Subgroups — C₃₁×#3222 #16843155

Order 3968.

396 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#535	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#464	C₂	1	2	yes	maximal, complemented
8	C₃₁×#460	C₂	1	2	yes	maximal, complemented
9	C₃₁×#452	C₂	1	2	yes	maximal, complemented
10	C₃₁×#450	C₂	1	2	yes	maximal, complemented
11	C₃₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
18	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₃₁×#127	C₂²	1	1	yes	characteristic
21	C₃₁×#126	—	2	2	no	—
22	C₃₁×#125	—	2	2	no	—
23	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#124	C₂²	1	1	yes	characteristic
25	C₃₁×#124	C₂²	1	1	yes	characteristic
26	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₃₁×#121	—	2	2	no	—
28	C₃₁×#120	—	2	2	no	—
29	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
30	C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₃₁×#103	C₂²	1	2	yes	complemented
33	C₃₁×#102	C₂²	1	2	yes	—
34	C₃₁×#102	C₂²	1	2	yes	—
35	C₃₁×#102	C₂²	1	2	yes	complemented
36	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₃₁	—	2	1	no	—
40	#41×C₃₁	—	4	2	no	—
41	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₂×D₄×C₃₁	—	2	1	no	—
44	C₂×D₄×C₃₁	—	2	1	no	—
45	C₂×D₄×C₃₁	—	2	2	no	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	4	2	no	—
48	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	Q₁₆×C₃₁	—	4	2	no	—
54	SD₁₆×C₃₁	—	4	2	no	—
55	SD₁₆×C₃₁	—	4	2	no	—
56	SD₁₆×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂×C₈×C₃₁	C₂³	1	2	yes	—
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₃₁	—	2	1	no	—
73	C₂³×C₃₁	—	2	1	no	—
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	—	4	2	no	—
76	C₂³×C₃₁	—	8	2	no	—
77	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₃₁	—	2	2	no	—
79	Q₈×C₃₁	—	2	2	no	—
80	Q₈×C₃₁	—	2	2	no	—
81	D₄×C₃₁	—	2	2	no	—
82	D₄×C₃₁	—	2	2	no	—
83	D₄×C₃₁	—	2	2	no	—
84	D₄×C₃₁	—	4	2	no	—
85	D₄×C₃₁	—	4	2	no	—
86	D₄×C₃₁	—	8	2	no	—
87	D₄×C₃₁	—	8	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	2	2	no	—
99	#3222	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	4	1	no	—
103	C₂²×C₃₁	—	4	1	no	—
104	C₂²×C₃₁	—	4	1	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	8	4	no	—
109	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	4	1	no	—
114	C₄×C₃₁	—	4	2	no	—
115	C₄×C₃₁	—	4	2	no	—
116	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
117	#541	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#535	C₂×C₃₁	1	1	yes	characteristic
119	#497	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#492	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#490	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#488	C₂×C₃₁	1	1	yes	characteristic, complemented
123	#464	C₂×C₃₁	1	2	yes	complemented
124	#460	C₂×C₃₁	1	2	yes	complemented
125	#452	C₂×C₃₁	1	2	yes	complemented
126	#450	C₂×C₃₁	1	2	yes	complemented
127	#433	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×C₃₁	—	4	1	no	—
129	C₂×C₃₁	—	8	1	no	—
130	C₂×C₃₁	—	8	1	no	—
131	C₂×C₃₁	—	8	1	no	—
132	C₂×C₃₁	#535	1	1	yes	characteristic
133	C₂×C₃₁	#577	1	2	yes	—
134	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
135	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
137	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
138	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
139	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
