Subgroups — C₃₁×#12849 #21469608

Order 7936.

462 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#349	C₂	1	2	yes	maximal
3	C₃₁×#1376	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1375	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₃₁	C₄	1	1	yes	characteristic
8	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
9	C₄×M₁₆×C₃₁	—	2	4	no	—
10	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
11	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#349	—	2	4	no	—
15	C₂³×C₄×C₃₁	—	2	1	no	—
16	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₃₁	—	2	2	no	—
18	C₂×M₁₆×C₃₁	—	2	2	no	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	—	2	2	no	—
28	C₂×C₄²×C₃₁	—	2	1	no	—
29	C₂×C₄²×C₃₁	C₈	1	4	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₃₁×M₃₂	—	2	4	no	—
33	C₃₁×M₃₂	—	2	4	no	—
34	C₂×C₁₆×C₃₁	—	2	4	no	—
35	C₃₁×#97	—	2	2	no	—
36	C₃₁×#97	—	2	2	no	—
37	C₄×C₈×C₃₁	—	2	2	no	—
38	C₄×C₈×C₃₁	—	2	2	no	—
39	C₂⁴×C₃₁	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	—	2	1	no	—
41	C₂²×C₄×C₃₁	—	2	1	no	—
42	C₂²×C₄×C₃₁	—	2	1	no	—
43	C₂²×C₄×C₃₁	—	2	1	no	—
44	C₂²×C₄×C₃₁	—	2	1	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	—	4	4	no	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₃₁	—	2	8	no	—
53	M₁₆×C₃₁	—	2	8	no	—
54	C₂×C₈×C₃₁	—	2	1	no	—
55	C₂×C₈×C₃₁	—	2	1	no	—
56	C₂×C₈×C₃₁	—	2	1	no	—
57	C₂×C₈×C₃₁	—	2	1	no	—
58	C₂×C₈×C₃₁	—	2	1	no	—
59	C₂×C₈×C₃₁	—	2	1	no	—
60	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
62	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
63	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₃₁	#41	1	2	yes	—
65	C₂×C₈×C₃₁	#41	1	2	yes	—
66	C₄²×C₃₁	—	2	4	no	—
67	C₄²×C₃₁	—	2	1	no	—
68	C₄²×C₃₁	—	2	1	no	—
69	C₄²×C₃₁	C₂×C₈	1	2	yes	—
70	C₄²×C₃₁	C₂×C₈	1	2	yes	—
71	C₁₆×C₃₁	—	4	4	no	—
72	#12849	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	—	2	1	no	—
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	—	4	2	no	—
76	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
77	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	4	8	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₃₁	—	2	2	no	—
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	2	2	no	—
99	C₈×C₃₁	#123	1	4	yes	—
100	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
101	C₂×#349	C₂×C₃₁	1	2	yes	—
102	#1376	C₂×C₃₁	1	2	yes	complemented
103	#1375	C₂×C₃₁	1	2	yes	complemented
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	2	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	4	2	no	—
109	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₂²×C₃₁	#432	1	1	yes	characteristic
111	C₂²×C₃₁	#432	1	1	yes	characteristic
112	C₄×C₃₁	—	2	4	no	—
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	—	4	2	no	—
115	C₄×C₃₁	#381	1	1	yes	characteristic
116	C₄×C₃₁	#407	1	1	yes	characteristic
117	C₄×C₃₁	#431	1	1	yes	characteristic
118	C₄×C₃₁	#431	1	1	yes	characteristic
119	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
120	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
121	C₂²×C₄²	C₄×C₃₁	1	1	yes	characteristic
122	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
123	C₄×M₁₆	—	2	4	no	—
124	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
125	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
126	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
128	#349	—	2	4	no	—
129	C₂×C₃₁	—	2	1	no	—
130	C₂×C₃₁	—	2	1	no	—
131	C₂×C₃₁	—	4	2	no	—
132	C₂×C₃₁	#1817	1	1	yes	characteristic
133	C₂×C₃₁	#2165	1	2	yes	—
134	C₂³×C₄	—	2	1	no	—
135	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
143	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
144	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
146	C₂×C₄²	—	2	2	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	C₈×C₃₁	1	4	yes	—
149	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
151	M₃₂	—	2	4	no	—
152	M₃₂	—	2	4	no	—
153	C₂×C₁₆	—	2	4	no	—
154	#97	—	2	2	no	—
155	#97	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₃₁	#12849	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	M₁₆×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
170	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
181	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
182	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
183	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
184	C₂×C₈	#41×C₃₁	1	2	yes	—
185	C₂×C₈	#41×C₃₁	1	2	yes	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	C₂×C₈×C₃₁	1	2	yes	—
190	C₄²	C₂×C₈×C₃₁	1	2	yes	—
191	C₁₆	—	4	4	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	4	2	no	—
195	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
196	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
211	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	C₃₁×#123	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
225	C₂²	C₃₁×#432	1	1	yes	characteristic
226	C₂²	C₃₁×#432	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	C₃₁×#381	1	1	yes	characteristic
231	C₄	C₃₁×#407	1	1	yes	characteristic
232	C₄	C₃₁×#431	1	1	yes	characteristic
233	C₄	C₃₁×#431	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	C₃₁×#1817	1	1	yes	characteristic
238	C₂	C₃₁×#2165	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	#432
prime radical	C₂⁴×C₃₁	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1376	C₂	semidirect
4. C₃₁×#1375	C₂	semidirect
72. #12849	C₃₁	direct (N × H)
102. #1376	C₂×C₃₁	semidirect
103. #1375	C₂×C₃₁	semidirect
158. C₃₁	#12849	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×M₁₆×C₃₁	2
C₂×C₃₁×#349	2
C₃₁×#1376	2
C₃₁×#1375	2
#12849	31

Subgroups — C31×#12849 #21469608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12849 #21469608