Subgroups — C₃₁×#13274 #21470033

Order 7936.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1382	C₂	1	4	yes	maximal
5	C₃₁×#533	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	1	yes	characteristic
8	C₃₁×#500	C₄	1	1	yes	characteristic
9	C₃₁×#499	—	2	1	no	—
10	C₃₁×#499	C₄	1	2	yes	—
11	C₃₁×#498	—	2	1	no	—
12	C₂×C₃₁×#108	C₂²	1	2	yes	—
13	C₃₁×#427	—	2	1	no	—
14	C₃₁×#427	C₂²	1	2	yes	—
15	C₄×M₁₆×C₃₁	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊C₁₆	—	2	1	no	—
18	C₃₁×C₄⋊C₁₆	—	2	1	no	—
19	C₃₁×#341	—	2	1	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	—	2	1	no	—
24	C₃₁×C₄⋊Q₈	—	2	1	no	—
25	C₃₁×#125	—	2	2	no	—
26	C₃₁×C₂²⋊Q₈	—	4	1	no	—
27	C₃₁×C₂²⋊Q₈	—	4	1	no	—
28	C₄×Q₈×C₃₁	—	4	1	no	—
29	C₄×Q₈×C₃₁	—	4	1	no	—
30	C₃₁×#117	—	2	2	no	—
31	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₃₁	—	2	2	no	—
33	C₂×C₁₆×C₃₁	—	2	4	no	—
34	C₃₁×#108	—	2	2	no	—
35	C₃₁×#108	Q₈	1	4	yes	complemented
36	C₃₁×#107	—	2	1	no	—
37	C₃₁×#107	—	2	1	no	—
38	C₃₁×#107	C₂×C₄	1	2	yes	—
39	C₃₁×#107	C₂×C₄	1	2	yes	—
40	C₃₁×#106	—	2	2	no	—
41	C₃₁×#106	—	2	2	no	—
42	C₃₁×#97	—	4	1	no	—
43	C₄×C₈×C₃₁	—	4	1	no	—
44	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₃₁	—	4	1	no	—
49	C₂×Q₈×C₃₁	—	4	1	no	—
50	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₃₁	—	2	4	no	—
54	M₁₆×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₃₁	#41	1	2	yes	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₄⋊C₄×C₃₁	—	4	2	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄⋊C₄×C₃₁	—	4	1	no	—
68	C₄⋊C₄×C₃₁	—	4	1	no	—
69	C₂²⋊C₄×C₃₁	—	4	2	no	—
70	C₄²×C₃₁	—	4	2	no	—
71	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₃₁	D₈	1	1	yes	characteristic
74	C₄²×C₃₁	D₈	1	1	yes	characteristic
75	C₁₆×C₃₁	—	4	2	no	—
76	#13274	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈×C₃₁	—	8	2	no	—
79	Q₈×C₃₁	—	8	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	4	1	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	—	4	2	no	—
84	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#137	1	2	yes	—
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	—	4	4	no	—
95	C₈×C₃₁	C₄×Q₈	1	2	yes	—
96	C₈×C₃₁	#124	1	1	yes	characteristic
97	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
98	#3160	C₂×C₃₁	1	1	yes	characteristic
99	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
100	C₄×#108	C₂×C₃₁	1	1	yes	characteristic
101	#1382	C₂×C₃₁	1	4	yes	—
102	C₂²×C₃₁	—	2	1	no	—
103	C₂²×C₃₁	—	2	1	no	—
104	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
105	C₂²×C₃₁	#475	1	2	yes	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	8	2	no	—
110	C₄×C₃₁	—	8	2	no	—
111	C₄×C₃₁	#359	1	2	yes	—
112	C₄×C₃₁	#391	1	1	yes	characteristic
113	C₄×C₃₁	#419	1	1	yes	characteristic
114	C₄×C₃₁	#474	1	2	yes	—
115	#533	—	2	1	no	—
116	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
117	#500	C₄×C₃₁	1	1	yes	characteristic
118	#500	C₄×C₃₁	1	1	yes	characteristic
119	#499	—	2	1	no	—
120	#499	C₄×C₃₁	1	2	yes	—
121	#498	—	2	1	no	—
122	C₂×#108	C₂²×C₃₁	1	2	yes	—
123	#427	—	2	1	no	—
124	#427	C₂²×C₃₁	1	2	yes	—
125	C₄×M₁₆	—	2	1	no	—
126	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	2	1	no	—
128	C₄⋊C₁₆	—	2	1	no	—
129	#341	—	2	1	no	—
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	#1988	1	1	yes	characteristic
132	C₂×C₃₁	C₂×#359	1	1	yes	characteristic
133	C₂×C₃₁	#2143	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	1	no	—
138	C₄⋊Q₈	—	2	1	no	—
139	#125	—	2	2	no	—
140	C₂²⋊Q₈	—	4	1	no	—
141	C₂²⋊Q₈	—	4	1	no	—
142	C₄×Q₈	—	4	1	no	—
143	C₄×Q₈	—	4	1	no	—
144	#117	—	2	2	no	—
145	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	4	no	—
148	#108	—	2	2	no	—
149	#108	Q₈×C₃₁	1	4	yes	complemented
150	#107	—	2	1	no	—
151	#107	—	2	1	no	—
152	#107	C₂×C₄×C₃₁	1	2	yes	—
153	#107	C₂×C₄×C₃₁	1	2	yes	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#97	—	4	1	no	—
157	C₄×C₈	—	4	1	no	—
158	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
159	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
160	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
161	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
162	C₃₁	#13274	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
176	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
177	C₂×C₈	#41×C₃₁	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
187	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₄²	D₈×C₃₁	1	1	yes	characteristic
189	C₄²	D₈×C₃₁	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
199	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
200	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
201	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
204	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₃₁×#137	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Q₈×C₃₁	1	2	yes	—
210	C₈	C₃₁×#124	1	1	yes	characteristic
211	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
215	C₂²	C₃₁×#475	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	C₃₁×#359	1	2	yes	—
222	C₄	C₃₁×#391	1	1	yes	characteristic
223	C₄	C₃₁×#419	1	1	yes	characteristic
224	C₄	C₃₁×#474	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	C₃₁×#1988	1	1	yes	characteristic
227	C₂	C₂×C₃₁×#359	1	1	yes	characteristic
228	C₂	C₃₁×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₂³×C₃₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
35. C₃₁×#108	Q₈	semidirect
76. #13274	C₃₁	direct (N × H)
149. #108	Q₈×C₃₁	semidirect
162. C₃₁	#13274	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×#3160	2
C₂×C₃₁×C₄⋊C₁₆	2
C₄×C₃₁×#108	2
C₃₁×#1382	2
#13274	31

Subgroups — C31×#13274 #21470033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13274 #21470033