₃₁	1	7	yes	—
184	C₂×C₄	C₂²×D₄×C₃₁	1	14	yes	—
185	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
186	C₈	—	2	8	no	—
187	C₂²	—	2	16	no	—
188	C₂²	—	4	224	no	—
189	C₂²	—	4	8	no	—
190	C₂²	—	4	8	no	—
191	C₂²	C₂×C₃₁×#102	1	112	yes	direct product factor
192	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	14	yes	—
193	C₂²	C₂²×D₈×C₃₁	1	14	yes	—
194	C₂²	C₂²×SD₁₆×C₃₁	1	14	yes	—
195	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
196	C₄	—	4	8	no	—
197	C₄	C₂²×C₂²⋊C₄×C₃₁	1	14	yes	—
198	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
199	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
200	C₂	—	4	16	no	—
201	C₂	C₂²×C₃₁×#102	1	28	yes	direct product factor
202	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
204	C₂	C₂³×SD₁₆×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₃₁	D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂⁴×D₄×C₃₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₈×C₃₁	C₂	semidirect
3. C₂³×C₄⋊C₄×C₃₁	C₂	semidirect
4. C₂²×C₃₁×#102	C₂	direct (N × H)
8. C₂³×D₄×C₃₁	C₄	semidirect
13. C₂³×C₈×C₃₁	C₂²	semidirect
15. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
16. C₂×C₃₁×#102	C₂²	direct (N × H)
25. C₂²×D₄×C₃₁	C₂×C₄	semidirect
34. C₂²×C₈×C₃₁	C₂³	semidirect
36. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
37. C₃₁×#102	C₂³	direct (N × H)
50. C₂×D₄×C₃₁	C₂²×C₄	semidirect
60. C₂×C₈×C₃₁	C₂⁴	semidirect
62. C₄⋊C₄×C₃₁	C₂⁴	semidirect
63. C₂³×#102	C₃₁	direct (N × H)
69. C₂³×C₃₁	#102	direct (N × H)
76. D₄×C₃₁	C₂³×C₄	semidirect
86. C₂⁴×C₈	C₂×C₃₁	semidirect
87. C₂³×C₄⋊C₄	C₂×C₃₁	semidirect
88. C₂²×#102	C₂×C₃₁	direct (N × H)
93. C₂²×C₃₁	C₂×#102	direct (N × H)
105. C₂³×D₄	C₄×C₃₁	semidirect
110. C₂³×C₈	C₂²×C₃₁	semidirect
112. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
113. C₂×#102	C₂²×C₃₁	direct (N × H)
115. C₂×C₃₁	C₂²×#102	direct (N × H)
127. C₂²×D₄	C₂×C₄×C₃₁	semidirect
136. C₂²×C₈	C₂³×C₃₁	semidirect
138. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
139. #102	C₂³×C₃₁	direct (N × H)
140. C₃₁	C₂³×#102	direct (N × H)
153. C₂×D₄	C₂²×C₄×C₃₁	semidirect
163. C₂×C₈	C₂⁴×C₃₁	semidirect
165. C₄⋊C₄	C₂⁴×C₃₁	semidirect
171. C₂³	C₃₁×#102	direct (N × H)
178. D₄	C₂³×C₄×C₃₁	semidirect
191. C₂²	C₂×C₃₁×#102	direct (N × H)
201. C₂	C₂²×C₃₁×#102	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
C₂⁴×C₈×C₃₁	2
C₂³×C₄⋊C₄×C₃₁	2
C₂²×C₃₁×#102	2
C₂³×#102	31

Subgroups — C23×C31×#102 #21516836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃₁×#102 #21516836