Subgroups — C₂×C₃₁×#2884 #21516828

Order 7936.

4818 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	C₂²×C₃₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#519	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2884	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#410	C₂	1	12	yes	maximal, complemented
5	C₂×C₃₁×2₋¹⁺⁴	—	2	8	no	—
6	C₂×C₃₁×2₋¹⁺⁴	—	2	8	no	—
7	C₂×C₃₁×2₋¹⁺⁴	C₄	1	8	yes	complemented
8	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	8	yes	—
9	C₂²×#41×C₃₁	C₄	1	4	yes	complemented
10	C₂²×#41×C₃₁	C₂²	1	6	yes	—
11	C₂³×Q₈×C₃₁	C₄	1	4	yes	complemented
12	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#519	—	2	8	no	—
14	C₃₁×#519	C₂²	1	16	yes	complemented
15	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
16	C₂×C₃₁×#117	C₂²	1	12	yes	—
17	C₃₁×#410	C₂²	1	96	yes	complemented
18	C₂×C₃₁×#99	C₂²	1	16	yes	—
19	C₃₁×2₋¹⁺⁴	—	2	64	no	—
20	C₃₁×2₋¹⁺⁴	—	2	64	no	—
21	C₂×#41×C₃₁	—	2	16	no	—
22	C₂×#41×C₃₁	—	2	16	no	—
23	C₂×#41×C₃₁	—	2	24	no	—
24	C₂×#41×C₃₁	—	2	24	no	—
25	C₂×#41×C₃₁	C₂×C₄	1	24	yes	—
26	C₂×#41×C₃₁	C₂×C₄	1	32	yes	complemented
27	C₂×#41×C₃₁	C₂³	1	24	yes	—
28	C₂²×Q₈×C₃₁	—	2	16	no	—
29	C₂²×Q₈×C₃₁	—	2	16	no	—
30	C₂²×Q₈×C₃₁	C₂×C₄	1	6	yes	—
31	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
32	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
33	C₂²×Q₈×C₃₁	C₂×C₄	1	32	yes	complemented
34	C₂²×Q₈×C₃₁	D₄	1	8	yes	—
35	C₂²×Q₈×C₃₁	D₄	1	8	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
37	C₂²×D₄×C₃₁	C₂×C₄	1	6	yes	—
38	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
39	C₂³×C₄×C₃₁	C₂×C₄	1	12	yes	—
40	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
41	C₄×Q₈×C₃₁	—	2	64	no	—
42	C₃₁×#117	—	2	24	no	—
43	C₃₁×#117	C₂³	1	48	yes	—
44	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
45	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
46	C₂×C₄²×C₃₁	—	2	12	no	—
47	C₃₁×#99	C₂³	1	64	yes	—
48	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
49	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
50	#41×C₃₁	—	2	128	no	—
51	#41×C₃₁	—	2	96	no	—
52	#41×C₃₁	—	2	96	no	—
53	C₂×Q₈×C₃₁	—	2	128	no	—
54	C₂×Q₈×C₃₁	—	2	12	no	—
55	C₂×Q₈×C₃₁	—	2	12	no	—
56	C₂×Q₈×C₃₁	C₂²⋊C₄	1	16	yes	—
57	C₂×Q₈×C₃₁	C₂²⋊C₄	1	16	yes	—
58	C₂×Q₈×C₃₁	C₂²⋊C₄	1	32	yes	—
59	C₂×Q₈×C₃₁	C₂²×C₄	1	24	yes	—
60	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	—
61	C₂×Q₈×C₃₁	C₂×D₄	1	32	yes	—
62	C₂×D₄×C₃₁	—	2	8	no	—
63	C₂×D₄×C₃₁	—	2	8	no	—
64	C₂×D₄×C₃₁	—	2	12	no	—
65	C₂×D₄×C₃₁	—	2	12	no	—
66	C₂×D₄×C₃₁	C₂²×C₄	1	24	yes	—
67	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
68	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₃₁	—	2	24	no	—
70	C₂²×C₄×C₃₁	—	2	24	no	—
71	C₂²×C₄×C₃₁	—	2	8	no	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₃₁	C₂²×C₄	1	48	yes	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
79	C₄⋊C₄×C₃₁	—	2	48	no	—
80	C₂²⋊C₄×C₃₁	—	2	8	no	—
81	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
82	C₄²×C₃₁	—	2	48	no	—
83	C₂×#2884	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	4	no	—
85	C₂³×C₃₁	—	2	4	no	—
86	C₂³×C₃₁	—	4	2	no	—
87	C₂³×C₃₁	—	4	8	no	—
88	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
89	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
90	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
