Subgroups — C₃₁×#13151 #21469910

Order 7936.

332 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#359	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2138	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
13	C₃₁×#427	C₂²	1	1	yes	characteristic
14	C₃₁×#424	—	2	1	no	—
15	C₂×C₃₁×#104	—	2	1	no	—
16	C₃₁×#408	—	2	1	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#359	—	2	2	no	—
19	C₃₁×#358	C₄	1	2	yes	complemented
20	C₃₁×#357	C₄	1	2	yes	complemented
21	C₄×C₁₆×C₃₁	—	2	2	no	—
22	C₃₁×#339	—	2	1	no	—
23	C₂×#41×C₃₁	—	2	1	no	—
24	C₃₁×#135	—	2	1	no	—
25	C₃₁×#135	—	2	1	no	—
26	C₃₁×#135	—	2	1	no	—
27	C₃₁×#135	—	2	1	no	—
28	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×SD₁₆×C₃₁	—	2	1	no	—
30	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#117	—	2	1	no	—
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₃₁	—	2	1	no	—
38	C₂×C₁₆×C₃₁	—	2	1	no	—
39	C₂×C₁₆×C₃₁	—	2	1	no	—
40	C₂×C₁₆×C₃₁	—	2	1	no	—
41	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
43	C₃₁×#108	—	2	1	no	—
44	C₃₁×#108	—	2	1	no	—
45	C₃₁×#108	—	2	1	no	—
46	C₃₁×#108	—	2	1	no	—
47	C₃₁×#107	C₂×C₄	1	2	yes	—
48	C₃₁×#106	—	2	1	no	—
49	C₃₁×C₄⋊C₈	—	4	2	no	—
50	C₃₁×#104	—	4	2	no	—
51	C₃₁×C₂²⋊C₈	—	4	2	no	—
52	C₄×C₈×C₃₁	—	2	1	no	—
53	C₄×C₈×C₃₁	—	2	1	no	—
54	#41×C₃₁	—	4	1	no	—
55	#41×C₃₁	—	4	1	no	—
56	#41×C₃₁	—	4	1	no	—
57	#41×C₃₁	—	4	1	no	—
58	C₂×Q₈×C₃₁	—	2	1	no	—
59	C₂×D₄×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	Q₁₆×C₃₁	—	2	1	no	—
65	Q₁₆×C₃₁	—	2	1	no	—
66	SD₁₆×C₃₁	—	4	1	no	—
67	SD₁₆×C₃₁	—	4	1	no	—
68	D₈×C₃₁	—	2	1	no	—
69	D₈×C₃₁	—	2	1	no	—
70	M₁₆×C₃₁	—	4	1	no	—
71	M₁₆×C₃₁	—	4	1	no	—
72	M₁₆×C₃₁	—	4	1	no	—
73	M₁₆×C₃₁	—	4	1	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	2	1	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	D₈	1	2	yes	—
79	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	1	no	—
87	C₄²×C₃₁	—	2	1	no	—
88	C₄²×C₃₁	—	4	1	no	—
89	C₄²×C₃₁	SD₁₆	1	2	yes	—
90	C₁₆×C₃₁	—	2	2	no	—
91	C₁₆×C₃₁	—	2	2	no	—
92	#13151	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	4	1	no	—
94	C₂³×C₃₁	#135	1	1	yes	characteristic
95	Q₈×C₃₁	—	4	1	no	—
96	Q₈×C₃₁	—	4	1	no	—
97	D₄×C₃₁	—	4	1	no	—
98	D₄×C₃₁	—	4	1	no	—
99	D₄×C₃₁	—	8	1	no	—
100	D₄×C₃₁	—	8	1	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	8	1	no	—
106	C₂×C₄×C₃₁	—	8	1	no	—
107	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
113	C₈×C₃₁	—	2	2	no	—
114	C₈×C₃₁	—	8	1	no	—
115	C₈×C₃₁	—	8	1	no	—
116	C₈×C₃₁	#117	1	1	yes	characteristic
117	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
118	C₈×C₃₁	#123	1	1	yes	characteristic
119	C₈×C₃₁	#126	1	1	yes	characteristic
120	C₂×#359	C₂×C₃₁	1	1	yes	characteristic
121	#2138	C₂×C₃₁	1	1	yes	characteristic
122	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
123	#1974	C₂×C₃₁	1	1	yes	characteristic
124	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#1382	C₂×C₃₁	1	1	yes	characteristic, complemented
126	#1381	C₂×C₃₁	1	1	yes	characteristic, complemented
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	8	2	no	—
129	C₂²×C₃₁	—	8	1	no	—
130	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
131	C₂²×C₃₁	#465	1	1	yes	characteristic
132	C₂²×C₃₁	#488	1	1	yes	characteristic
133	C₄×C₃₁	—	2	4	no	—
134	C₄×C₃₁	—	8	2	no	—
135	C₄×C₃₁	—	8	1	no	—
136	C₄×C₃₁	#388	1	1	yes	characteristic
137	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
138	C₄×C₃₁	#466	1	1	yes	characteristic
139	C₄×C₃₁	#486	1	1	yes	characteristic
