Subgroups — C₂×C₃₁×#539 #16843384

Order 3968.

1144 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#539	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃₁×#126	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#125	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#123	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#123	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#126	C₂²	1	16	yes	complemented
19	C₃₁×#125	C₂²	1	16	yes	complemented
20	C₃₁×#123	C₂²	1	16	yes	complemented
21	C₃₁×#123	C₂²	1	16	yes	complemented
22	C₃₁×C₂²⋊Q₈	C₂²	1	8	yes	complemented
23	C₃₁×C₂²⋊Q₈	C₂²	1	16	yes	complemented
24	C₃₁×#121	C₂²	1	8	yes	—
25	C₄×D₄×C₃₁	C₂²	1	16	yes	complemented
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
35	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
41	C₂⁴×C₃₁	C₂³	1	2	yes	—
42	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₃₁	—	4	2	no	—
44	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
45	C₂×D₄×C₃₁	—	2	8	no	—
46	C₂×D₄×C₃₁	—	4	2	no	—
47	C₂×D₄×C₃₁	C₂³	1	8	yes	—
48	C₂×D₄×C₃₁	C₂³	1	4	yes	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	—	4	4	no	—
52	C₂²×C₄×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
54	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
63	C₂²×C₄×C₃₁	C₂³	1	8	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
65	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
66	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
67	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
68	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
69	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
70	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
71	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
72	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
73	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
74	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
75	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
77	C₄²×C₃₁	C₂³	1	8	yes	—
78	C₂³×C₃₁	—	2	2	no	—
79	C₂³×C₃₁	—	2	2	no	—
80	C₂³×C₃₁	—	2	4	no	—
81	C₂³×C₃₁	—	2	4	no	—
82	C₂³×C₃₁	—	2	4	no	—
83	C₂³×C₃₁	#41	1	4	yes	—
84	C₂³×C₃₁	C₂⁴	1	4	yes	—
85	C₂³×C₃₁	C₂⁴	1	2	yes	—
86	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
87	Q₈×C₃₁	—	4	4	no	—
88	D₄×C₃₁	—	2	16	no	—
89	D₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	2	8	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	—	4	8	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
102	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
103	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
104	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
105	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
106	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
107	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
108	C₂×#539	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₃₁	—	2	8	no	—
110	C₂²×C₃₁	—	2	2	no	—
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	2	4	no	—
113	C₂²×C₃₁	—	2	4	no	—
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	—	4	8	no	—
116	C₂²×C₃₁	C₂×#41	1	4	yes	—
117	C₂²×C₃₁	C₂×#41	1	2	yes	—
118	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
120	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
121	C₄×C₃₁	—	2	4	no	—
122	C₄×C₃₁	—	2	4	no	—
123	C₄×C₃₁	—	2	4	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	2	4	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	2	4	no	—
128	#539	C₂×C₃₁	1	16	yes	direct product factor
129	C₂×#126	C₂×C₃₁	1	2	yes	complemented
130	C₂×#125	C₂×C₃₁	1	2	yes	complemented
131	C₂×#123	C₂×C₃₁	1	2	yes	complemented
132	C₂×#123	C₂×C₃₁	1	2	yes	complemented
133	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
134	C₂×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
135	C₂×#121	C₂×C₃₁	1	1	yes	characteristic
136	C₂×C₄×D₄	C₂×C₃₁	1	2	yes	complemented
137	C₂²×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂×C₃₁	—	2	4	no	—
139	C₂×C₃₁	—	4	4	no	—
140	C₂×C₃₁	#539	1	4	yes	direct product factor
141	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
142	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
144	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
146	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
147	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
151	#126	C₂²×C₃₁	1	16	yes	complemented
152	#125	C₂²×C₃₁	1	16	yes	complemented
153	#123	C₂²×C₃₁	1	16	yes	complemented
