Subgroups — C₂×C₃₁×#1991 #21478076

Order 7936.

1246 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	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1991	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	4	yes	maximal
6	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#125	—	2	2	no	—
8	C₂×C₃₁×#125	—	2	4	no	—
9	C₂×C₃₁×#125	C₄	1	8	yes	complemented
10	C₂×C₃₁×#125	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	—	2	4	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
17	C₃₁×#382	—	2	2	no	—
18	C₃₁×#382	C₂²	1	4	yes	—
19	C₂×C₃₁×#95	—	2	2	no	—
20	C₃₁×#340	C₂²	1	16	yes	—
21	C₂³×C₄×C₃₁	—	2	2	no	—
22	C₂³×C₄×C₃₁	—	2	2	no	—
23	C₂³×C₄×C₃₁	D₄	1	2	yes	—
24	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	—	2	4	no	—
26	C₂²×C₈×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	Q₈	1	8	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
29	C₃₁×#125	—	2	16	no	—
30	C₃₁×#125	—	2	8	no	—
31	C₃₁×#125	C₂×C₄	1	16	yes	complemented
32	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	D₄	1	4	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
45	C₄×C₈×C₃₁	—	2	8	no	—
46	C₃₁×#95	—	2	8	no	—
47	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	8	no	—
49	C₂²×C₄×C₃₁	—	2	8	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	4	8	no	—
53	C₂²×C₄×C₃₁	—	4	4	no	—
54	C₂²×C₄×C₃₁	—	4	8	no	—
55	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₃₁	#41	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₃₁	—	2	16	no	—
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
67	C₄⋊C₄×C₃₁	—	2	32	no	—
68	C₄⋊C₄×C₃₁	—	2	8	no	—
69	C₄⋊C₄×C₃₁	—	2	4	no	—
70	C₄⋊C₄×C₃₁	—	2	8	no	—
71	C₄⋊C₄×C₃₁	—	4	4	no	—
72	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
73	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
74	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
75	C₄²×C₃₁	C₂×D₄	1	4	yes	—
76	C₂×#1991	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂³×C₃₁	#124	1	2	yes	—
79	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
80	C₂³×C₃₁	#135	1	8	yes	—
81	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	—	2	8	no	—
83	C₂×C₄×C₃₁	—	2	16	no	—
84	C₂×C₄×C₃₁	—	2	8	no	—
85	C₂×C₄×C₃₁	—	4	8	no	—
86	C₂×C₄×C₃₁	—	4	16	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	—	4	8	no	—
89	C₂×C₄×C₃₁	—	8	8	no	—
90	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
91	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
93	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
94	C₂×C₄×C₃₁	#124	1	2	yes	—
95	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
96	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
99	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
100	C₈×C₃₁	—	2	8	no	—
101	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic
102	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
103	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
104	#1991	C₂×C₃₁	1	8	yes	direct product factor
105	C₂×#340	C₂×C₃₁	1	4	yes	—
106	C₂²×C₃₁	#391	1	4	yes	—
107	C₂²×C₃₁	#416	1	8	yes	—
108	C₂²×C₃₁	#488	1	8	yes	—
109	C₂²×C₃₁	#499	1	8	yes	—
110	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
112	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×#135	1	4	yes	—
114	C₄×C₃₁	—	2	8	no	—
115	C₄×C₃₁	—	4	8	no	—
116	C₄×C₃₁	—	8	4	no	—
117	C₄×C₃₁	#391	1	4	yes	—
118	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
119	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
120	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
121	C₂×#125	—	2	2	no	—
122	C₂×#125	—	2	4	no	—
123	C₂×#125	C₄×C₃₁	1	8	yes	complemented
124	C₂×#125	C₂²×C₃₁	1	4	yes	—
125	C₂²×C₄⋊C₄	—	2	2	no	—
126	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
128	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
129	C₂×C₄×C₈	—	2	4	no	—
130	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
131	#382	—	2	2	no	—
132	#382	C₂²×C₃₁	1	4	yes	—
133	C₂×#95	—	2	2	no	—
134	#340	C₂²×C₃₁	1	16	yes	—
135	C₂×C₃₁	#1991	1	8	yes	direct product factor
136	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
137	C₂×C₃₁	C₂×#416	1	2	yes	—
138	C₂×C₃₁	C₂×#488	1	2	yes	—
139	C₂×C₃₁	C₂×#499	1	2	yes	—
140	C₂³×C₄	—	2	2	no	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	D₄×C₃₁	1	2	yes	—
143	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	Q₈×C₃₁	1	8	yes	—
147	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
148	#125	—	2	16	no	—
149	#125	—	2	8	no	—
150	#125	C₂×C₄×C₃₁	1	16	yes	complemented
151	C₂×C₄⋊C₄	—	2	16	no	—
152	C₂×C₄⋊C₄	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	8	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
158	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
159	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
160	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
161	C₂×C₄²	D₄×C₃₁	1	2	yes	—
162	C₂×C₄²	D₄×C₃₁	1	4	yes	—
163	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
164	C₄×C₈	—	2	8	no	—
165	#95	—	2	8	no	—
166	C₃₁	C₂×#1991	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	—	4	8	no	—
175	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
176	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
178	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
180	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
181	C₂²×C₄	#41×C₃₁	1	4	yes	—
182	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
183	C₂×C₈	—	2	16	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
193	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
194	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
195	C₄²	C₂×D₄×C₃₁	1	4	yes	—
196	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
197	C₂³	C₃₁×#124	1	2	yes	—
198	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
199	C₂³	C₃₁×#135	1	8	yes	—
200	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄	—	2	16	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	4	16	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
210	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
211	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
212	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
213	C₂×C₄	C₃₁×#124	1	2	yes	—
214	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
215	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
218	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₂²	C₃₁×#391	1	4	yes	—
221	C₂²	C₃₁×#416	1	8	yes	—
222	C₂²	C₃₁×#488	1	8	yes	—
223	C₂²	C₃₁×#499	1	8	yes	—
224	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
225	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
226	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
227	C₂²	C₂×C₃₁×#135	1	4	yes	—
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	—	8	4	no	—
231	C₄	C₃₁×#391	1	4	yes	—
232	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
233	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
234	C₂	C₃₁×#1991	1	8	yes	direct product factor
235	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
236	C₂	C₂×C₃₁×#416	1	2	yes	—
237	C₂	C₂×C₃₁×#488	1	2	yes	—
238	C₂	C₂×C₃₁×#499	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1991	C₂	direct (N × H)
9. C₂×C₃₁×#125	C₄	semidirect
31. C₃₁×#125	C₂×C₄	semidirect
76. C₂×#1991	C₃₁	direct (N × H)
104. #1991	C₂×C₃₁	direct (N × H)
123. C₂×#125	C₄×C₃₁	semidirect
135. C₂×C₃₁	#1991	direct (N × H)
150. #125	C₂×C₄×C₃₁	semidirect
166. C₃₁	C₂×#1991	direct (N × H)
234. C₂	C₃₁×#1991	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#125	2
C₂²×C₄×C₈×C₃₁	2
C₂×C₃₁×#382	2
C₃₁×#1991	2
C₂×C₃₁×#340	2
C₂×#1991	31

Subgroups — C2×C31×#1991 #21478076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1991 #21478076