140	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
141	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
145	#427	C₂²×C₃₁	1	1	yes	characteristic
146	#424	—	2	1	no	—
147	C₂×#104	—	2	1	no	—
148	#408	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	#359	—	2	2	no	—
151	#358	C₄×C₃₁	1	2	yes	complemented
152	#357	C₄×C₃₁	1	2	yes	complemented
153	C₄×C₁₆	—	2	2	no	—
154	#339	—	2	1	no	—
155	C₂×C₃₁	—	2	1	no	—
156	C₂×C₃₁	—	2	1	no	—
157	C₂×C₃₁	—	16	1	no	—
158	C₂×C₃₁	#1934	1	1	yes	characteristic
159	C₂×C₃₁	#2178	1	2	yes	—
160	C₂×#41	—	2	1	no	—
161	#135	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	—	2	1	no	—
165	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂×SD₁₆	—	2	1	no	—
167	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	#117	—	2	1	no	—
173	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
179	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
180	#108	—	2	1	no	—
181	#108	—	2	1	no	—
182	#108	—	2	1	no	—
183	#108	—	2	1	no	—
184	#107	C₂×C₄×C₃₁	1	2	yes	—
185	#106	—	2	1	no	—
186	C₄⋊C₈	—	4	2	no	—
187	#104	—	4	2	no	—
188	C₂²⋊C₈	—	4	2	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₃₁	#13151	1	1	yes	characteristic, direct product factor, Sylow
192	#41	—	4	1	no	—
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	#41	—	4	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
202	Q₁₆	—	2	1	no	—
203	Q₁₆	—	2	1	no	—
204	SD₁₆	—	4	1	no	—
205	SD₁₆	—	4	1	no	—
206	D₈	—	2	1	no	—
207	D₈	—	2	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	D₈×C₃₁	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
220	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
221	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	4	1	no	—
227	C₄²	SD₁₆×C₃₁	1	2	yes	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	C₃₁×#135	1	1	yes	characteristic
232	Q₈	—	4	1	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	8	1	no	—
237	D₄	—	8	1	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
246	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
247	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	1	no	—
253	C₈	C₃₁×#117	1	1	yes	characteristic
254	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
255	C₈	C₃₁×#123	1	1	yes	characteristic
256	C₈	C₃₁×#126	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
261	C₂²	C₃₁×#465	1	1	yes	characteristic
262	C₂²	C₃₁×#488	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	1	no	—
266	C₄	C₃₁×#388	1	1	yes	characteristic
267	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
268	C₄	C₃₁×#466	1	1	yes	characteristic
269	C₄	C₃₁×#486	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	16	1	no	—
273	C₂	C₃₁×#1934	1	1	yes	characteristic
274	C₂	C₃₁×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×D₈
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1945	C₂	semidirect
6. C₃₁×#1382	C₂	semidirect
7. C₃₁×#1381	C₂	semidirect
19. C₃₁×#358	C₄	semidirect
20. C₃₁×#357	C₄	semidirect
92. #13151	C₃₁	direct (N × H)
122. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
124. #1945	C₂×C₃₁	semidirect
125. #1382	C₂×C₃₁	semidirect
126. #1381	C₂×C₃₁	semidirect
151. #358	C₄×C₃₁	semidirect
152. #357	C₄×C₃₁	semidirect
191. C₃₁	#13151	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×#359	2
C₃₁×#2138	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1974	2
C₃₁×#1945	2
C₃₁×#1382	2
C₃₁×#1381	2
#13151	31

Subgroups — C31×#13151 #21469910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13151 #21469910