92	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
93	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
94	Q₈×C₃₁	—	2	48	no	—
95	Q₈×C₃₁	C₂×C₂²⋊C₄	1	64	yes	—
96	D₄×C₃₁	—	2	16	no	—
97	D₄×C₃₁	—	2	16	no	—
98	D₄×C₃₁	—	2	48	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	96	no	—
102	C₂×C₄×C₃₁	—	2	16	no	—
103	C₂×C₄×C₃₁	—	4	16	no	—
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	24	yes	—
106	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
107	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₃₁	C₂³×C₄	1	12	yes	—
109	C₂×C₄×C₃₁	C₂²×D₄	1	24	yes	—
110	C₂²×2₋¹⁺⁴	C₂×C₃₁	1	1	yes	characteristic
111	C₂×#519	C₂×C₃₁	1	2	yes	complemented
112	#2884	C₂×C₃₁	1	16	yes	direct product factor
113	C₂×#410	C₂×C₃₁	1	12	yes	complemented
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	2	8	no	—
117	C₂²×C₃₁	—	4	1	no	—
118	C₂²×C₃₁	—	4	2	no	—
119	C₂²×C₃₁	—	4	1	no	—
120	C₂²×C₃₁	—	4	8	no	—
121	C₂²×C₃₁	—	4	8	no	—
122	C₂²×C₃₁	—	4	8	no	—
123	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	2	yes	—
125	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
127	C₂²×C₃₁	C₂³×D₄	1	2	yes	—
128	C₄×C₃₁	—	2	8	no	—
129	C₄×C₃₁	—	4	16	no	—
130	C₄×C₃₁	C₂²×C₂²⋊C₄	1	24	yes	—
131	C₂×2₋¹⁺⁴	—	2	8	no	—
132	C₂×2₋¹⁺⁴	—	2	8	no	—
133	C₂×2₋¹⁺⁴	C₄×C₃₁	1	8	yes	complemented
134	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	8	yes	—
135	C₂²×#41	C₄×C₃₁	1	4	yes	complemented
136	C₂²×#41	C₂²×C₃₁	1	6	yes	—
137	C₂³×Q₈	C₄×C₃₁	1	4	yes	complemented
138	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
139	#519	—	2	8	no	—
140	#519	C₂²×C₃₁	1	16	yes	complemented
141	C₂×C₄×Q₈	—	2	8	no	—
142	C₂×#117	C₂²×C₃₁	1	12	yes	—
143	#410	C₂²×C₃₁	1	96	yes	complemented
144	C₂×#99	C₂²×C₃₁	1	16	yes	—
145	C₂×C₃₁	—	2	2	no	—
146	C₂×C₃₁	—	2	1	no	—
147	C₂×C₃₁	—	2	1	no	—
148	C₂×C₃₁	—	2	2	no	—
149	C₂×C₃₁	—	4	8	no	—
150	C₂×C₃₁	#2884	1	2	yes	direct product factor
151	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
152	2₋¹⁺⁴	—	2	64	no	—
153	2₋¹⁺⁴	—	2	64	no	—
154	C₂×#41	—	2	16	no	—
155	C₂×#41	—	2	16	no	—
156	C₂×#41	—	2	24	no	—
157	C₂×#41	—	2	24	no	—
158	C₂×#41	C₂×C₄×C₃₁	1	24	yes	—
159	C₂×#41	C₂×C₄×C₃₁	1	32	yes	complemented
160	C₂×#41	C₂³×C₃₁	1	24	yes	—
161	C₂²×Q₈	—	2	16	no	—
162	C₂²×Q₈	—	2	16	no	—
163	C₂²×Q₈	C₂×C₄×C₃₁	1	6	yes	—
164	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
165	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
166	C₂²×Q₈	C₂×C₄×C₃₁	1	32	yes	complemented
167	C₂²×Q₈	D₄×C₃₁	1	8	yes	—
168	C₂²×Q₈	D₄×C₃₁	1	8	yes	—
169	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
170	C₂²×D₄	C₂×C₄×C₃₁	1	6	yes	—
171	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
172	C₂³×C₄	C₂×C₄×C₃₁	1	12	yes	—
173	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
174	C₄×Q₈	—	2	64	no	—
175	#117	—	2	24	no	—
176	#117	C₂³×C₃₁	1	48	yes	—
177	C₂×C₄⋊C₄	—	2	12	no	—
178	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
179	C₂×C₄²	—	2	12	no	—
180	#99	C₂³×C₃₁	1	64	yes	—
181	C₃₁	C₂×#2884	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
183	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
184	#41	—	2	128	no	—
185	#41	—	2	96	no	—
186	#41	—	2	96	no	—
187	C₂×Q₈	—	2	128	no	—
188	C₂×Q₈	—	2	12	no	—
189	C₂×Q₈	—	2	12	no	—
190	C₂×Q₈	C₂²⋊C₄×C₃₁	1	16	yes	—
191	C₂×Q₈	C₂²⋊C₄×C₃₁	1	16	yes	—
192	C₂×Q₈	C₂²⋊C₄×C₃₁	1	32	yes	—
193	C₂×Q₈	C₂²×C₄×C₃₁	1	24	yes	—
