Subgroups — C₃₁×#33385 #21490144

Order 7936.

1114 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3584	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2143	C₂	1	6	yes	maximal, complemented
5	C₂×C₃₁×#359	C₂	1	6	yes	maximal, complemented
6	C₃₁×#585	—	2	1	no	—
7	C₃₁×#578	C₄	1	4	yes	—
8	C₃₁×#577	—	2	4	no	—
9	C₃₁×#576	C₄	1	4	yes	—
10	C₃₁×#575	—	2	3	no	—
11	C₂×C₃₁×#135	C₂²	1	6	yes	—
12	C₃₁×#568	—	2	1	no	—
13	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
14	C₃₁×#504	—	2	4	no	—
15	C₂×C₃₁×M₃₂	C₂²	1	3	yes	complemented
16	C₂²×C₁₆×C₃₁	C₂²	1	3	yes	complemented
17	C₃₁×#430	—	2	3	no	—
18	C₂×C₃₁×#108	C₂²	1	6	yes	complemented
19	C₃₁×#359	C₂²	1	4	yes	complemented
20	C₃₁×#359	C₂²	1	12	yes	complemented
21	C₃₁×2₋¹⁺⁴	—	2	4	no	—
22	C₃₁×2₋¹⁺⁴	—	4	1	no	—
23	C₃₁×H(5,𝔽₂)	—	2	4	no	—
24	C₃₁×H(5,𝔽₂)	—	4	3	no	—
25	C₂×#41×C₃₁	—	2	6	no	—
26	C₂×#41×C₃₁	—	4	4	no	—
27	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×#137	—	2	12	no	—
29	C₃₁×AGL(1,ℤ/8)	—	2	12	no	—
30	C₃₁×#135	—	2	6	no	—
31	C₃₁×#135	—	2	6	no	—
32	C₃₁×#135	C₂×C₄	1	12	yes	—
33	C₃₁×#135	C₂³	1	4	yes	—
34	C₂×Q₁₆×C₃₁	C₂×C₄	1	6	yes	—
35	C₂×SD₁₆×C₃₁	—	2	6	no	—
36	C₂×D₈×C₃₁	C₂×C₄	1	6	yes	—
37	C₃₁×#131	—	4	4	no	—
38	C₃₁×#131	C₂×C₄	1	4	yes	—
39	C₃₁×#131	D₄	1	4	yes	—
40	C₂×M₁₆×C₃₁	—	2	6	no	—
41	C₂×M₁₆×C₃₁	D₄	1	3	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
43	C₃₁×M₃₂	—	2	6	no	—
44	C₂×C₁₆×C₃₁	—	2	6	no	—
45	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₃₁	C₂³	1	3	yes	—
47	C₃₁×#108	—	2	6	no	—
48	C₃₁×#108	C₂³	1	4	yes	—
49	#41×C₃₁	—	2	12	no	—
50	#41×C₃₁	—	2	4	no	—
51	#41×C₃₁	—	4	4	no	—
52	#41×C₃₁	—	4	12	no	—
53	#41×C₃₁	—	4	6	no	—
54	#41×C₃₁	—	4	6	no	—
55	#41×C₃₁	C₂²⋊C₄	1	4	yes	—
56	#41×C₃₁	D₈	1	4	yes	—
57	C₂×Q₈×C₃₁	—	2	6	no	—
58	C₂×Q₈×C₃₁	—	4	4	no	—
59	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
60	C₂×D₄×C₃₁	—	2	6	no	—
61	C₂×D₄×C₃₁	—	4	6	no	—
62	C₂×D₄×C₃₁	—	4	12	no	—
63	C₂×D₄×C₃₁	SD₁₆	1	3	yes	—
64	C₂²×C₄×C₃₁	—	4	6	no	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
66	Q₁₆×C₃₁	—	2	6	no	—
67	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	—
68	SD₁₆×C₃₁	—	2	12	no	—
69	SD₁₆×C₃₁	—	2	4	no	—
70	D₈×C₃₁	—	2	6	no	—
71	D₈×C₃₁	C₂²×C₄	1	4	yes	—
72	M₁₆×C₃₁	—	2	4	no	—
73	M₁₆×C₃₁	—	4	6	no	—
74	M₁₆×C₃₁	C₂²⋊C₄	1	12	yes	—
75	C₂×C₈×C₃₁	—	4	6	no	—
76	C₂×C₈×C₃₁	C₂²×C₄	1	6	yes	—
77	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
78	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
79	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₁₆×C₃₁	—	2	1	no	—
81	C₁₆×C₃₁	—	2	3	no	—
82	#33385	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	4	6	no	—
84	C₂³×C₃₁	C₂×SD₁₆	1	3	yes	—
85	Q₈×C₃₁	—	2	4	no	—
86	Q₈×C₃₁	—	4	6	no	—
87	Q₈×C₃₁	#102	1	4	yes	—
