Subgroups — C₃₁×#33456 #21490215

Order 7936.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2948	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2177	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₁×D₁₆	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2140	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#518	—	2	2	no	—
10	C₂×C₃₁×D₁₆	C₄	1	8	yes	complemented
11	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
12	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
13	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#442	—	2	4	no	—
16	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
17	C₃₁×#427	C₂²	1	2	yes	—
18	C₃₁×#418	—	2	2	no	—
19	C₃₁×#405	C₂²	1	1	yes	characteristic
20	C₃₁×#367	C₂²	1	2	yes	complemented
21	C₃₁×#366	C₂²	1	2	yes	complemented
22	C₃₁×#357	C₂²	1	8	yes	complemented
23	C₃₁×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
25	C₂²×D₄×C₃₁	—	2	2	no	—
26	C₂×D₈×C₃₁	—	2	2	no	—
27	C₂×D₈×C₃₁	—	2	4	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
29	C₂×D₈×C₃₁	C₂³	1	4	yes	—
30	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₄×D₄×C₃₁	—	2	8	no	—
33	C₄×D₄×C₃₁	—	4	4	no	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
37	C₃₁×D₁₆	C₂×C₄	1	8	yes	complemented
38	C₃₁×D₁₆	C₂×C₄	1	8	yes	complemented
39	C₃₁×M₃₂	—	2	2	no	—
40	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
41	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
42	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×#107	C₂³	1	4	yes	—
45	C₃₁×#106	—	2	2	no	—
46	C₃₁×#102	—	2	8	no	—
47	C₃₁×#97	D₄	1	2	yes	—
48	C₄×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂⁴×C₃₁	—	4	2	no	—
50	C₂×D₄×C₃₁	—	2	8	no	—
51	C₂×D₄×C₃₁	—	2	4	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂×D₄×C₃₁	—	4	4	no	—
54	C₂²×C₄×C₃₁	—	4	8	no	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	D₈×C₃₁	—	2	8	no	—
57	D₈×C₃₁	—	2	4	no	—
58	D₈×C₃₁	C₂²×C₄	1	8	yes	—
59	M₁₆×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	#41	1	2	yes	—
64	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
67	C₂²⋊C₄×C₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₃₁	—	4	8	no	—
69	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
70	C₄²×C₃₁	—	4	2	no	—
71	C₄²×C₃₁	C₂×D₄	1	2	yes	—
72	C₁₆×C₃₁	—	2	2	no	—
73	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
74	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
75	#33456	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	—	4	4	no	—
77	C₂³×C₃₁	—	4	8	no	—
78	C₂³×C₃₁	—	16	4	no	—
79	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
80	D₄×C₃₁	—	2	8	no	—
81	D₄×C₃₁	—	4	4	no	—
82	D₄×C₃₁	—	4	8	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	16	8	no	—
85	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
87	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#135	1	2	yes	—
89	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
90	C₈×C₃₁	—	2	2	no	—
91	C₈×C₃₁	C₄×D₄	1	2	yes	—
92	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
94	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic
95	#2948	C₂×C₃₁	1	2	yes	complemented
96	#2177	C₂×C₃₁	1	4	yes	complemented
97	C₄×D₁₆	C₂×C₃₁	1	4	yes	complemented
98	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
99	#2140	C₂×C₃₁	1	2	yes	complemented
100	#2108	C₂×C₃₁	1	1	yes	characteristic, complemented
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	4	8	no	—
103	C₂²×C₃₁	—	8	8	no	—
104	C₂²×C₃₁	—	16	8	no	—
105	C₂²×C₃₁	C₄×D₈	1	2	yes	—
106	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₃₁	—	2	4	no	—
108	C₄×C₃₁	—	8	4	no	—
109	C₄×C₃₁	C₄×D₈	1	2	yes	—
110	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
112	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
113	#518	—	2	2	no	—
114	C₂×D₁₆	C₄×C₃₁	1	8	yes	complemented
115	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
116	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
117	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
119	#442	—	2	4	no	—
120	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
121	#427	C₂²×C₃₁	1	2	yes	—
122	#418	—	2	2	no	—
123	#405	C₂²×C₃₁	1	1	yes	characteristic
