Subgroups — C₃₁×#17958 #21474717

Order 7936.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	2	yes	maximal
3	C₃₁×#1591	C₂	1	4	yes	maximal, complemented
4	C₈×Q₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1493	C₂	1	2	yes	maximal
7	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂²	1	2	yes	—
9	C₃₁×#500	C₄	1	1	yes	characteristic
10	C₃₁×#494	C₄	1	1	yes	characteristic
11	C₃₁×#487	C₄	1	2	yes	—
12	C₃₁×#467	C₄	1	4	yes	—
13	C₃₁×#446	—	2	2	no	—
14	C₃₁×#446	C₂²	1	4	yes	—
15	C₈×Q₈×C₃₁	—	2	2	no	—
16	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
18	C₃₁×#424	—	2	2	no	—
19	C₃₁×#423	C₂²	1	2	yes	—
20	C₃₁×#405	—	2	2	no	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#335	C₂²	1	4	yes	—
25	C₃₁×Q₈⋊C₈	C₂²	1	8	yes	—
26	C₈²×C₃₁	C₂²	1	2	yes	—
27	C₈²×C₃₁	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₃₁×#125	—	2	2	no	—
37	C₃₁×C₂²⋊Q₈	—	2	2	no	—
38	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₃₁	—	2	2	no	—
40	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
41	C₃₁×#117	—	2	2	no	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#107	C₂×C₄	1	4	yes	—
44	C₃₁×C₄⋊C₈	—	2	4	no	—
45	C₃₁×C₄⋊C₈	—	2	4	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
48	C₃₁×#103	C₂×C₄	1	8	yes	—
49	C₃₁×#97	—	2	4	no	—
50	C₄×C₈×C₃₁	—	2	4	no	—
51	C₄×C₈×C₃₁	—	2	1	no	—
52	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	C₂³	1	2	yes	—
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	C₂×Q₈×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
65	Q₁₆×C₃₁	—	2	8	no	—
66	M₁₆×C₃₁	—	4	4	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	Q₁₆	1	4	yes	complemented
73	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×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	#17958	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	8	no	—
94	Q₈×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	2	2	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	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₃₁	#135	1	2	yes	—
104	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
107	C₈×C₃₁	—	2	4	no	—
108	C₈×C₃₁	—	2	2	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	4	4	no	—
112	C₈×C₃₁	#131	1	4	yes	—
113	C₈×C₃₁	C₂×Q₁₆	1	2	yes	—
114	C₈×C₃₁	C₂×Q₁₆	1	2	yes	—
115	#3170	C₂×C₃₁	1	1	yes	characteristic
116	#2960	C₂×C₃₁	1	2	yes	—
117	#1591	C₂×C₃₁	1	4	yes	complemented
118	C₈×Q₁₆	C₂×C₃₁	1	4	yes	complemented
119	#1567	C₂×C₃₁	1	1	yes	characteristic
120	#1493	C₂×C₃₁	1	2	yes	—
121	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	2	1	no	—
124	C₂²×C₃₁	C₄×Q₁₆	1	2	yes	—
125	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	4	4	no	—
129	C₄×C₃₁	—	4	4	no	—
130	C₄×C₃₁	C₄×Q₁₆	1	2	yes	—
131	C₄×C₃₁	#443	1	2	yes	—
132	C₄×C₃₁	C₂×#131	1	2	yes	—
133	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
134	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
135	#533	C₂²×C₃₁	1	2	yes	—
136	#500	C₄×C₃₁	1	1	yes	characteristic
137	#494	C₄×C₃₁	1	1	yes	characteristic
138	#487	C₄×C₃₁	1	2	yes	—
139	#467	C₄×C₃₁	1	4	yes	—
140	#446	—	2	2	no	—
141	#446	C₂²×C₃₁	1	4	yes	—
142	C₈×Q₈	—	2	2	no	—
143	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
144	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
145	#424	—	2	2	no	—
146	#423	C₂²×C₃₁	1	2	yes	—
147	#405	—	2	2	no	—
148	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	#335	C₂²×C₃₁	1	4	yes	—
152	Q₈⋊C₈	C₂²×C₃₁	1	8	yes	—
153	C₈²	C₂²×C₃₁	1	2	yes	—
154	C₈²	C₂²×C₃₁	1	2	yes	—
155	C₂×C₃₁	—	2	2	no	—
156	C₂×C₃₁	#2931	1	1	yes	characteristic
157	C₂×C₃₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
158	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
159	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
163	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
165	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
166	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
167	#125	—	2	2	no	—
168	C₂²⋊Q₈	—	2	2	no	—
169	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
170	C₄×Q₈	—	2	2	no	—
171	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
172	#117	—	2	2	no	—
173	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
174	#107	C₂×C₄×C₃₁	1	4	yes	—
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
179	#103	C₂×C₄×C₃₁	1	8	yes	—
180	#97	—	2	4	no	—
181	C₄×C₈	—	2	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
186	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₃₁	1	2	yes	—
188	C₄×C₈	C₂³×C₃₁	1	2	yes	—
189	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
191	C₃₁	#17958	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
194	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
197	Q₁₆	—	2	8	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	Q₁₆×C₃₁	1	4	yes	complemented
205	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₄²	#41×C₃₁	1	1	yes	characteristic
222	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
223	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	Q₈	—	4	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
234	C₂×C₄	C₃₁×#135	1	2	yes	—
235	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	4	no	—
243	C₈	C₃₁×#131	1	4	yes	—
244	C₈	C₂×Q₁₆×C₃₁	1	2	yes	—
245	C₈	C₂×Q₁₆×C₃₁	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Q₁₆×C₃₁	1	2	yes	—
249	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	C₄×Q₁₆×C₃₁	1	2	yes	—
255	C₄	C₃₁×#443	1	2	yes	—
256	C₄	C₂×C₃₁×#131	1	2	yes	—
257	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
258	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	C₃₁×#2931	1	1	yes	characteristic
261	C₂	C₂×C₄×Q₁₆×C₃₁	1	1	yes	characteristic
262	C₂	C₂×C₃₁×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×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×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1591	C₂	semidirect
4. C₈×Q₁₆×C₃₁	C₂	semidirect
72. C₂×C₈×C₃₁	Q₁₆	semidirect
91. #17958	C₃₁	direct (N × H)
117. #1591	C₂×C₃₁	semidirect
118. C₈×Q₁₆	C₂×C₃₁	semidirect
191. C₃₁	#17958	direct (N × H)
204. C₂×C₈	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3170	2
C₃₁×#2960	2
C₃₁×#1591	2
C₈×Q₁₆×C₃₁	2
C₃₁×#1567	2
C₃₁×#1493	2
C₂×C₈²×C₃₁	2
#17958	31

Subgroups — C31×#17958 #21474717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17958 #21474717