Subgroups — C₂×C₃₁×#2090 #21480432

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#428	C₂	1	2	yes	maximal
2	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2090	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#343	C₂	1	2	yes	maximal
6	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#428	—	2	4	no	—
13	C₃₁×#428	C₂²	1	8	yes	—
14	C₂×C₃₁×#107	—	2	2	no	—
15	C₂×C₃₁×#106	—	2	2	no	—
16	C₂×C₃₁×#106	—	2	2	no	—
17	C₂×C₃₁×#106	—	2	2	no	—
18	C₂×C₃₁×#106	C₂²	1	4	yes	—
19	C₃₁×#424	—	2	2	no	—
20	C₃₁×#424	C₂²	1	4	yes	—
21	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
22	C₃₁×#343	C₂²	1	8	yes	—
23	C₃₁×#341	C₂²	1	4	yes	—
24	C₂³×C₄×C₃₁	—	2	2	no	—
25	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	4	no	—
28	C₂×M₁₆×C₃₁	—	2	1	no	—
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	Q₈	1	8	yes	—
31	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
32	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	D₄	1	2	yes	—
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×#117	—	2	1	no	—
40	C₃₁×#117	—	2	1	no	—
41	C₃₁×#117	D₄	1	2	yes	—
42	C₃₁×#117	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄²×C₃₁	—	2	1	no	—
52	C₃₁×#107	—	2	8	no	—
53	C₃₁×#106	—	2	8	no	—
54	C₃₁×#106	—	2	8	no	—
55	C₃₁×C₄⋊C₈	—	2	8	no	—
56	C₂⁴×C₃₁	#41	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	—	8	4	no	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	#41	1	4	yes	—
69	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₃₁	—	2	8	no	—
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	—	2	4	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	C₂×Q₈	1	8	yes	—
75	C₂×C₈×C₃₁	—	2	8	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	4	2	no	—
87	C₄⋊C₄×C₃₁	—	4	2	no	—
88	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
89	C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₄²×C₃₁	—	2	2	no	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₂×#2090	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	4	no	—
95	C₂³×C₃₁	#121	1	1	yes	characteristic
96	C₂³×C₃₁	#121	1	1	yes	characteristic
97	C₂³×C₃₁	#125	1	4	yes	—
98	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	8	4	no	—
103	C₂×C₄×C₃₁	—	8	4	no	—
104	C₂×C₄×C₃₁	—	8	4	no	—
105	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₃₁	#123	1	2	yes	—
110	C₂×C₄×C₃₁	#123	1	2	yes	—
111	C₂×C₄×C₃₁	#125	1	4	yes	—
112	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
118	C₈×C₃₁	—	2	8	no	—
119	C₈×C₃₁	—	2	4	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₂×#428	C₂×C₃₁	1	2	yes	—
122	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic
123	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
124	#2090	C₂×C₃₁	1	8	yes	direct product factor
125	C₂×#343	C₂×C₃₁	1	2	yes	—
126	C₂×#341	C₂×C₃₁	1	1	yes	characteristic
127	C₂²×C₃₁	—	2	8	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	#398	1	2	yes	—
131	C₂²×C₃₁	#398	1	2	yes	—
132	C₂²×C₃₁	#471	1	4	yes	—
133	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×#125	1	2	yes	—
135	C₄×C₃₁	—	4	4	no	—
136	C₄×C₃₁	—	8	4	no	—
137	C₄×C₃₁	#398	1	2	yes	—
138	C₄×C₃₁	#398	1	2	yes	—
139	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
140	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
141	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
143	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
144	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
146	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
147	#428	—	2	4	no	—
148	#428	C₂²×C₃₁	1	8	yes	—
149	C₂×#107	—	2	2	no	—
150	C₂×#106	—	2	2	no	—
151	C₂×#106	—	2	2	no	—
152	C₂×#106	—	2	2	no	—
153	C₂×#106	C₂²×C₃₁	1	4	yes	—
154	#424	—	2	2	no	—
155	#424	C₂²×C₃₁	1	4	yes	—
156	C₂×C₄⋊C₈	—	2	2	no	—
157	#343	C₂²×C₃₁	1	8	yes	—
158	#341	C₂²×C₃₁	1	4	yes	—
159	C₂×C₃₁	—	2	2	no	—
160	C₂×C₃₁	—	2	2	no	—
161	C₂×C₃₁	#2090	1	4	yes	direct product factor
162	C₂×C₃₁	C₂×#398	1	1	yes	characteristic
163	C₂×C₃₁	C₂×#471	1	2	yes	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	Q₈×C₃₁	1	8	yes	—
171	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
172	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	D₄×C₃₁	1	2	yes	—
177	C₂²×C₈	D₄×C₃₁	1	2	yes	—
178	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
179	#117	—	2	1	no	—
180	#117	—	2	1	no	—
181	#117	D₄×C₃₁	1	2	yes	—
182	#117	C₂³×C₃₁	1	2	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	#107	—	2	8	no	—
193	#106	—	2	8	no	—
194	#106	—	2	8	no	—
195	C₄⋊C₈	—	2	8	no	—
196	C₃₁	C₂×#2090	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	#41×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	4	no	—
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
208	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	#41×C₃₁	1	4	yes	—
210	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
211	M₁₆	—	2	8	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	C₂×Q₈×C₃₁	1	8	yes	—
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	C₃₁×#121	1	1	yes	characteristic
236	C₂³	C₃₁×#121	1	1	yes	characteristic
237	C₂³	C₃₁×#125	1	4	yes	—
238	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
246	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
247	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
248	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
249	C₂×C₄	C₃₁×#123	1	2	yes	—
250	C₂×C₄	C₃₁×#123	1	2	yes	—
251	C₂×C₄	C₃₁×#125	1	4	yes	—
252	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
253	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
257	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
258	C₈	—	2	8	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₃₁×#398	1	2	yes	—
265	C₂²	C₃₁×#398	1	2	yes	—
266	C₂²	C₃₁×#471	1	4	yes	—
267	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
268	C₂²	C₂×C₃₁×#125	1	2	yes	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₃₁×#398	1	2	yes	—
272	C₄	C₃₁×#398	1	2	yes	—
273	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
274	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
275	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	2	2	no	—
278	C₂	C₃₁×#2090	1	4	yes	direct product factor
279	C₂	C₂×C₃₁×#398	1	1	yes	characteristic
280	C₂	C₂×C₃₁×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2090	C₂	direct (N × H)
93. C₂×#2090	C₃₁	direct (N × H)
124. #2090	C₂×C₃₁	direct (N × H)
161. C₂×C₃₁	#2090	direct (N × H)
196. C₃₁	C₂×#2090	direct (N × H)
278. C₂	C₃₁×#2090	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×#428	2
C₂²×C₃₁×#106	2
C₂×C₃₁×#424	2
C₃₁×#2090	2
C₂×C₃₁×#343	2
C₂×C₃₁×#341	2
C₂×#2090	31

Subgroups — C2×C31×#2090 #21480432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2090 #21480432