194	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	—
195	C₂×Q₈	C₂×D₄×C₃₁	1	32	yes	—
196	C₂×D₄	—	2	8	no	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	2	12	no	—
199	C₂×D₄	—	2	12	no	—
200	C₂×D₄	C₂²×C₄×C₃₁	1	24	yes	—
201	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
202	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
203	C₂²×C₄	—	2	24	no	—
204	C₂²×C₄	—	2	24	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₃₁	1	48	yes	—
208	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
209	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
211	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
212	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
213	C₄⋊C₄	—	2	48	no	—
214	C₂²⋊C₄	—	2	8	no	—
215	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
216	C₄²	—	2	48	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	8	no	—
221	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
222	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
223	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
224	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
225	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
226	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
227	Q₈	—	2	48	no	—
228	Q₈	C₂×C₂²⋊C₄×C₃₁	1	64	yes	—
229	D₄	—	2	16	no	—
230	D₄	—	2	16	no	—
231	D₄	—	2	48	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	96	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	4	16	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	24	yes	—
240	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
241	C₂×C₄	C₂³×C₄×C₃₁	1	12	yes	—
242	C₂×C₄	C₂²×D₄×C₃₁	1	24	yes	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	8	no	—
252	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
253	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	2	yes	—
254	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
255	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
256	C₂²	C₂³×D₄×C₃₁	1	2	yes	—
257	C₄	—	2	8	no	—
258	C₄	—	4	16	no	—
259	C₄	C₂²×C₂²⋊C₄×C₃₁	1	24	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	2	no	—
264	C₂	—	4	8	no	—
265	C₂	C₃₁×#2884	1	2	yes	direct product factor
266	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×2₋¹⁺⁴	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#519	C₂	semidirect
3. C₃₁×#2884	C₂	direct (N × H)
4. C₂×C₃₁×#410	C₂	semidirect
7. C₂×C₃₁×2₋¹⁺⁴	C₄	semidirect
9. C₂²×#41×C₃₁	C₄	semidirect
11. C₂³×Q₈×C₃₁	C₄	semidirect
14. C₃₁×#519	C₂²	semidirect
17. C₃₁×#410	C₂²	semidirect
26. C₂×#41×C₃₁	C₂×C₄	semidirect
33. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
83. C₂×#2884	C₃₁	direct (N × H)
111. C₂×#519	C₂×C₃₁	semidirect
112. #2884	C₂×C₃₁	direct (N × H)
113. C₂×#410	C₂×C₃₁	semidirect
133. C₂×2₋¹⁺⁴	C₄×C₃₁	semidirect
135. C₂²×#41	C₄×C₃₁	semidirect
137. C₂³×Q₈	C₄×C₃₁	semidirect
140. #519	C₂²×C₃₁	semidirect
143. #410	C₂²×C₃₁	semidirect
150. C₂×C₃₁	#2884	direct (N × H)
159. C₂×#41	C₂×C₄×C₃₁	semidirect
166. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
181. C₃₁	C₂×#2884	direct (N × H)
265. C₂	C₃₁×#2884	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×C₃₁×2₋¹⁺⁴	2
C₂×C₃₁×#519	2
C₃₁×#2884	2
C₂×C₃₁×#410	2
C₂×#2884	31

Subgroups — C2×C31×#2884 #21516828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2884 #21516828