154	#123	C₂²×C₃₁	1	16	yes	complemented
155	C₂²⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
156	C₂²⋊Q₈	C₂²×C₃₁	1	16	yes	complemented
157	#121	C₂²×C₃₁	1	8	yes	—
158	C₄×D₄	C₂²×C₃₁	1	16	yes	complemented
159	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
168	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
173	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
174	C₃₁	C₂×#539	1	1	yes	characteristic, direct product factor, Sylow
175	C₂⁴	C₂³×C₃₁	1	2	yes	—
176	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	C₂³×C₃₁	1	8	yes	—
182	C₂×D₄	C₂³×C₃₁	1	4	yes	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
188	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
189	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
191	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
195	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
196	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
197	C₂²×C₄	C₂³×C₃₁	1	8	yes	—
198	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
199	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
200	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
201	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
202	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
203	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
204	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
205	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
206	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
207	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
208	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
209	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
210	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
211	C₄²	C₂³×C₃₁	1	8	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	4	no	—
217	C₂³	#41×C₃₁	1	4	yes	—
218	C₂³	C₂⁴×C₃₁	1	4	yes	—
219	C₂³	C₂⁴×C₃₁	1	2	yes	—
220	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	D₄	—	2	16	no	—
223	D₄	—	4	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
236	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
237	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
238	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
239	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
240	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
241	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	4	8	no	—
249	C₂²	C₂×#41×C₃₁	1	4	yes	—
250	C₂²	C₂×#41×C₃₁	1	2	yes	—
251	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
252	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
253	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₂	—	2	4	no	—
262	C₂	—	4	4	no	—
263	C₂	C₃₁×#539	1	4	yes	direct product factor
264	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
265	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#539	C₂	direct (N × H)
2. C₂×C₃₁×#126	C₂	semidirect
3. C₂×C₃₁×#125	C₂	semidirect
4. C₂×C₃₁×#123	C₂	semidirect
5. C₂×C₃₁×#123	C₂	semidirect
6. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
7. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
9. C₂×C₄×D₄×C₃₁	C₂	semidirect
10. C₂²×C₄⋊C₄×C₃₁	C₂	semidirect
18. C₃₁×#126	C₂²	semidirect
19. C₃₁×#125	C₂²	semidirect
20. C₃₁×#123	C₂²	semidirect
21. C₃₁×#123	C₂²	semidirect
22. C₃₁×C₂²⋊Q₈	C₂²	semidirect
23. C₃₁×C₂²⋊Q₈	C₂²	semidirect
25. C₄×D₄×C₃₁	C₂²	semidirect
34. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
108. C₂×#539	C₃₁	direct (N × H)
128. #539	C₂×C₃₁	direct (N × H)
129. C₂×#126	C₂×C₃₁	semidirect
130. C₂×#125	C₂×C₃₁	semidirect
131. C₂×#123	C₂×C₃₁	semidirect
132. C₂×#123	C₂×C₃₁	semidirect
133. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
134. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
136. C₂×C₄×D₄	C₂×C₃₁	semidirect
137. C₂²×C₄⋊C₄	C₂×C₃₁	semidirect
140. C₂×C₃₁	#539	direct (N × H)
151. #126	C₂²×C₃₁	semidirect
152. #125	C₂²×C₃₁	semidirect
153. #123	C₂²×C₃₁	semidirect
154. #123	C₂²×C₃₁	semidirect
155. C₂²⋊Q₈	C₂²×C₃₁	semidirect
156. C₂²⋊Q₈	C₂²×C₃₁	semidirect
158. C₄×D₄	C₂²×C₃₁	semidirect
167. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
174. C₃₁	C₂×#539	direct (N × H)
263. C₂	C₃₁×#539	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×#539	2
C₂×C₃₁×#126	2
C₂×C₃₁×#125	2
C₂×C₃₁×#123	2
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₃₁×#121	2
C₂×C₄×D₄×C₃₁	2
C₂²×C₄⋊C₄×C₃₁	2
C₂×#539	31

Subgroups — C2×C31×#539 #16843384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#539 #16843384