Subgroups — C₂×C₃₁×#2086 #21480422

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#389	C₂	1	2	yes	maximal
4	C₃₁×#2086	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	2	yes	maximal
6	C₂×C₃₁×#340	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#107	—	2	4	no	—
14	C₂×C₃₁×#107	C₂²	1	4	yes	—
15	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
17	C₃₁×#389	—	2	4	no	—
18	C₃₁×#389	C₂²	1	8	yes	—
19	C₂×C₃₁×#95	—	2	2	no	—
20	C₃₁×#340	C₂²	1	8	yes	—
21	C₃₁×#340	C₂²	1	4	yes	—
22	C₂³×C₄×C₃₁	—	2	2	no	—
23	C₂³×C₄×C₃₁	—	2	2	no	—
24	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	—	2	4	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	4	no	—
30	C₂²×C₈×C₃₁	D₄	1	4	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
40	C₂×C₄⋊C₄×C₃₁	Q₈	1	8	yes	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
43	C₂×C₄²×C₃₁	—	2	2	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	D₄	1	2	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#107	—	2	8	no	—
48	C₃₁×C₄⋊C₈	—	2	8	no	—
49	C₃₁×#95	—	2	8	no	—
50	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	8	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	—	4	4	no	—
60	C₂²×C₄×C₃₁	—	4	4	no	—
61	C₂²×C₄×C₃₁	—	4	4	no	—
62	C₂²×C₄×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	D₈	1	4	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₃₁	#41	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₃₁	—	2	8	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	4	4	no	—
77	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₃₁	—	2	8	no	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	4	4	no	—
83	C₄⋊C₄×C₃₁	—	4	16	no	—
84	C₄⋊C₄×C₃₁	C₂×Q₈	1	8	yes	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	2	2	no	—
87	C₄²×C₃₁	—	2	2	no	—
88	C₂×#2086	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	#121	1	2	yes	—
90	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
91	C₂³×C₃₁	C₂×D₈	1	2	yes	—
92	C₂³×C₃₁	#135	1	2	yes	—
93	C₂³×C₃₁	#137	1	4	yes	—
94	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	8	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	8	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	16	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	8	8	no	—
109	C₂×C₄×C₃₁	#121	1	2	yes	—
110	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₃₁	#123	1	4	yes	—
112	C₂×C₄×C₃₁	#125	1	8	yes	—
113	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
114	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
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	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
119	C₈×C₃₁	—	2	4	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic
122	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
123	C₂×#389	C₂×C₃₁	1	2	yes	—
124	#2086	C₂×C₃₁	1	8	yes	direct product factor
125	C₂×#340	C₂×C₃₁	1	2	yes	—
126	C₂×#340	C₂×C₃₁	1	1	yes	characteristic
127	C₂²×C₃₁	#398	1	4	yes	—
128	C₂²×C₃₁	#466	1	4	yes	—
129	C₂²×C₃₁	#470	1	4	yes	—
130	C₂²×C₃₁	#474	1	8	yes	—
131	C₂²×C₃₁	#479	1	8	yes	—
132	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
134	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
135	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×#137	1	2	yes	—
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	4	2	no	—
139	C₄×C₃₁	—	4	8	no	—
140	C₄×C₃₁	—	8	4	no	—
141	C₄×C₃₁	#398	1	4	yes	—
142	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
143	C₄×C₃₁	C₂×#125	1	2	yes	—
144	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂²×C₄⋊C₄	—	2	2	no	—
148	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
149	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
150	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
151	C₂×#107	—	2	4	no	—
152	C₂×#107	C₂²×C₃₁	1	4	yes	—
153	C₂×C₄⋊C₈	—	2	4	no	—
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
155	#389	—	2	4	no	—
156	#389	C₂²×C₃₁	1	8	yes	—
157	C₂×#95	—	2	2	no	—
158	#340	C₂²×C₃₁	1	8	yes	—
159	#340	C₂²×C₃₁	1	4	yes	—
160	C₂×C₃₁	#2086	1	8	yes	direct product factor
161	C₂×C₃₁	C₂×#398	1	1	yes	characteristic
162	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
163	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#474	1	2	yes	—
165	C₂×C₃₁	C₂×#479	1	2	yes	—
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	—	2	2	no	—
168	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
169	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	D₄×C₃₁	1	4	yes	—
175	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
176	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	4	8	no	—
184	C₂×C₄⋊C₄	Q₈×C₃₁	1	8	yes	—
185	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	D₄×C₃₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
191	#107	—	2	8	no	—
192	C₄⋊C₈	—	2	8	no	—
193	#95	—	2	8	no	—
194	C₃₁	C₂×#2086	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	D₈×C₃₁	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
213	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
214	C₂²×C₄	#41×C₃₁	1	2	yes	—
215	C₂²×C₄	#41×C₃₁	1	4	yes	—
216	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	16	no	—
229	C₄⋊C₄	C₂×Q₈×C₃₁	1	8	yes	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₂³	C₃₁×#121	1	2	yes	—
234	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
235	C₂³	C₂×D₈×C₃₁	1	2	yes	—
236	C₂³	C₃₁×#135	1	2	yes	—
237	C₂³	C₃₁×#137	1	4	yes	—
238	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	16	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	8	no	—
253	C₂×C₄	C₃₁×#121	1	2	yes	—
254	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
255	C₂×C₄	C₃₁×#123	1	4	yes	—
256	C₂×C₄	C₃₁×#125	1	8	yes	—
257	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
258	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
259	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
261	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
262	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
263	C₈	—	2	4	no	—
264	C₈	—	4	2	no	—
265	C₂²	C₃₁×#398	1	4	yes	—
266	C₂²	C₃₁×#466	1	4	yes	—
267	C₂²	C₃₁×#470	1	4	yes	—
268	C₂²	C₃₁×#474	1	8	yes	—
269	C₂²	C₃₁×#479	1	8	yes	—
270	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
271	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
272	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
273	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
274	C₂²	C₂×C₃₁×#137	1	2	yes	—
275	C₄	—	2	4	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	8	no	—
278	C₄	—	8	4	no	—
279	C₄	C₃₁×#398	1	4	yes	—
280	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
281	C₄	C₂×C₃₁×#125	1	2	yes	—
282	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
283	C₂	C₃₁×#2086	1	8	yes	direct product factor
284	C₂	C₂×C₃₁×#398	1	1	yes	characteristic
285	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
286	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
287	C₂	C₂×C₃₁×#474	1	2	yes	—
288	C₂	C₂×C₃₁×#479	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×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×#2086	C₂	direct (N × H)
88. C₂×#2086	C₃₁	direct (N × H)
124. #2086	C₂×C₃₁	direct (N × H)
160. C₂×C₃₁	#2086	direct (N × H)
194. C₃₁	C₂×#2086	direct (N × H)
283. C₂	C₃₁×#2086	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×#107	2
C₂²×C₃₁×C₄⋊C₈	2
C₂×C₃₁×#389	2
C₃₁×#2086	2
C₂×C₃₁×#340	2
C₂×#2086	31

Subgroups — C2×C31×#2086 #21480422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2086 #21480422