Subgroups — C₃₁×#10837 #21467596

Order 7936.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 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	C₃₁×#2285	C₂	1	2	yes	maximal
3	C₂×C₃₁×H(3,ℤ/4)	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
5	C₂×C₃₁×#117	C₂²	1	4	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₃₁	—	2	2	no	—
8	C₄×M₁₆×C₃₁	—	2	2	no	—
9	C₂×C₃₁×#97	C₄	1	2	yes	complemented
10	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	complemented
11	C₃₁×#384	—	2	2	no	—
12	C₃₁×#382	—	2	2	no	—
13	C₃₁×#380	—	2	4	no	—
14	C₄×C₄⋊C₄×C₃₁	C₄	1	8	yes	complemented
15	C₃₁×H(3,ℤ/4)	—	2	8	no	—
16	C₂³×C₄×C₃₁	D₄	1	2	yes	—
17	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₃₁	—	2	4	no	—
19	C₂×M₁₆×C₃₁	—	2	4	no	—
20	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
21	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
23	C₃₁×#117	—	2	8	no	—
24	C₃₁×#117	—	2	8	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
27	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
35	C₃₁×#97	—	2	4	no	—
36	C₄×C₈×C₃₁	—	2	2	no	—
37	C₄×C₈×C₃₁	—	2	2	no	—
38	C₃₁×#95	—	2	8	no	—
39	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	—	2	8	no	—
41	C₂²×C₄×C₃₁	—	4	4	no	—
42	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₃₁	#41	1	4	yes	—
53	M₁₆×C₃₁	—	2	8	no	—
54	M₁₆×C₃₁	—	2	8	no	—
55	C₂×C₈×C₃₁	—	2	4	no	—
56	C₂×C₈×C₃₁	—	2	4	no	—
57	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	complemented
58	C₄⋊C₄×C₃₁	—	4	16	no	—
59	C₄⋊C₄×C₃₁	—	4	8	no	—
60	C₂²⋊C₄×C₃₁	—	4	4	no	—
61	C₂²⋊C₄×C₃₁	—	4	4	no	—
62	C₄²×C₃₁	—	2	16	no	—
63	C₄²×C₃₁	C₄²	1	2	yes	—
64	C₄²×C₃₁	C₄²	1	2	yes	—
65	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
69	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
70	#10837	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	—	4	2	no	—
72	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₃₁	#124	1	2	yes	—
75	C₂³×C₃₁	#127	1	1	yes	characteristic
76	C₂³×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₃₁	—	2	8	no	—
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	4	4	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	16	no	—
82	C₂×C₄×C₃₁	—	4	8	no	—
83	C₂×C₄×C₃₁	#95	1	2	yes	—
84	C₂×C₄×C₃₁	#95	1	4	yes	—
85	C₂×C₄×C₃₁	#95	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₃₁	#117	1	2	yes	—
91	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
93	C₂×C₄×C₃₁	#121	1	4	yes	—
94	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
95	C₂×C₄×C₃₁	#125	1	2	yes	—
96	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
97	C₈×C₃₁	—	2	8	no	—
98	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
99	#2285	C₂×C₃₁	1	2	yes	—
100	C₂×H(3,ℤ/4)	C₂×C₃₁	1	4	yes	—
101	C₂²×C₃₁	—	2	4	no	—
102	C₂²×C₃₁	—	2	2	no	—
103	C₂²×C₃₁	—	4	2	no	—
104	C₂²×C₃₁	—	4	1	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
107	C₂²×C₃₁	#382	1	1	yes	characteristic
108	C₂²×C₃₁	#384	1	1	yes	characteristic
109	C₂²×C₃₁	#390	1	2	yes	—
110	C₂²×C₃₁	#391	1	2	yes	—
111	C₄×C₃₁	—	2	4	no	—
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	8	8	no	—
114	C₄×C₃₁	H(3,ℤ/4)	1	4	yes	complemented
115	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
116	C₄×C₃₁	#380	1	2	yes	—
117	C₄×C₃₁	#389	1	4	yes	—
118	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
119	C₂×#117	C₂²×C₃₁	1	4	yes	—
120	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
121	C₄×M₁₆	—	2	2	no	—
122	C₄×M₁₆	—	2	2	no	—
123	C₂×#97	C₄×C₃₁	1	2	yes	complemented
124	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	complemented
125	#384	—	2	2	no	—
