Subgroups — C₃₁×#3375 #16843308

Order 3968.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#559	C₂	1	1	yes	maximal, characteristic
2	Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#489	C₂	1	2	yes	maximal, complemented
4	C₃₁×#486	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#474	C₂	1	2	yes	maximal, complemented
7	C₃₁×#464	C₂	1	2	yes	maximal, complemented
8	C₃₁×#460	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#440	C₂	1	2	yes	maximal, complemented
11	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×C₄⋊Q₈	—	2	1	no	—
15	C₃₁×C₄⋊Q₈	—	2	2	no	—
16	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×#127	C₂²	1	1	yes	characteristic
19	C₃₁×#124	—	2	1	no	—
20	C₃₁×#124	C₂²	1	2	yes	—
21	C₃₁×#123	—	2	2	no	—
22	C₄×Q₈×C₃₁	—	2	1	no	—
23	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₃₁	—	2	1	no	—
28	C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#107	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
33	C₃₁×#103	C₂²	1	1	yes	characteristic
34	C₃₁×#103	C₂²	1	1	yes	characteristic
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#102	C₂²	1	2	yes	—
37	C₃₁×#102	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	—	2	2	no	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	Q₁₆×C₃₁	—	2	2	no	—
51	SD₁₆×C₃₁	—	2	4	no	—
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂³	1	2	yes	—
55	C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
62	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	1	no	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	—	2	1	no	—
71	C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₄²×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₃₁	—	2	2	no	—
74	Q₈×C₃₁	—	2	2	no	—
75	Q₈×C₃₁	—	2	1	no	—
76	Q₈×C₃₁	—	2	1	no	—
77	Q₈×C₃₁	—	2	1	no	—
78	Q₈×C₃₁	—	8	2	no	—
79	Q₈×C₃₁	—	8	1	no	—
80	Q₈×C₃₁	#41	1	2	yes	—
81	D₄×C₃₁	—	2	2	no	—
82	D₄×C₃₁	—	2	2	no	—
83	D₄×C₃₁	—	4	2	no	—
84	D₄×C₃₁	—	8	1	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	8	2	no	—
91	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₈×C₃₁	—	2	2	no	—
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	1	no	—
99	#3375	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	2	no	—
105	C₄×C₃₁	—	2	2	no	—
106	C₄×C₃₁	—	2	2	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	4	1	no	—
109	C₄×C₃₁	—	4	1	no	—
110	C₄×C₃₁	—	4	1	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	#137	1	2	yes	—
113	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
114	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
115	#559	C₂×C₃₁	1	1	yes	characteristic
116	Q₈²	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#489	C₂×C₃₁	1	2	yes	complemented
118	#486	C₂×C₃₁	1	1	yes	characteristic
119	#477	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#474	C₂×C₃₁	1	2	yes	complemented
121	#464	C₂×C₃₁	1	2	yes	complemented
122	#460	C₂×C₃₁	1	1	yes	characteristic
123	#446	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#440	C₂×C₃₁	1	2	yes	complemented
125	C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂×C₃₁	—	8	2	no	—
127	C₂×C₃₁	#548	1	1	yes	characteristic
128	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
129	C₂×C₃₁	#576	1	1	yes	characteristic
130	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
132	C₄⋊Q₈	—	2	1	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
135	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
136	#127	C₂²×C₃₁	1	1	yes	characteristic
137	#124	—	2	1	no	—
138	#124	C₂²×C₃₁	1	2	yes	—
139	#123	—	2	2	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
142	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
143	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	C₂²×C₃₁	1	2	yes	—
147	#107	C₂²×C₃₁	1	1	yes	characteristic
148	#107	C₂²×C₃₁	1	2	yes	—
149	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
151	#103	C₂²×C₃₁	1	1	yes	characteristic
152	#103	C₂²×C₃₁	1	1	yes	characteristic
153	#103	C₂²×C₃₁	1	2	yes	—
154	#102	C₂²×C₃₁	1	2	yes	—
155	#102	C₂²×C₃₁	1	2	yes	—
156	#102	C₂²×C₃₁	1	2	yes	—
157	#97	C₂²×C₃₁	1	2	yes	—
158	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₃₁	#3375	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	C₂³×C₃₁	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	Q₁₆	—	2	2	no	—
170	SD₁₆	—	2	4	no	—
171	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₃₁	1	2	yes	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
180	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
181	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
182	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	C₂³×C₃₁	1	1	yes	characteristic
191	C₄²	C₂³×C₃₁	1	2	yes	—
192	C₂³	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	2	1	no	—
196	Q₈	—	2	1	no	—
197	Q₈	—	8	2	no	—
198	Q₈	—	8	1	no	—
199	Q₈	#41×C₃₁	1	2	yes	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	8	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
211	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	C₃₁×#137	1	2	yes	—
231	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
232	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
233	C₂	—	8	2	no	—
234	C₂	C₃₁×#548	1	1	yes	characteristic
235	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
236	C₂	C₃₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈²×C₃₁	C₂	semidirect
3. C₃₁×#489	C₂	semidirect
5. C₃₁×#477	C₂	semidirect
6. C₃₁×#474	C₂	semidirect
7. C₃₁×#464	C₂	semidirect
9. C₃₁×#446	C₂	semidirect
10. C₃₁×#440	C₂	semidirect
11. C₄×Q₁₆×C₃₁	C₂	semidirect
99. #3375	C₃₁	direct (N × H)
116. Q₈²	C₂×C₃₁	semidirect
117. #489	C₂×C₃₁	semidirect
119. #477	C₂×C₃₁	semidirect
120. #474	C₂×C₃₁	semidirect
121. #464	C₂×C₃₁	semidirect
123. #446	C₂×C₃₁	semidirect
124. #440	C₂×C₃₁	semidirect
125. C₄×Q₁₆	C₂×C₃₁	semidirect
159. C₃₁	#3375	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×#559	2
Q₈²×C₃₁	2
C₃₁×#489	2
C₃₁×#486	2
C₃₁×#477	2
C₃₁×#474	2
C₃₁×#464	2
C₃₁×#460	2
C₃₁×#446	2
C₃₁×#440	2
C₄×Q₁₆×C₃₁	2
#3375	31

Subgroups — C31×#3375 #16843308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3375 #16843308