Subgroups — C₃₁×#11671 #21468430

Order 7936.

378 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1331	C₂	1	4	yes	maximal, complemented
5	C₃₁×#532	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	—	2	1	no	—
7	C₃₁×#500	C₄	1	1	yes	characteristic
8	C₃₁×#493	C₄	1	1	yes	characteristic
9	C₃₁×#486	—	2	1	no	—
10	C₃₁×#466	—	2	1	no	—
11	C₃₁×#466	C₄	1	2	yes	—
12	C₄×D₈×C₃₁	—	2	1	no	—
13	C₄×D₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₃₁	—	2	1	no	—
17	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#335	—	2	1	no	—
19	C₃₁×#335	C₂²	1	2	yes	—
20	C₂×#41×C₃₁	—	4	1	no	—
21	C₂×D₈×C₃₁	—	2	1	no	—
22	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₃₁	—	4	1	no	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₃₁×C₄⋊Q₈	—	2	1	no	—
27	C₃₁×#127	—	2	1	no	—
28	C₃₁×#124	—	4	1	no	—
29	C₃₁×#121	—	2	2	no	—
30	C₃₁×#121	—	4	1	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	—	4	1	no	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₃₁	—	2	4	no	—
35	C₂×C₁₆×C₃₁	—	2	2	no	—
36	C₃₁×#107	—	2	1	no	—
37	C₃₁×#107	C₂×C₄	1	2	yes	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	4	1	no	—
40	C₃₁×#102	—	2	2	no	—
41	C₃₁×#102	—	2	2	no	—
42	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	#41×C₃₁	—	8	2	no	—
47	#41×C₃₁	—	8	2	no	—
48	C₂×Q₈×C₃₁	—	4	1	no	—
49	C₂×D₄×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	—	4	2	no	—
51	C₂×D₄×C₃₁	—	4	1	no	—
52	C₂×D₄×C₃₁	—	4	1	no	—
53	C₂²×C₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
57	D₈×C₃₁	—	2	4	no	—
58	D₈×C₃₁	—	4	2	no	—
59	M₁₆×C₃₁	—	8	2	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	4	2	no	—
63	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₄⋊C₄×C₃₁	—	4	1	no	—
69	C₂²⋊C₄×C₃₁	—	4	2	no	—
70	C₂²⋊C₄×C₃₁	—	4	2	no	—
71	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₃₁	D₈	1	1	yes	characteristic
73	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
74	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₃₁	—	2	4	no	—
76	#11671	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	—	4	2	no	—
78	C₂³×C₃₁	#102	1	1	yes	characteristic
79	Q₈×C₃₁	—	8	2	no	—
80	D₄×C₃₁	—	4	2	no	—
81	D₄×C₃₁	—	4	2	no	—
82	D₄×C₃₁	—	8	4	no	—
83	D₄×C₃₁	—	8	2	no	—
84	D₄×C₃₁	—	8	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	8	2	no	—
90	C₂×C₄×C₃₁	—	8	2	no	—
91	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
94	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
95	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	—	2	1	no	—
101	C₈×C₃₁	—	8	2	no	—
102	C₈×C₃₁	#104	1	2	yes	—
103	C₈×C₃₁	#131	1	2	yes	—
104	#3169	C₂×C₃₁	1	1	yes	characteristic
105	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
106	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
107	#1331	C₂×C₃₁	1	4	yes	complemented
108	C₂²×C₃₁	—	2	1	no	—
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	—	8	2	no	—
111	C₂²×C₃₁	—	8	2	no	—
112	C₂²×C₃₁	—	8	2	no	—
113	C₂²×C₃₁	#357	1	2	yes	—
114	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
115	C₄×C₃₁	—	2	1	no	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	8	2	no	—
118	C₄×C₃₁	#357	1	2	yes	—
119	C₄×C₃₁	#359	1	2	yes	—
120	C₄×C₃₁	#408	1	1	yes	characteristic
121	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
122	C₄×C₃₁	C₂×D₁₆	1	1	yes	characteristic
123	C₄×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
124	#532	—	2	1	no	—
125	C₂²×C₁₆	—	2	1	no	—
126	#500	C₄×C₃₁	1	1	yes	characteristic
127	#493	C₄×C₃₁	1	1	yes	characteristic
128	#486	—	2	1	no	—
129	#466	—	2	1	no	—
130	#466	C₄×C₃₁	1	2	yes	—
131	C₄×D₈	—	2	1	no	—
132	C₄×D₈	C₂²×C₃₁	1	2	yes	—
133	#423	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
135	C₄×C₁₆	—	2	1	no	—
136	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
137	#335	—	2	1	no	—
138	#335	C₂²×C₃₁	1	2	yes	—
139	C₂×C₃₁	—	2	2	no	—
140	C₂×C₃₁	—	16	2	no	—
141	C₂×C₃₁	#1492	1	1	yes	characteristic
142	C₂×C₃₁	C₂×#357	1	1	yes	characteristic
143	C₂×C₃₁	C₂×#359	1	1	yes	characteristic
144	C₂×#41	—	4	1	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×M₁₆	—	4	1	no	—
148	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
149	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	1	no	—
151	#127	—	2	1	no	—
152	#124	—	4	1	no	—
153	#121	—	2	2	no	—
154	#121	—	4	1	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	—	4	1	no	—
157	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	—	2	2	no	—
160	#107	—	2	1	no	—
161	#107	C₂×C₄×C₃₁	1	2	yes	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	4	1	no	—
164	#102	—	2	2	no	—
165	#102	—	2	2	no	—
166	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
168	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
169	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
170	C₃₁	#11671	1	1	yes	characteristic, direct product factor, Sylow
171	#41	—	8	2	no	—
172	#41	—	8	2	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
182	D₈	—	2	4	no	—
183	D₈	—	4	2	no	—
184	M₁₆	—	8	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
197	C₄²	D₈×C₃₁	1	1	yes	characteristic
198	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₁₆	—	2	4	no	—
201	C₂³	—	4	2	no	—
202	C₂³	C₃₁×#102	1	1	yes	characteristic
203	Q₈	—	8	2	no	—
204	D₄	—	4	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	4	no	—
207	D₄	—	8	2	no	—
208	D₄	—	8	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
216	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
217	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
218	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
219	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	8	2	no	—
226	C₈	C₃₁×#104	1	2	yes	—
227	C₈	C₃₁×#131	1	2	yes	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	C₃₁×#357	1	2	yes	—
234	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	8	2	no	—
238	C₄	C₃₁×#357	1	2	yes	—
239	C₄	C₃₁×#359	1	2	yes	—
240	C₄	C₃₁×#408	1	1	yes	characteristic
241	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
242	C₄	C₂×C₃₁×D₁₆	1	1	yes	characteristic
243	C₄	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	C₃₁×#1492	1	1	yes	characteristic
247	C₂	C₂×C₃₁×#357	1	1	yes	characteristic
248	C₂	C₂×C₃₁×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₃₁×#3169	C₂
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
2. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
3. C₃₁×#1567	C₂	semidirect
4. C₃₁×#1331	C₂	semidirect
76. #11671	C₃₁	direct (N × H)
105. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
106. #1567	C₂×C₃₁	semidirect
107. #1331	C₂×C₃₁	semidirect
170. C₃₁	#11671	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×#3169	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1567	2
C₃₁×#1331	2
#11671	31

Subgroups — C31×#11671 #21468430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11671 #21468430