Subgroups — C₃₁×#29290 #21486049

Order 7936.

756 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃₁×#2087	C₂	1	4	yes	maximal
3	C₃₁×#1987	C₂	1	4	yes	maximal
4	C₃₁×#1947	C₂	1	4	yes	maximal
5	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
8	C₂×C₃₁×#107	C₂²	1	4	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃₁×#391	—	2	2	no	—
12	C₃₁×#389	—	2	2	no	—
13	C₃₁×#389	C₂²	1	4	yes	—
14	C₃₁×#384	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#340	C₂²	1	8	yes	—
19	C₃₁×#340	C₂²	1	4	yes	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
22	C₃₁×C₄⋊Q₈	—	2	4	no	—
23	C₃₁×C₄⋊Q₈	Q₈	1	8	yes	complemented
24	C₄×Q₈×C₃₁	—	4	8	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
31	C₂×C₄²×C₃₁	—	2	4	no	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
37	C₃₁×#107	—	4	4	no	—
38	C₃₁×C₄⋊C₈	—	2	8	no	—
39	C₃₁×C₄⋊C₈	—	4	2	no	—
40	C₃₁×#95	—	2	4	no	—
41	C₃₁×#95	—	2	4	no	—
42	C₂×Q₈×C₃₁	—	2	8	no	—
43	C₂×Q₈×C₃₁	—	4	4	no	—
44	C₂×Q₈×C₃₁	—	4	4	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₃₁	—	2	8	no	—
53	C₂×C₈×C₃₁	—	2	4	no	—
54	C₄⋊C₄×C₃₁	—	2	8	no	—
55	C₄⋊C₄×C₃₁	—	2	4	no	—
56	C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₄⋊C₄×C₃₁	—	4	4	no	—
59	C₄⋊C₄×C₃₁	—	4	8	no	—
60	C₄⋊C₄×C₃₁	—	4	4	no	—
61	C₄⋊C₄×C₃₁	C₂×Q₈	1	8	yes	—
62	C₄²×C₃₁	—	2	4	no	—
63	C₄²×C₃₁	—	2	1	no	—
64	C₄²×C₃₁	—	2	1	no	—
65	C₄²×C₃₁	—	4	1	no	—
66	C₄²×C₃₁	—	4	8	no	—
67	C₄²×C₃₁	Q₁₆	1	4	yes	—
68	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
69	C₄²×C₃₁	#41	1	4	yes	—
70	#29290	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
72	Q₈×C₃₁	—	4	16	no	—
73	C₂×C₄×C₃₁	—	2	1	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	1	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	8	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
86	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
87	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
89	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
90	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
92	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
94	C₈×C₃₁	—	4	4	no	—
95	C₄×C₄⋊Q₈	C₂×C₃₁	1	2	yes	—
96	#2087	C₂×C₃₁	1	4	yes	—
97	#1987	C₂×C₃₁	1	4	yes	—
98	#1947	C₂×C₃₁	1	4	yes	—
99	#1842	C₂×C₃₁	1	1	yes	characteristic
100	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
101	C₂²×C₃₁	#537	1	1	yes	characteristic
102	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
103	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
104	C₂²×C₃₁	#575	1	1	yes	characteristic
105	C₄×C₃₁	—	2	4	no	—
106	C₄×C₃₁	—	2	2	no	—
107	C₄×C₃₁	—	4	8	no	—
108	C₄×C₃₁	—	4	1	no	—
109	C₄×C₃₁	—	8	4	no	—
110	C₄×C₃₁	#477	1	8	yes	—
111	C₄×C₃₁	#557	1	2	yes	—
112	C₄×C₃₁	Q₈²	1	1	yes	characteristic
113	C₄×C₃₁	#559	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
115	C₂×C₄×Q₈	—	2	2	no	—
116	C₂×#107	C₂²×C₃₁	1	4	yes	—
117	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
118	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
119	#391	—	2	2	no	—
120	#389	—	2	2	no	—
121	#389	C₂²×C₃₁	1	4	yes	—
122	#384	C₂²×C₃₁	1	2	yes	—
123	C₄×C₄⋊C₄	—	2	2	no	—
124	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
125	C₄³	C₂²×C₃₁	1	1	yes	characteristic
126	#340	C₂²×C₃₁	1	8	yes	—
127	#340	C₂²×C₃₁	1	4	yes	—
128	C₂×C₃₁	#2582	1	1	yes	characteristic
129	C₂×C₃₁	C₂×#477	1	2	yes	—
130	C₂×C₃₁	#3077	1	2	yes	—
131	C₂×C₃₁	#3249	1	1	yes	characteristic
132	C₂×C₃₁	#3253	1	1	yes	characteristic
133	C₂²×Q₈	—	2	2	no	—
134	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
135	C₄⋊Q₈	—	2	4	no	—
136	C₄⋊Q₈	Q₈×C₃₁	1	8	yes	complemented
137	C₄×Q₈	—	4	8	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
143	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
144	C₂×C₄²	—	2	4	no	—
145	C₂×C₄²	D₄×C₃₁	1	2	yes	—
146	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
147	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
149	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
150	#107	—	4	4	no	—
151	C₄⋊C₈	—	2	8	no	—
152	C₄⋊C₈	—	4	2	no	—
153	#95	—	2	4	no	—
154	#95	—	2	4	no	—
155	C₃₁	#29290	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×Q₈	—	2	8	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
165	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	C₂×Q₈×C₃₁	1	8	yes	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	—	4	8	no	—
181	C₄²	Q₁₆×C₃₁	1	4	yes	—
182	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
183	C₄²	#41×C₃₁	1	4	yes	—
184	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
185	Q₈	—	4	16	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
199	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
200	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
201	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
202	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
203	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
204	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
205	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
207	C₈	—	4	4	no	—
208	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
209	C₂²	C₃₁×#537	1	1	yes	characteristic
210	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
211	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
212	C₂²	C₃₁×#575	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	8	no	—
216	C₄	—	4	1	no	—
217	C₄	—	8	4	no	—
218	C₄	C₃₁×#477	1	8	yes	—
219	C₄	C₃₁×#557	1	2	yes	—
220	C₄	Q₈²×C₃₁	1	1	yes	characteristic
221	C₄	C₃₁×#559	1	1	yes	characteristic
222	C₂	C₃₁×#2582	1	1	yes	characteristic
223	C₂	C₂×C₃₁×#477	1	2	yes	—
224	C₂	C₃₁×#3077	1	2	yes	—
225	C₂	C₃₁×#3249	1	1	yes	characteristic
226	C₂	C₃₁×#3253	1	1	yes	characteristic

	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
23. C₃₁×C₄⋊Q₈	Q₈	semidirect
70. #29290	C₃₁	direct (N × H)
136. C₄⋊Q₈	Q₈×C₃₁	semidirect
155. C₃₁	#29290	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2087	2
C₃₁×#1987	2
C₃₁×#1947	2
C₃₁×#1842	2
#29290	31

Subgroups — C31×#29290 #21486049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29290 #21486049