Subgroups — C₃₁×#17897 #21474656

Order 7936.

614 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2983	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1593	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1591	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1479	C₂	1	2	yes	maximal
7	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#446	—	2	4	no	—
11	C₃₁×#446	C₂²	1	8	yes	—
12	C₄×Q₁₆×C₃₁	C₄	1	2	yes	—
13	C₄×Q₁₆×C₃₁	C₄	1	2	yes	—
14	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#432	—	2	4	no	—
17	C₃₁×#431	C₂²	1	2	yes	—
18	C₃₁×#431	C₂²	1	1	yes	characteristic
19	C₃₁×#431	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
21	C₂×C₃₁×#103	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#335	C₂²	1	2	yes	—
24	C₃₁×#334	C₂²	1	2	yes	—
25	C₃₁×Q₈⋊C₈	C₂²	1	8	yes	—
26	C₃₁×#322	C₂²	1	2	yes	—
27	C₈²×C₃₁	C₂²	1	2	yes	—
28	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×Q₁₆×C₃₁	—	2	4	no	—
30	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₃₁	—	2	2	no	—
35	C₄×Q₈×C₃₁	—	2	2	no	—
36	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	—	2	2	no	—
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#107	C₂×C₄	1	2	yes	—
43	C₃₁×#107	C₂×C₄	1	2	yes	—
44	C₃₁×C₄⋊C₈	—	2	8	no	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
47	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
48	C₃₁×#103	C₂×C₄	1	4	yes	—
49	C₃₁×#103	C₂×C₄	1	4	yes	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₃₁×C₂²⋊C₈	—	2	1	no	—
52	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
53	C₃₁×#97	—	2	8	no	—
54	C₄×C₈×C₃₁	—	2	2	no	—
55	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₃₁	C₂³	1	2	yes	—
58	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×Q₈×C₃₁	—	2	2	no	—
62	C₂×Q₈×C₃₁	—	2	4	no	—
63	C₂×Q₈×C₃₁	—	2	4	no	—
64	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
70	Q₁₆×C₃₁	—	2	4	no	—
71	Q₁₆×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
76	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	2	4	no	—
87	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₁	#41	1	1	yes	characteristic
90	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
91	#17897	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	4	no	—
94	Q₈×C₃₁	—	2	4	no	—
95	Q₈×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	4	1	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	8	2	no	—
102	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₃₁	#131	1	2	yes	—
107	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	2	1	no	—
112	C₈×C₃₁	—	2	4	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₈×C₃₁	—	4	4	no	—
115	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
116	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
117	#2983	C₂×C₃₁	1	2	yes	—
118	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
119	#1593	C₂×C₃₁	1	4	yes	complemented
120	#1591	C₂×C₃₁	1	4	yes	complemented
121	#1564	C₂×C₃₁	1	1	yes	characteristic
122	#1479	C₂×C₃₁	1	2	yes	—
123	#1450	C₂×C₃₁	1	1	yes	characteristic
124	C₂²×C₃₁	—	2	1	no	—
125	C₂²×C₃₁	—	2	1	no	—
126	C₂²×C₃₁	#436	1	2	yes	—
127	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	4	4	no	—
132	C₄×C₃₁	—	4	4	no	—
133	C₄×C₃₁	#436	1	2	yes	—
134	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
135	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
136	C₄×C₃₁	#578	1	2	yes	—
137	C₂²×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
139	#446	—	2	4	no	—
140	#446	C₂²×C₃₁	1	8	yes	—
141	C₄×Q₁₆	C₄×C₃₁	1	2	yes	—
142	C₄×Q₁₆	C₄×C₃₁	1	2	yes	—
143	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
144	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
145	#432	—	2	4	no	—
146	#431	C₂²×C₃₁	1	2	yes	—
147	#431	C₂²×C₃₁	1	1	yes	characteristic
148	#431	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
150	C₂×#103	C₄×C₃₁	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
152	#335	C₂²×C₃₁	1	2	yes	—
153	#334	C₂²×C₃₁	1	2	yes	—
154	Q₈⋊C₈	C₂²×C₃₁	1	8	yes	—
155	#322	C₂²×C₃₁	1	2	yes	—
156	C₈²	C₂²×C₃₁	1	2	yes	—
157	C₂×C₃₁	—	2	2	no	—
158	C₂×C₃₁	C₂×#436	1	1	yes	characteristic
159	C₂×C₃₁	#2950	1	1	yes	characteristic
160	C₂×C₃₁	#2957	1	1	yes	characteristic
161	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
164	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
170	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
175	#107	C₂×C₄×C₃₁	1	2	yes	—
176	#107	C₂×C₄×C₃₁	1	2	yes	—
177	C₄⋊C₈	—	2	8	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
180	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
181	#103	C₂×C₄×C₃₁	1	4	yes	—
182	#103	C₂×C₄×C₃₁	1	4	yes	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₂²⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
186	#97	—	2	8	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
189	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₃₁	1	2	yes	—
191	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₃₁	1	2	yes	—
194	C₃₁	#17897	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
210	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
222	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₄²	#41×C₃₁	1	1	yes	characteristic
224	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
225	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	2	4	no	—
228	Q₈	—	4	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
239	C₂×C₄	C₃₁×#131	1	2	yes	—
240	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	4	no	—
248	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
249	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₃₁×#436	1	2	yes	—
253	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	C₃₁×#436	1	2	yes	—
260	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
261	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
262	C₄	C₃₁×#578	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₃₁×#436	1	1	yes	characteristic
265	C₂	C₃₁×#2950	1	1	yes	characteristic
266	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1593	C₂	semidirect
4. C₃₁×#1591	C₂	semidirect
91. #17897	C₃₁	direct (N × H)
119. #1593	C₂×C₃₁	semidirect
120. #1591	C₂×C₃₁	semidirect
194. C₃₁	#17897	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×#2983	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#1593	2
C₃₁×#1591	2
C₃₁×#1564	2
C₃₁×#1479	2
C₃₁×#1450	2
#17897	31

Subgroups — C31×#17897 #21474656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17897 #21474656