Subgroups — C₂×C₃₁×#1561 #21474495

Order 7936.

1042 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	2	yes	maximal
3	C₃₁×#1561	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#332	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	—	2	2	no	—
6	C₃₁×#533	—	2	4	no	—
7	C₃₁×#533	C₄	1	4	yes	—
8	C₃₁×#533	C₂²	1	4	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₃₁×#125	C₄	1	2	yes	—
11	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
12	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
13	C₂×C₃₁×#117	C₄	1	2	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#423	—	2	4	no	—
16	C₃₁×#423	C₂²	1	8	yes	—
17	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
19	C₃₁×#332	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	D₄	1	2	yes	—
23	C₂×M₁₆×C₃₁	—	2	8	no	—
24	C₂×M₁₆×C₃₁	—	4	8	no	—
25	C₂²×C₈×C₃₁	—	2	4	no	—
26	C₃₁×C₄⋊Q₈	—	2	2	no	—
27	C₃₁×C₄⋊Q₈	—	2	2	no	—
28	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
29	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
30	C₃₁×#125	—	2	4	no	—
31	C₃₁×#125	C₂×C₄	1	8	yes	—
32	C₃₁×C₂²⋊Q₈	—	2	8	no	—
33	C₃₁×C₂²⋊Q₈	—	2	8	no	—
34	C₄×Q₈×C₃₁	—	2	8	no	—
35	C₄×Q₈×C₃₁	—	2	8	no	—
36	C₃₁×#117	—	2	4	no	—
37	C₃₁×#117	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₃₁	D₄	1	2	yes	—
46	C₂×C₄²×C₃₁	D₄	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
49	C₃₁×C₄⋊C₈	—	2	8	no	—
50	C₃₁×C₄⋊C₈	—	2	4	no	—
51	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
52	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Q₈×C₃₁	—	2	4	no	—
54	C₂×Q₈×C₃₁	—	2	4	no	—
55	C₂×Q₈×C₃₁	—	4	4	no	—
56	C₂×Q₈×C₃₁	—	4	4	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	M₁₆×C₃₁	—	4	16	no	—
70	C₂×C₈×C₃₁	—	2	8	no	—
71	C₂×C₈×C₃₁	—	4	8	no	—
72	C₄⋊C₄×C₃₁	—	2	16	no	—
73	C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₄⋊C₄×C₃₁	—	2	4	no	—
76	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
77	C₂²⋊C₄×C₃₁	—	2	8	no	—
78	C₄²×C₃₁	—	2	8	no	—
79	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
82	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	4	yes	—
84	C₄²×C₃₁	C₂×D₄	1	2	yes	—
85	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
86	C₂×#1561	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	—	2	4	no	—
89	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₃₁	—	4	4	no	—
92	Q₈×C₃₁	—	4	4	no	—
93	Q₈×C₃₁	—	4	8	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	8	no	—
101	C₂×C₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	4	8	no	—
103	C₂×C₄×C₃₁	#101	1	4	yes	—
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
109	C₂×C₄×C₃₁	#137	1	4	yes	—
110	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
112	C₈×C₃₁	—	4	8	no	—
113	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
114	C₂×#423	C₂×C₃₁	1	2	yes	—
115	#1561	C₂×C₃₁	1	8	yes	direct product factor
116	C₂×#332	C₂×C₃₁	1	4	yes	complemented
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	2	4	no	—
119	C₂²×C₃₁	—	4	4	no	—
120	C₂²×C₃₁	C₂×#101	1	2	yes	—
121	C₂²×C₃₁	#417	1	4	yes	—
122	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	4	no	—
126	C₄×C₃₁	—	4	8	no	—
127	C₄×C₃₁	C₂×#101	1	4	yes	—
128	C₄×C₃₁	#417	1	4	yes	—
129	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	C₄×C₃₁	C₂×#137	1	2	yes	—
131	C₂²×M₁₆	—	2	2	no	—
132	#533	—	2	4	no	—
133	#533	C₄×C₃₁	1	4	yes	—
134	#533	C₂²×C₃₁	1	4	yes	—
135	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
136	C₂×#125	C₄×C₃₁	1	2	yes	—
137	C₂×C₂²⋊Q₈	—	2	2	no	—
138	C₂×C₄×Q₈	—	2	2	no	—
139	C₂×#117	C₄×C₃₁	1	2	yes	—
140	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
141	#423	—	2	4	no	—
142	#423	C₂²×C₃₁	1	8	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
145	#332	C₂²×C₃₁	1	16	yes	complemented
146	C₂×C₃₁	—	4	2	no	—
147	C₂×C₃₁	#1561	1	4	yes	direct product factor
148	C₂×C₃₁	C₂²×#101	1	1	yes	characteristic
149	C₂×C₃₁	C₂×#417	1	2	yes	—
150	C₂²×Q₈	—	2	2	no	—
151	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
152	C₂³×C₄	D₄×C₃₁	1	2	yes	—
153	C₂×M₁₆	—	2	8	no	—
154	C₂×M₁₆	—	4	8	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
159	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
160	#125	—	2	4	no	—
161	#125	C₂×C₄×C₃₁	1	8	yes	—
162	C₂²⋊Q₈	—	2	8	no	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₄×Q₈	—	2	8	no	—
165	C₄×Q₈	—	2	8	no	—
166	#117	—	2	4	no	—
167	#117	C₂×C₄×C₃₁	1	8	yes	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×C₄²	D₄×C₃₁	1	2	yes	—
176	C₂×C₄²	D₄×C₃₁	1	4	yes	—
177	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
179	C₄⋊C₈	—	2	8	no	—
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
182	C₃₁	C₂×#1561	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
199	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
200	M₁₆	—	4	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	4	8	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
208	C₂²⋊C₄	—	2	8	no	—
209	C₄²	—	2	8	no	—
210	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
211	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
213	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₃₁	1	4	yes	—
215	C₄²	C₂×D₄×C₃₁	1	2	yes	—
216	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
220	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
221	Q₈	—	4	8	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	4	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	C₃₁×#101	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
238	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄	C₃₁×#137	1	4	yes	—
240	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
242	C₈	—	4	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×C₃₁×#101	1	2	yes	—
247	C₂²	C₃₁×#417	1	4	yes	—
248	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×C₃₁×#101	1	4	yes	—
254	C₄	C₃₁×#417	1	4	yes	—
255	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
256	C₄	C₂×C₃₁×#137	1	2	yes	—
257	C₂	—	4	2	no	—
258	C₂	C₃₁×#1561	1	4	yes	direct product factor
259	C₂	C₂²×C₃₁×#101	1	1	yes	characteristic
260	C₂	C₂×C₃₁×#417	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1561	C₂	direct (N × H)
4. C₂×C₃₁×#332	C₂	semidirect
19. C₃₁×#332	C₂²	semidirect
86. C₂×#1561	C₃₁	direct (N × H)
115. #1561	C₂×C₃₁	direct (N × H)
116. C₂×#332	C₂×C₃₁	semidirect
145. #332	C₂²×C₃₁	semidirect
147. C₂×C₃₁	#1561	direct (N × H)
182. C₃₁	C₂×#1561	direct (N × H)
258. C₂	C₃₁×#1561	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×C₃₁×#533	2
C₂×C₃₁×#423	2
C₃₁×#1561	2
C₂×C₃₁×#332	2
C₂×#1561	31

Subgroups — C2×C31×#1561 #21474495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1561 #21474495