124	#367	C₂²×C₃₁	1	2	yes	complemented
125	#366	C₂²×C₃₁	1	2	yes	complemented
126	#357	C₂²×C₃₁	1	8	yes	complemented
127	#346	C₂²×C₃₁	1	2	yes	complemented
128	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
129	C₂×C₃₁	—	2	2	no	—
130	C₂×C₃₁	—	8	8	no	—
131	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
132	C₂×C₃₁	#3415	1	2	yes	—
133	C₂²×D₄	—	2	2	no	—
134	C₂×D₈	—	2	2	no	—
135	C₂×D₈	—	2	4	no	—
136	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
137	C₂×D₈	C₂³×C₃₁	1	4	yes	—
138	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
140	C₄×D₄	—	2	8	no	—
141	C₄×D₄	—	4	4	no	—
142	#117	—	2	2	no	—
143	#117	D₄×C₃₁	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	4	4	no	—
145	D₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
146	D₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
147	M₃₂	—	2	2	no	—
148	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
149	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
150	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
151	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
152	#107	C₂³×C₃₁	1	4	yes	—
153	#106	—	2	2	no	—
154	#102	—	2	8	no	—
155	#97	D₄×C₃₁	1	2	yes	—
156	C₄×C₈	C₂³×C₃₁	1	2	yes	—
157	C₃₁	#33456	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	4	8	no	—
164	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
165	D₈	—	2	8	no	—
166	D₈	—	2	4	no	—
167	D₈	C₂²×C₄×C₃₁	1	8	yes	—
168	M₁₆	—	2	2	no	—
169	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
170	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
171	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	#41×C₃₁	1	2	yes	—
173	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₂²⋊C₄	—	4	8	no	—
178	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
179	C₄²	—	4	2	no	—
180	C₄²	C₂×D₄×C₃₁	1	2	yes	—
181	C₁₆	—	2	2	no	—
182	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
183	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	8	no	—
190	D₄	—	4	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	16	8	no	—
193	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
194	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
195	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₃₁×#135	1	2	yes	—
197	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₄×D₄×C₃₁	1	2	yes	—
200	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
201	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	C₄×D₈×C₃₁	1	2	yes	—
207	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	8	4	no	—
210	C₄	C₄×D₈×C₃₁	1	2	yes	—
211	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
212	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
216	C₂	C₃₁×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×C₃₁×D₁₆	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2948	C₂	semidirect
3. C₃₁×#2177	C₂	semidirect
4. C₄×C₃₁×D₁₆	C₂	semidirect
5. C₃₁×#2158	C₂	semidirect
6. C₃₁×#2140	C₂	semidirect
7. C₃₁×#2108	C₂	semidirect
10. C₂×C₃₁×D₁₆	C₄	semidirect
16. C₄×D₈×C₃₁	C₂²	semidirect
20. C₃₁×#367	C₂²	semidirect
21. C₃₁×#366	C₂²	semidirect
22. C₃₁×#357	C₂²	semidirect
23. C₃₁×#346	C₂²	semidirect
24. C₄×C₁₆×C₃₁	C₂²	semidirect
28. C₂×D₈×C₃₁	C₂×C₄	semidirect
37. C₃₁×D₁₆	C₂×C₄	semidirect
38. C₃₁×D₁₆	C₂×C₄	semidirect
40. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
75. #33456	C₃₁	direct (N × H)
95. #2948	C₂×C₃₁	semidirect
96. #2177	C₂×C₃₁	semidirect
97. C₄×D₁₆	C₂×C₃₁	semidirect
98. #2158	C₂×C₃₁	semidirect
99. #2140	C₂×C₃₁	semidirect
100. #2108	C₂×C₃₁	semidirect
114. C₂×D₁₆	C₄×C₃₁	semidirect
120. C₄×D₈	C₂²×C₃₁	semidirect
124. #367	C₂²×C₃₁	semidirect
125. #366	C₂²×C₃₁	semidirect
126. #357	C₂²×C₃₁	semidirect
127. #346	C₂²×C₃₁	semidirect
128. C₄×C₁₆	C₂²×C₃₁	semidirect
136. C₂×D₈	C₂×C₄×C₃₁	semidirect
145. D₁₆	C₂×C₄×C₃₁	semidirect
146. D₁₆	C₂×C₄×C₃₁	semidirect
148. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
157. C₃₁	#33456	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×D₁₆	2
C₃₁×#2948	2
C₃₁×#2177	2
C₄×C₃₁×D₁₆	2
C₃₁×#2158	2
C₃₁×#2140	2
C₃₁×#2108	2
#33456	31

Subgroups — C31×#33456 #21490215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33456 #21490215