Subgroups — C₃₁×#3214 #16843147

Order 3968.

434 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#496	C₂	1	2	yes	maximal, complemented
4	C₃₁×#489	C₂	1	2	yes	maximal, complemented
5	C₃₁×#461	C₂	1	2	yes	maximal, complemented
6	C₃₁×#459	C₂	1	2	yes	maximal, complemented
7	C₃₁×#449	C₂	1	4	yes	maximal, complemented
8	C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₃₁	—	2	1	no	—
10	C₂×#41×C₃₁	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	—
12	C₂×D₈×C₃₁	C₂²	1	4	yes	—
13	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₃₁×#124	—	2	1	no	—
18	C₃₁×#124	C₂²	1	2	yes	—
19	C₃₁×#121	—	2	1	no	—
20	C₃₁×#121	—	2	1	no	—
21	C₃₁×#121	—	2	1	no	—
22	C₃₁×#121	C₂²	1	2	yes	—
23	C₄×D₄×C₃₁	—	2	1	no	—
24	C₄×D₄×C₃₁	—	2	1	no	—
25	C₄×D₄×C₃₁	—	2	1	no	—
26	C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
29	C₃₁×#103	C₂²	1	4	yes	—
30	C₃₁×#102	C₂²	1	4	yes	—
31	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₃₁×#97	C₂²	1	2	yes	—
33	#41×C₃₁	—	4	4	no	—
34	#41×C₃₁	—	4	4	no	—
35	C₂×Q₈×C₃₁	—	2	1	no	—
36	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂×D₄×C₃₁	—	2	2	no	—
38	C₂×D₄×C₃₁	—	2	2	no	—
39	C₂×D₄×C₃₁	—	2	1	no	—
40	C₂×D₄×C₃₁	—	2	1	no	—
41	C₂×D₄×C₃₁	—	2	1	no	—
42	C₂×D₄×C₃₁	—	2	1	no	—
43	C₂×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	SD₁₆×C₃₁	—	4	4	no	—
51	D₈×C₃₁	—	4	4	no	—
52	C₂×C₈×C₃₁	C₂³	1	4	yes	—
53	C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₄²×C₃₁	D₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	2	2	no	—
67	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₃₁	—	2	4	no	—
69	Q₈×C₃₁	—	4	2	no	—
70	D₄×C₃₁	—	2	2	no	—
71	D₄×C₃₁	—	2	4	no	—
72	D₄×C₃₁	—	2	2	no	—
73	D₄×C₃₁	—	4	2	no	—
74	D₄×C₃₁	—	4	2	no	—
75	D₄×C₃₁	—	4	2	no	—
76	D₄×C₃₁	—	4	4	no	—
77	D₄×C₃₁	—	4	4	no	—
78	D₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	4	1	no	—
84	C₂×C₄×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	—	4	4	no	—
86	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₈×C₃₁	—	2	4	no	—
93	#3214	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₃₁	—	2	1	no	—
95	C₂²×C₃₁	—	2	1	no	—
96	C₂²×C₃₁	—	2	1	no	—
97	C₂²×C₃₁	—	4	2	no	—
98	C₂²×C₃₁	—	4	2	no	—
99	C₂²×C₃₁	—	4	2	no	—
100	C₂²×C₃₁	—	4	2	no	—
101	C₂²×C₃₁	—	4	2	no	—
102	C₂²×C₃₁	—	4	2	no	—
103	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	2	1	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	4	2	no	—
109	C₄×C₃₁	—	4	2	no	—
110	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
111	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
112	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
113	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
114	#496	C₂×C₃₁	1	2	yes	complemented
115	#489	C₂×C₃₁	1	2	yes	complemented
116	#461	C₂×C₃₁	1	2	yes	complemented
117	#459	C₂×C₃₁	1	2	yes	complemented
118	#449	C₂×C₃₁	1	4	yes	complemented
119	#432	C₂×C₃₁	1	1	yes	characteristic, complemented
120	C₂×C₃₁	—	4	1	no	—
121	C₂×C₃₁	—	8	2	no	—
122	C₂×C₃₁	—	8	2	no	—
123	C₂×C₃₁	#535	1	1	yes	characteristic
124	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₃₁	#575	1	1	yes	characteristic
126	C₂×#41	—	2	1	no	—
127	C₂×#41	C₂²×C₃₁	1	2	yes	—
128	C₂×SD₁₆	C₂²×C₃₁	1	4	yes	—
129	C₂×D₈	C₂²×C₃₁	1	4	yes	—
130	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
131	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
132	#127	C₂²×C₃₁	1	1	yes	characteristic
133	#127	C₂²×C₃₁	1	1	yes	characteristic
134	#124	—	2	1	no	—
135	#124	C₂²×C₃₁	1	2	yes	—
136	#121	—	2	1	no	—
137	#121	—	2	1	no	—
138	#121	—	2	1	no	—
139	#121	C₂²×C₃₁	1	2	yes	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	C₂²×C₃₁	1	2	yes	—
144	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
145	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
146	#103	C₂²×C₃₁	1	4	yes	—
147	#102	C₂²×C₃₁	1	4	yes	—
148	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
149	#97	C₂²×C₃₁	1	2	yes	—
150	C₃₁	#3214	1	1	yes	characteristic, direct product factor, Sylow
151	#41	—	4	4	no	—
152	#41	—	4	4	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	C₂³×C₃₁	1	2	yes	—
162	C₂×D₄	C₂³×C₃₁	1	2	yes	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
168	SD₁₆	—	4	4	no	—
169	D₈	—	4	4	no	—
170	C₂×C₈	C₂³×C₃₁	1	4	yes	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₄²	D₄×C₃₁	1	1	yes	characteristic
180	C₄²	D₄×C₃₁	1	1	yes	characteristic
181	C₄²	C₂³×C₃₁	1	1	yes	characteristic
182	C₄²	C₂³×C₃₁	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
186	Q₈	—	2	4	no	—
187	Q₈	—	4	2	no	—
188	D₄	—	2	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	4	2	no	—
192	D₄	—	4	2	no	—
193	D₄	—	4	2	no	—
194	D₄	—	4	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
210	C₈	—	2	4	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
228	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
229	C₂	—	4	1	no	—
230	C₂	—	8	2	no	—
231	C₂	—	8	2	no	—
232	C₂	C₃₁×#535	1	1	yes	characteristic
233	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂	C₃₁×#575	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#532	C₂	semidirect
2. C₃₁×#532	C₂	semidirect
3. C₃₁×#496	C₂	semidirect
4. C₃₁×#489	C₂	semidirect
5. C₃₁×#461	C₂	semidirect
6. C₃₁×#459	C₂	semidirect
7. C₃₁×#449	C₂	semidirect
8. C₃₁×#432	C₂	semidirect
93. #3214	C₃₁	direct (N × H)
112. #532	C₂×C₃₁	semidirect
113. #532	C₂×C₃₁	semidirect
114. #496	C₂×C₃₁	semidirect
115. #489	C₂×C₃₁	semidirect
116. #461	C₂×C₃₁	semidirect
117. #459	C₂×C₃₁	semidirect
118. #449	C₂×C₃₁	semidirect
119. #432	C₂×C₃₁	semidirect
150. C₃₁	#3214	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×#532	2
C₃₁×#496	2
C₃₁×#489	2
C₃₁×#461	2
C₃₁×#459	2
C₃₁×#449	2
C₃₁×#432	2
#3214	31

Subgroups — C31×#3214 #16843147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3214 #16843147