Subgroups — C₃₁×#33297 #21490056

Order 7936.

750 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	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1993	C₂	1	4	yes	maximal
4	C₃₁×#1992	C₂	1	4	yes	maximal
5	C₃₁×#1834	C₂	1	4	yes	maximal
6	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃₁×#125	C₂²	1	2	yes	—
10	C₂×C₃₁×#97	C₂²	1	2	yes	—
11	C₂×C₃₁×#97	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#389	—	2	2	no	—
14	C₃₁×#389	—	2	2	no	—
15	C₃₁×#384	—	2	2	no	—
16	C₃₁×#384	C₂²	1	2	yes	—
17	C₃₁×#384	C₂²	1	2	yes	—
18	C₃₁×#384	C₂²	1	2	yes	—
19	C₃₁×#382	C₂²	1	2	yes	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#340	C₂²	1	16	yes	—
22	C₂²×Q₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
24	C₃₁×C₄⋊Q₈	—	2	8	no	—
25	C₃₁×C₄⋊Q₈	—	4	8	no	—
26	C₃₁×#125	—	2	8	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×#97	—	2	4	no	—
41	C₃₁×#97	—	2	4	no	—
42	C₄×C₈×C₃₁	Q₈	1	8	yes	complemented
43	C₃₁×#95	—	2	2	no	—
44	C₃₁×#95	—	2	2	no	—
45	C₂×Q₈×C₃₁	—	4	8	no	—
46	C₂×Q₈×C₃₁	—	4	4	no	—
47	C₂²×C₄×C₃₁	—	2	4	no	—
48	C₂²×C₄×C₃₁	—	2	4	no	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×C₈×C₃₁	—	2	4	no	—
55	C₂×C₈×C₃₁	—	2	4	no	—
56	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
57	C₄⋊C₄×C₃₁	—	2	8	no	—
58	C₄⋊C₄×C₃₁	—	2	8	no	—
59	C₄⋊C₄×C₃₁	—	4	8	no	—
60	C₄⋊C₄×C₃₁	—	4	8	no	—
61	C₄⋊C₄×C₃₁	—	4	4	no	—
62	C₄⋊C₄×C₃₁	—	4	4	no	—
63	C₄⋊C₄×C₃₁	—	4	8	no	—
64	C₄⋊C₄×C₃₁	—	4	4	no	—
65	C₄²×C₃₁	—	2	4	no	—
66	C₄²×C₃₁	—	2	4	no	—
67	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
68	C₄²×C₃₁	#41	1	4	yes	—
69	C₄²×C₃₁	#41	1	4	yes	—
70	#33297	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
72	Q₈×C₃₁	—	8	8	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	4	4	no	—
78	C₂×C₄×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	—	4	8	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	#124	1	2	yes	—
83	C₂×C₄×C₃₁	#124	1	4	yes	—
84	C₂×C₄×C₃₁	#124	1	2	yes	—
85	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
86	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
87	C₂×C₄×C₃₁	#137	1	4	yes	—
88	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
91	C₈×C₃₁	—	2	8	no	—
92	#2868	C₂×C₃₁	1	1	yes	characteristic
93	#2859	C₂×C₃₁	1	1	yes	characteristic
94	#1993	C₂×C₃₁	1	4	yes	—
95	#1992	C₂×C₃₁	1	4	yes	—
96	#1834	C₂×C₃₁	1	4	yes	—
97	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
98	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
99	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
100	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
102	C₂²×C₃₁	C₂×#137	1	2	yes	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	—	2	2	no	—
105	C₄×C₃₁	—	2	4	no	—
106	C₄×C₃₁	—	8	4	no	—
107	C₄×C₃₁	—	8	4	no	—
108	C₄×C₃₁	#501	1	8	yes	—
109	C₄×C₃₁	#533	1	4	yes	—
110	C₂×C₄⋊Q₈	—	2	2	no	—
111	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
112	C₂×#125	C₂²×C₃₁	1	2	yes	—
113	C₂×#97	C₂²×C₃₁	1	2	yes	—
114	C₂×#97	C₂²×C₃₁	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
116	#389	—	2	2	no	—
117	#389	—	2	2	no	—
118	#384	—	2	2	no	—
119	#384	C₂²×C₃₁	1	2	yes	—
120	#384	C₂²×C₃₁	1	2	yes	—
121	#384	C₂²×C₃₁	1	2	yes	—
122	#382	C₂²×C₃₁	1	2	yes	—
123	C₄³	C₂²×C₃₁	1	1	yes	characteristic
124	#340	C₂²×C₃₁	1	16	yes	—
125	C₂×C₃₁	#2862	1	1	yes	characteristic
126	C₂×C₃₁	#3138	1	1	yes	characteristic
127	C₂×C₃₁	#3139	1	1	yes	characteristic
128	C₂×C₃₁	#3141	1	2	yes	—
129	C₂×C₃₁	C₂×#501	1	2	yes	—
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
132	C₄⋊Q₈	—	2	8	no	—
133	C₄⋊Q₈	—	4	8	no	—
134	#125	—	2	8	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
142	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
143	C₂×C₄²	D₄×C₃₁	1	2	yes	—
144	C₂×C₄²	D₄×C₃₁	1	2	yes	—
145	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
146	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
148	#97	—	2	4	no	—
149	#97	—	2	4	no	—
150	C₄×C₈	Q₈×C₃₁	1	8	yes	complemented
151	#95	—	2	2	no	—
152	#95	—	2	2	no	—
153	C₃₁	#33297	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×Q₈	—	4	8	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
162	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
166	C₄⋊C₄	—	2	8	no	—
167	C₄⋊C₄	—	2	8	no	—
168	C₄⋊C₄	—	4	8	no	—
169	C₄⋊C₄	—	4	8	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	8	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
177	C₄²	#41×C₃₁	1	4	yes	—
178	C₄²	#41×C₃₁	1	4	yes	—
179	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	8	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	C₃₁×#124	1	2	yes	—
191	C₂×C₄	C₃₁×#124	1	4	yes	—
192	C₂×C₄	C₃₁×#124	1	2	yes	—
193	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
194	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
195	C₂×C₄	C₃₁×#137	1	4	yes	—
196	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
197	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
198	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
199	C₈	—	2	8	no	—
200	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
201	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
202	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
203	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
204	C₂²	C₂×C₃₁×#137	1	2	yes	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	4	no	—
208	C₄	—	8	4	no	—
209	C₄	—	8	4	no	—
210	C₄	C₃₁×#501	1	8	yes	—
211	C₄	C₃₁×#533	1	4	yes	—
212	C₂	C₃₁×#2862	1	1	yes	characteristic
213	C₂	C₃₁×#3138	1	1	yes	characteristic
214	C₂	C₃₁×#3139	1	1	yes	characteristic
215	C₂	C₃₁×#3141	1	2	yes	—
216	C₂	C₂×C₃₁×#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
42. C₄×C₈×C₃₁	Q₈	semidirect
70. #33297	C₃₁	direct (N × H)
150. C₄×C₈	Q₈×C₃₁	semidirect
153. C₃₁	#33297	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×#2868	2
C₃₁×#2859	2
C₃₁×#1993	2
C₃₁×#1992	2
C₃₁×#1834	2
C₄×C₃₁×#97	2
#33297	31

Subgroups — C31×#33297 #21490056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33297 #21490056