Subgroups — C₃₁×#22281 #21479040

Order 7936.

818 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3150	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2931	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1963	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1876	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1838	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#135	C₂²	1	4	yes	—
8	C₂×C₃₁×#131	C₂²	1	4	yes	—
9	C₃₁×#532	C₂²	1	2	yes	—
10	C₃₁×#497	C₄	1	4	yes	complemented
11	C₃₁×#496	C₄	1	4	yes	complemented
12	C₃₁×#436	—	2	2	no	—
13	C₃₁×#436	—	2	2	no	—
14	C₈×D₄×C₃₁	—	2	2	no	—
15	C₈×D₄×C₃₁	—	2	2	no	—
16	C₃₁×#423	C₂²	1	2	yes	—
17	C₂×C₃₁×#104	C₂²	1	8	yes	—
18	C₃₁×#408	C₂²	1	4	yes	—
19	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#376	C₂²	1	1	yes	characteristic
22	C₃₁×#341	C₂²	1	4	yes	—
23	C₃₁×#336	C₂²	1	4	yes	—
24	C₂×#41×C₃₁	C₂³	1	4	yes	—
25	C₃₁×#135	—	4	8	no	—
26	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
27	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	—
28	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
29	C₃₁×#131	—	4	8	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₃₁×#127	C₂×C₄	1	2	yes	—
37	C₃₁×#124	C₂×C₄	1	4	yes	—
38	C₃₁×#121	—	2	2	no	—
39	C₃₁×#121	—	2	2	no	—
40	C₄×D₄×C₃₁	—	2	2	no	—
41	C₄×D₄×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₃₁×#104	—	4	8	no	—
47	C₃₁×C₂²⋊C₈	—	2	4	no	—
48	C₃₁×C₂²⋊C₈	—	2	4	no	—
49	C₃₁×#97	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	D₄	1	2	yes	—
52	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₃₁×#95	—	2	2	no	—
54	#41×C₃₁	—	2	8	no	—
55	#41×C₃₁	—	2	8	no	—
56	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
57	C₂×D₄×C₃₁	—	2	4	no	—
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	#41	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
65	Q₁₆×C₃₁	—	4	4	no	—
66	SD₁₆×C₃₁	—	4	8	no	—
67	D₈×C₃₁	—	4	4	no	—
68	M₁₆×C₃₁	—	2	8	no	—
69	M₁₆×C₃₁	—	4	8	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	4	8	no	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₂²⋊C₄×C₃₁	—	2	4	no	—
80	C₂²⋊C₄×C₃₁	—	2	4	no	—
81	C₄²×C₃₁	—	2	4	no	—
82	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
85	#22281	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	—	2	4	no	—
87	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
88	Q₈×C₃₁	—	2	8	no	—
89	D₄×C₃₁	—	2	8	no	—
90	D₄×C₃₁	—	4	4	no	—
91	D₄×C₃₁	—	4	4	no	—
92	D₄×C₃₁	—	4	8	no	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
104	C₈×C₃₁	—	2	4	no	—
105	C₈×C₃₁	—	2	2	no	—
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	—	4	4	no	—
108	#3150	C₂×C₃₁	1	1	yes	characteristic
109	#2931	C₂×C₃₁	1	2	yes	complemented
110	#1963	C₂×C₃₁	1	2	yes	complemented
111	#1876	C₂×C₃₁	1	8	yes	complemented
112	#1838	C₂×C₃₁	1	1	yes	characteristic, complemented
113	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
114	C₂²×C₃₁	—	2	2	no	—
115	C₂²×C₃₁	—	4	4	no	—
116	C₂²×C₃₁	—	4	4	no	—
117	C₂²×C₃₁	—	4	4	no	—
118	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₃₁	#540	1	2	yes	—
120	C₄×C₃₁	—	2	2	no	—
121	C₄×C₃₁	—	2	2	no	—
122	C₄×C₃₁	—	4	4	no	—
123	C₄×C₃₁	—	8	2	no	—
124	C₄×C₃₁	#518	1	2	yes	—
125	C₄×C₃₁	#535	1	2	yes	—
126	C₂×#135	C₂²×C₃₁	1	4	yes	—
127	C₂×#131	C₂²×C₃₁	1	4	yes	—
128	#532	C₂²×C₃₁	1	2	yes	—
129	#497	C₄×C₃₁	1	4	yes	complemented
130	#496	C₄×C₃₁	1	4	yes	complemented
131	#436	—	2	2	no	—