88	D₄×C₃₁	—	2	4	no	—
89	D₄×C₃₁	—	4	4	no	—
90	D₄×C₃₁	—	4	12	no	—
91	D₄×C₃₁	—	4	6	no	—
92	D₄×C₃₁	#102	1	12	yes	—
93	C₂×C₄×C₃₁	—	4	6	no	—
94	C₂×C₄×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	4	12	no	—
96	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
97	C₂×C₄×C₃₁	C₂×D₈	1	6	yes	—
98	C₂×C₄×C₃₁	C₂×SD₁₆	1	3	yes	—
99	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₃₁	—	4	4	no	—
101	C₈×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
102	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	#3584	C₂×C₃₁	1	1	yes	characteristic
105	C₂×#504	C₂×C₃₁	1	1	yes	characteristic, complemented
106	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
107	#2143	C₂×C₃₁	1	6	yes	complemented
108	C₂×#359	C₂×C₃₁	1	6	yes	complemented
109	C₂²×C₃₁	—	4	3	no	—
110	C₂²×C₃₁	—	4	4	no	—
111	C₂²×C₃₁	—	16	6	no	—
112	C₂²×C₃₁	C₂×#102	1	6	yes	—
113	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
114	C₄×C₃₁	—	4	4	no	—
115	C₄×C₃₁	C₂×#102	1	6	yes	—
116	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
118	#585	—	2	1	no	—
119	#578	C₄×C₃₁	1	4	yes	—
120	#577	—	2	4	no	—
121	#576	C₄×C₃₁	1	4	yes	—
122	#575	—	2	3	no	—
123	C₂×#135	C₂²×C₃₁	1	6	yes	—
124	#568	—	2	1	no	—
125	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
126	#504	—	2	4	no	—
127	C₂×M₃₂	C₂²×C₃₁	1	3	yes	complemented
128	C₂²×C₁₆	C₂²×C₃₁	1	3	yes	complemented
129	#430	—	2	3	no	—
130	C₂×#108	C₂²×C₃₁	1	6	yes	complemented
131	#359	C₂²×C₃₁	1	4	yes	complemented
132	#359	C₂²×C₃₁	1	12	yes	complemented
133	C₂×C₃₁	—	2	1	no	—
134	C₂×C₃₁	—	2	6	no	—
135	C₂×C₃₁	—	8	4	no	—
136	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
137	2₋¹⁺⁴	—	2	4	no	—
138	2₋¹⁺⁴	—	4	1	no	—
139	H(5,𝔽₂)	—	2	4	no	—
140	H(5,𝔽₂)	—	4	3	no	—
141	C₂×#41	—	2	6	no	—
142	C₂×#41	—	4	4	no	—
143	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
144	#137	—	2	12	no	—
145	AGL(1,ℤ/8)	—	2	12	no	—
146	#135	—	2	6	no	—
147	#135	—	2	6	no	—
148	#135	C₂×C₄×C₃₁	1	12	yes	—
149	#135	C₂³×C₃₁	1	4	yes	—
150	C₂×Q₁₆	C₂×C₄×C₃₁	1	6	yes	—
151	C₂×SD₁₆	—	2	6	no	—
152	C₂×D₈	C₂×C₄×C₃₁	1	6	yes	—
153	#131	—	4	4	no	—
154	#131	C₂×C₄×C₃₁	1	4	yes	—
155	#131	D₄×C₃₁	1	4	yes	—
156	C₂×M₁₆	—	2	6	no	—
157	C₂×M₁₆	D₄×C₃₁	1	3	yes	—
158	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
159	M₃₂	—	2	6	no	—
160	C₂×C₁₆	—	2	6	no	—
161	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₃₁	1	3	yes	—
163	#108	—	2	6	no	—
164	#108	C₂³×C₃₁	1	4	yes	—
165	C₃₁	#33385	1	1	yes	characteristic, direct product factor, Sylow
166	#41	—	2	12	no	—
167	#41	—	2	4	no	—
168	#41	—	4	4	no	—
169	#41	—	4	12	no	—
170	#41	—	4	6	no	—
171	#41	—	4	6	no	—
172	#41	C₂²⋊C₄×C₃₁	1	4	yes	—
173	#41	D₈×C₃₁	1	4	yes	—
174	C₂×Q₈	—	2	6	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