140	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
141	#127	C₂²×C₃₁	1	1	yes	characteristic
142	#127	C₂²×C₃₁	1	1	yes	characteristic
143	#126	—	2	2	no	—
144	#125	—	2	2	no	—
145	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
146	#124	C₂²×C₃₁	1	1	yes	characteristic
147	#124	C₂²×C₃₁	1	1	yes	characteristic
148	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
149	#121	—	2	2	no	—
150	#120	—	2	2	no	—
151	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
152	#117	C₂²×C₃₁	1	1	yes	characteristic
153	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
154	#103	C₂²×C₃₁	1	2	yes	complemented
155	#102	C₂²×C₃₁	1	2	yes	—
156	#102	C₂²×C₃₁	1	2	yes	—
157	#102	C₂²×C₃₁	1	2	yes	complemented
158	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
159	#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
160	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
161	C₃₁	#3222	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	—	2	1	no	—
163	#41	—	4	2	no	—
164	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
172	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
173	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
176	Q₁₆	—	4	2	no	—
177	SD₁₆	—	4	2	no	—
178	SD₁₆	—	4	2	no	—
179	SD₁₆	—	4	2	no	—
180	C₂×C₈	C₂³×C₃₁	1	2	yes	—
181	C₂×C₈	C₂³×C₃₁	1	2	yes	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
187	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	C₂³×C₃₁	1	1	yes	characteristic
194	C₄²	C₂³×C₃₁	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	8	2	no	—
200	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
201	Q₈	—	2	2	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	2	no	—
210	D₄	—	8	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
239	C₂	—	4	1	no	—
240	C₂	—	8	1	no	—
241	C₂	—	8	1	no	—
242	C₂	—	8	1	no	—
243	C₂	C₃₁×#535	1	1	yes	characteristic
244	C₂	C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#541	C₂	semidirect
3. C₃₁×#497	C₂	semidirect
4. C₃₁×#492	C₂	semidirect
5. C₃₁×#490	C₂	semidirect
6. C₃₁×#488	C₂	semidirect
7. C₃₁×#464	C₂	semidirect
8. C₃₁×#460	C₂	semidirect
9. C₃₁×#452	C₂	semidirect
10. C₃₁×#450	C₂	semidirect
11. C₃₁×#433	C₂	semidirect
14. C₂×Q₁₆×C₃₁	C₂²	semidirect
17. C₂×SD₁₆×C₃₁	C₂²	semidirect
18. C₃₁×C₄⋊Q₈	C₂²	semidirect
23. C₃₁×#125	C₂²	semidirect
29. C₄×Q₈×C₃₁	C₂²	semidirect
31. C₃₁×C₄⋊C₈	C₂²	semidirect
32. C₃₁×#103	C₂²	semidirect
35. C₃₁×#102	C₂²	semidirect
37. C₃₁×#97	C₂²	semidirect
38. C₄×C₈×C₃₁	C₂²	semidirect
99. #3222	C₃₁	direct (N × H)
117. #541	C₂×C₃₁	semidirect
119. #497	C₂×C₃₁	semidirect
120. #492	C₂×C₃₁	semidirect
121. #490	C₂×C₃₁	semidirect
122. #488	C₂×C₃₁	semidirect
123. #464	C₂×C₃₁	semidirect
124. #460	C₂×C₃₁	semidirect
125. #452	C₂×C₃₁	semidirect
126. #450	C₂×C₃₁	semidirect
127. #433	C₂×C₃₁	semidirect
136. C₂×Q₁₆	C₂²×C₃₁	semidirect
139. C₂×SD₁₆	C₂²×C₃₁	semidirect
140. C₄⋊Q₈	C₂²×C₃₁	semidirect
145. #125	C₂²×C₃₁	semidirect
151. C₄×Q₈	C₂²×C₃₁	semidirect
153. C₄⋊C₈	C₂²×C₃₁	semidirect
154. #103	C₂²×C₃₁	semidirect
157. #102	C₂²×C₃₁	semidirect
159. #97	C₂²×C₃₁	semidirect
160. C₄×C₈	C₂²×C₃₁	semidirect
161. C₃₁	#3222	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×#541	2
C₃₁×#535	2
C₃₁×#497	2
C₃₁×#492	2
C₃₁×#490	2
C₃₁×#488	2
C₃₁×#464	2
C₃₁×#460	2
C₃₁×#452	2
C₃₁×#450	2
C₃₁×#433	2
#3222	31

Subgroups — C31×#3222 #16843155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3222 #16843155