Subgroups — C₃₁×#9739 #21466498

Order 7936.

442 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1987	C₂	1	2	yes	maximal
3	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#333	C₂	1	2	yes	maximal
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₃₁×#500	—	2	2	no	—
8	C₃₁×#500	C₄	1	4	yes	complemented
9	C₂×C₃₁×#107	—	2	1	no	—
10	C₂×C₃₁×#107	—	2	1	no	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#340	—	2	4	no	—
17	C₃₁×#336	—	2	1	no	—
18	C₃₁×#334	—	2	2	no	—
19	C₃₁×#333	—	2	4	no	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	—	2	1	no	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
25	C₃₁×C₄⋊Q₈	—	2	2	no	—
26	C₃₁×C₄⋊Q₈	—	2	2	no	—
27	C₃₁×C₄⋊Q₈	—	2	2	no	—
28	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	—	2	1	no	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×#107	—	2	4	no	—
35	C₃₁×#107	—	2	4	no	—
36	C₃₁×C₄⋊C₈	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
40	C₃₁×#97	—	4	2	no	—
41	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₃₁	D₄	1	2	yes	—
43	C₃₁×#95	—	2	1	no	—
44	C₂×Q₈×C₃₁	—	2	4	no	—
45	C₂×Q₈×C₃₁	—	4	2	no	—
46	C₂×Q₈×C₃₁	—	8	4	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	4	2	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	#41	1	1	yes	characteristic
52	C₂×C₈×C₃₁	—	2	1	no	—
53	C₂×C₈×C₃₁	—	2	1	no	—
54	C₂×C₈×C₃₁	—	2	2	no	—
55	C₂×C₈×C₃₁	—	4	1	no	—
56	C₂×C₈×C₃₁	—	4	1	no	—
57	C₂×C₈×C₃₁	—	4	2	no	—
58	C₂×C₈×C₃₁	—	4	4	no	—
59	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
62	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₄²×C₃₁	—	4	1	no	—
69	C₄²×C₃₁	—	4	1	no	—
70	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₃₁	#41	1	1	yes	characteristic
74	#9739	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	#121	1	1	yes	characteristic
76	Q₈×C₃₁	—	8	4	no	—
77	Q₈×C₃₁	—	8	4	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	4	4	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	—	8	4	no	—
84	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
89	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
94	C₂×C₄×C₃₁	#137	1	4	yes	—
95	C₈×C₃₁	—	2	2	no	—
96	C₈×C₃₁	—	4	2	no	—
97	C₈×C₃₁	—	8	2	no	—
98	C₈×C₃₁	#103	1	4	yes	—
99	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
100	#1987	C₂×C₃₁	1	2	yes	—
101	#1770	C₂×C₃₁	1	1	yes	characteristic
102	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
103	C₂×#333	C₂×C₃₁	1	2	yes	—
104	C₂²×C₃₁	#390	1	1	yes	characteristic
105	C₂²×C₃₁	#461	1	1	yes	characteristic
106	C₂²×C₃₁	#462	1	1	yes	characteristic
107	C₂²×C₃₁	#470	1	2	yes	—
108	C₂²×C₃₁	#472	1	2	yes	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	—	8	2	no	—
112	C₄×C₃₁	—	8	4	no	—
113	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
114	C₄×C₃₁	#419	1	1	yes	characteristic
115	C₄×C₃₁	#441	1	2	yes	—
116	C₄×C₃₁	#461	1	1	yes	characteristic
117	C₄×C₃₁	#462	1	1	yes	characteristic
118	C₄×C₃₁	#497	1	2	yes	—
119	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
120	#500	—	2	2	no	—
121	#500	C₄×C₃₁	1	4	yes	complemented
122	C₂×#107	—	2	1	no	—
123	C₂×#107	—	2	1	no	—
124	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
125	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₂×#97	—	2	1	no	—
127	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
128	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
129	#340	—	2	4	no	—
130	#336	—	2	1	no	—
131	#334	—	2	2	no	—
132	#333	—	2	4	no	—
133	C₂×C₃₁	#1695	1	2	yes	—
134	C₂×C₃₁	#1696	1	2	yes	—
135	C₂×C₃₁	#1971	1	1	yes	characteristic
136	C₂×C₃₁	#1978	1	1	yes	characteristic
137	C₂×C₃₁	#1981	1	1	yes	characteristic
138	C₂²×Q₈	—	2	2	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
142	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
152	#107	—	2	4	no	—
153	#107	—	2	4	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
158	#97	—	4	2	no	—
159	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
160	C₄×C₈	D₄×C₃₁	1	2	yes	—
161	#95	—	2	1	no	—
162	C₃₁	#9739	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
179	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
180	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
181	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	1	no	—
189	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
192	C₄²	#41×C₃₁	1	1	yes	characteristic
193	C₂³	C₃₁×#121	1	1	yes	characteristic
194	Q₈	—	8	4	no	—
195	Q₈	—	8	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	8	4	no	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
206	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
207	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
208	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
209	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
211	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
212	C₂×C₄	C₃₁×#137	1	4	yes	—
213	C₈	—	2	2	no	—
214	C₈	—	4	2	no	—
215	C₈	—	8	2	no	—
216	C₈	C₃₁×#103	1	4	yes	—
217	C₂²	C₃₁×#390	1	1	yes	characteristic
218	C₂²	C₃₁×#461	1	1	yes	characteristic
219	C₂²	C₃₁×#462	1	1	yes	characteristic
220	C₂²	C₃₁×#470	1	2	yes	—
221	C₂²	C₃₁×#472	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	4	no	—
226	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
227	C₄	C₃₁×#419	1	1	yes	characteristic
228	C₄	C₃₁×#441	1	2	yes	—
229	C₄	C₃₁×#461	1	1	yes	characteristic
230	C₄	C₃₁×#462	1	1	yes	characteristic
231	C₄	C₃₁×#497	1	2	yes	—
232	C₂	C₃₁×#1695	1	2	yes	—
233	C₂	C₃₁×#1696	1	2	yes	—
234	C₂	C₃₁×#1971	1	1	yes	characteristic
235	C₂	C₃₁×#1978	1	1	yes	characteristic
236	C₂	C₃₁×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₃₁×#500	C₄	semidirect
74. #9739	C₃₁	direct (N × H)
121. #500	C₄×C₃₁	semidirect
162. C₃₁	#9739	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₃₁×#1987	2
C₃₁×#1770	2
C₂×C₃₁×#334	2
C₂×C₃₁×#333	2
#9739	31

Subgroups — C31×#9739 #21466498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9739 #21466498