177	C₂×D₄	—	2	6	no	—
178	C₂×D₄	—	4	6	no	—
179	C₂×D₄	—	4	12	no	—
180	C₂×D₄	SD₁₆×C₃₁	1	3	yes	—
181	C₂²×C₄	—	4	6	no	—
182	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
183	Q₁₆	—	2	6	no	—
184	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
185	SD₁₆	—	2	12	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	6	no	—
188	D₈	C₂²×C₄×C₃₁	1	4	yes	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	6	no	—
191	M₁₆	C₂²⋊C₄×C₃₁	1	12	yes	—
192	C₂×C₈	—	4	6	no	—
193	C₂×C₈	C₂²×C₄×C₃₁	1	6	yes	—
194	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
195	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
196	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
197	C₁₆	—	2	1	no	—
198	C₁₆	—	2	3	no	—
199	C₂³	—	4	6	no	—
200	C₂³	C₂×SD₁₆×C₃₁	1	3	yes	—
201	Q₈	—	2	4	no	—
202	Q₈	—	4	6	no	—
203	Q₈	C₃₁×#102	1	4	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	4	12	no	—
206	D₄	—	4	4	no	—
207	D₄	—	4	6	no	—
208	D₄	C₃₁×#102	1	12	yes	—
209	C₂×C₄	—	4	6	no	—
210	C₂×C₄	—	4	12	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
213	C₂×C₄	C₂×D₈×C₃₁	1	6	yes	—
214	C₂×C₄	C₂×SD₁₆×C₃₁	1	3	yes	—
215	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
216	C₈	—	4	4	no	—
217	C₈	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
218	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
219	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
220	C₂²	—	4	3	no	—
221	C₂²	—	4	4	no	—
222	C₂²	—	16	6	no	—
223	C₂²	C₂×C₃₁×#102	1	6	yes	—
224	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
225	C₄	—	4	4	no	—
226	C₄	C₂×C₃₁×#102	1	6	yes	—
227	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
228	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	6	no	—
231	C₂	—	8	4	no	—
232	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂²×#102
prime radical	C₃₁×#3584	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#504	C₂	semidirect
3. C₃₁×#2916	C₂	semidirect
4. C₃₁×#2143	C₂	semidirect
5. C₂×C₃₁×#359	C₂	semidirect
15. C₂×C₃₁×M₃₂	C₂²	semidirect
16. C₂²×C₁₆×C₃₁	C₂²	semidirect
18. C₂×C₃₁×#108	C₂²	semidirect
19. C₃₁×#359	C₂²	semidirect
20. C₃₁×#359	C₂²	semidirect
82. #33385	C₃₁	direct (N × H)
105. C₂×#504	C₂×C₃₁	semidirect
106. #2916	C₂×C₃₁	semidirect
107. #2143	C₂×C₃₁	semidirect
108. C₂×#359	C₂×C₃₁	semidirect
127. C₂×M₃₂	C₂²×C₃₁	semidirect
128. C₂²×C₁₆	C₂²×C₃₁	semidirect
130. C₂×#108	C₂²×C₃₁	semidirect
131. #359	C₂²×C₃₁	semidirect
132. #359	C₂²×C₃₁	semidirect
165. C₃₁	#33385	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×#3584	2
C₂×C₃₁×#504	2
C₃₁×#2916	2
C₃₁×#2143	2
C₂×C₃₁×#359	2
#33385	31

Subgroups — C31×#33385 #21490144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33385 #21490144