132	#436	—	2	2	no	—
133	C₈×D₄	—	2	2	no	—
134	C₈×D₄	—	2	2	no	—
135	#423	C₂²×C₃₁	1	2	yes	—
136	C₂×#104	C₂²×C₃₁	1	8	yes	—
137	#408	C₂²×C₃₁	1	4	yes	—
138	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
140	#376	C₂²×C₃₁	1	1	yes	characteristic
141	#341	C₂²×C₃₁	1	4	yes	—
142	#336	C₂²×C₃₁	1	4	yes	—
143	C₂×C₃₁	—	2	2	no	—
144	C₂×C₃₁	—	8	4	no	—
145	C₂×C₃₁	#2328	1	1	yes	characteristic
146	C₂×C₃₁	#2955	1	2	yes	—
147	C₂×#41	C₂³×C₃₁	1	4	yes	—
148	#135	—	4	8	no	—
149	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
150	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	—
151	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
152	#131	—	4	8	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
157	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
158	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
159	#127	C₂×C₄×C₃₁	1	2	yes	—
160	#124	C₂×C₄×C₃₁	1	4	yes	—
161	#121	—	2	2	no	—
162	#121	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
166	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	#104	—	4	8	no	—
170	C₂²⋊C₈	—	2	4	no	—
171	C₂²⋊C₈	—	2	4	no	—
172	#97	C₂×C₄×C₃₁	1	4	yes	—
173	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
174	C₄×C₈	D₄×C₃₁	1	2	yes	—
175	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
176	#95	—	2	2	no	—
177	C₃₁	#22281	1	1	yes	characteristic, direct product factor, Sylow
178	#41	—	2	8	no	—
179	#41	—	2	8	no	—
180	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	#41×C₃₁	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
189	Q₁₆	—	4	4	no	—
190	SD₁₆	—	4	8	no	—
191	D₈	—	4	4	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	—	4	8	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	8	no	—
198	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
199	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
207	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂³	—	2	4	no	—
210	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
211	Q₈	—	2	8	no	—
212	D₄	—	2	8	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
223	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
224	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
227	C₈	—	2	4	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	4	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	4	no	—
234	C₂²	—	4	4	no	—
235	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
236	C₂²	C₃₁×#540	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	4	no	—
240	C₄	—	8	2	no	—
241	C₄	C₃₁×#518	1	2	yes	—
242	C₄	C₃₁×#535	1	2	yes	—
243	C₂	—	2	2	no	—
244	C₂	—	8	4	no	—
245	C₂	C₃₁×#2328	1	1	yes	characteristic
246	C₂	C₃₁×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2931	C₂	semidirect
3. C₃₁×#1963	C₂	semidirect
4. C₃₁×#1876	C₂	semidirect
5. C₃₁×#1838	C₂	semidirect
6. C₃₁×#1746	C₂	semidirect
10. C₃₁×#497	C₄	semidirect
11. C₃₁×#496	C₄	semidirect
85. #22281	C₃₁	direct (N × H)
109. #2931	C₂×C₃₁	semidirect
110. #1963	C₂×C₃₁	semidirect
111. #1876	C₂×C₃₁	semidirect
112. #1838	C₂×C₃₁	semidirect
113. #1746	C₂×C₃₁	semidirect
129. #497	C₄×C₃₁	semidirect
130. #496	C₄×C₃₁	semidirect
177. C₃₁	#22281	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×#3150	2
C₃₁×#2931	2
C₃₁×#1963	2
C₃₁×#1876	2
C₃₁×#1838	2
C₃₁×#1746	2
#22281	31

Subgroups — C31×#22281 #21479040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22281 #21479040