126	#382	—	2	2	no	—
127	#380	—	2	4	no	—
128	C₄×C₄⋊C₄	C₄×C₃₁	1	8	yes	complemented
129	H(3,ℤ/4)	—	2	8	no	—
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	—	2	1	no	—
132	C₂×C₃₁	—	2	1	no	—
133	C₂×C₃₁	—	2	2	no	—
134	C₂×C₃₁	C₄⋊#95	1	1	yes	characteristic
135	C₂×C₃₁	C₂×H(3,ℤ/4)	1	2	yes	—
136	C₂³×C₄	D₄×C₃₁	1	2	yes	—
137	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
138	C₂×M₁₆	—	2	4	no	—
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
141	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
143	#117	—	2	8	no	—
144	#117	—	2	8	no	—
145	C₂×C₄⋊C₄	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₂²⋊C₄	—	2	4	no	—
148	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
149	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
152	C₂×C₄²	D₄×C₃₁	1	2	yes	—
153	C₂×C₄²	D₄×C₃₁	1	2	yes	—
154	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	#95	—	2	8	no	—
159	C₃₁	#10837	1	1	yes	characteristic, direct product factor, Sylow
160	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
165	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
166	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
173	C₂²×C₄	#41×C₃₁	1	4	yes	—
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	complemented
179	C₄⋊C₄	—	4	16	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₂²⋊C₄	—	4	4	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₄²	—	2	16	no	—
184	C₄²	C₄²×C₃₁	1	2	yes	—
185	C₄²	C₄²×C₃₁	1	2	yes	—
186	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
187	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
189	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
190	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
193	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
194	C₂³	C₃₁×#124	1	2	yes	—
195	C₂³	C₃₁×#127	1	1	yes	characteristic
196	C₂³	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	16	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	C₃₁×#95	1	2	yes	—
204	C₂×C₄	C₃₁×#95	1	4	yes	—
205	C₂×C₄	C₃₁×#95	1	2	yes	—
206	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
210	C₂×C₄	C₃₁×#117	1	2	yes	—
211	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
212	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
213	C₂×C₄	C₃₁×#121	1	4	yes	—
214	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
215	C₂×C₄	C₃₁×#125	1	2	yes	—
216	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
217	C₈	—	2	8	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
224	C₂²	C₃₁×#382	1	1	yes	characteristic
225	C₂²	C₃₁×#384	1	1	yes	characteristic
226	C₂²	C₃₁×#390	1	2	yes	—
227	C₂²	C₃₁×#391	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	8	8	no	—
231	C₄	C₃₁×H(3,ℤ/4)	1	4	yes	complemented
232	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
233	C₄	C₃₁×#380	1	2	yes	—
234	C₄	C₃₁×#389	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	C₃₁×C₄⋊#95	1	1	yes	characteristic
240	C₂	C₂×C₃₁×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₃₁×#97	C₄	semidirect
10. C₂×C₄×C₈×C₃₁	C₄	semidirect
14. C₄×C₄⋊C₄×C₃₁	C₄	semidirect
57. C₂×C₈×C₃₁	C₄⋊C₄	semidirect
70. #10837	C₃₁	direct (N × H)
114. C₄×C₃₁	H(3,ℤ/4)	semidirect
123. C₂×#97	C₄×C₃₁	semidirect
124. C₂×C₄×C₈	C₄×C₃₁	semidirect
128. C₄×C₄⋊C₄	C₄×C₃₁	semidirect
159. C₃₁	#10837	direct (N × H)
178. C₂×C₈	C₄⋊C₄×C₃₁	semidirect
231. C₄	C₃₁×H(3,ℤ/4)	semidirect

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
C₃₁×#2285	2
C₂×C₃₁×H(3,ℤ/4)	2
#10837	31

Subgroups — C31×#10837 #21467596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10837 #21467596