Subgroups — C₃₁×#33286 #21490045

Order 7936.

834 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1990	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1957	C₂	1	2	yes	maximal
5	C₃₁×#1936	C₂	1	8	yes	maximal, complemented
6	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#125	—	2	1	no	—
10	C₂×C₃₁×#124	—	2	2	no	—
11	C₂×C₃₁×#102	C₂²	1	8	yes	—
12	C₂×C₃₁×#97	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#390	—	2	1	no	—
15	C₃₁×#390	—	2	1	no	—
16	C₃₁×#390	C₂²	1	4	yes	—
17	C₃₁×#389	—	2	2	no	—
18	C₃₁×#389	C₂²	1	4	yes	—
19	C₃₁×#384	C₂²	1	2	yes	—
20	C₃₁×#382	—	2	1	no	—
21	C₄³×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#340	C₂²	1	8	yes	—
23	C₂²×Q₈×C₃₁	—	2	1	no	—
24	C₂²×D₄×C₃₁	—	2	1	no	—
25	C₂²×D₄×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
28	C₃₁×C₄⋊Q₈	—	2	4	no	—
29	C₃₁×#127	—	2	4	no	—
30	C₃₁×#125	—	4	4	no	—
31	C₃₁×#124	—	4	8	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
38	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄²×C₃₁	D₄	1	2	yes	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×#102	—	2	16	no	—
46	C₃₁×#97	—	2	8	no	—
47	C₄×C₈×C₃₁	—	2	4	no	—
48	C₃₁×#95	—	2	4	no	—
49	C₂⁴×C₃₁	—	2	2	no	—
50	C₂×Q₈×C₃₁	—	4	2	no	—
51	C₂×Q₈×C₃₁	—	4	4	no	—
52	C₂×D₄×C₃₁	—	2	8	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂×D₄×C₃₁	—	4	2	no	—
56	C₂×D₄×C₃₁	—	4	4	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₃₁	—	2	4	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	#41	1	8	yes	—
69	C₄⋊C₄×C₃₁	—	2	8	no	—
70	C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₄⋊C₄×C₃₁	—	4	8	no	—
72	C₄⋊C₄×C₃₁	—	4	2	no	—
73	C₄⋊C₄×C₃₁	—	4	4	no	—
74	C₄⋊C₄×C₃₁	—	4	4	no	—
75	C₂²⋊C₄×C₃₁	—	4	8	no	—
76	C₂²⋊C₄×C₃₁	—	4	8	no	—
77	C₄²×C₃₁	—	2	4	no	—
78	C₄²×C₃₁	—	2	2	no	—
79	C₄²×C₃₁	—	2	2	no	—
80	C₄²×C₃₁	—	2	4	no	—
81	C₄²×C₃₁	#41	1	4	yes	—
82	#33286	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	4	2	no	—
84	C₂³×C₃₁	—	4	4	no	—
85	C₂³×C₃₁	—	4	4	no	—
86	C₂³×C₃₁	—	4	2	no	—
87	C₂³×C₃₁	—	4	2	no	—
88	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₃₁	—	8	4	no	—
90	D₄×C₃₁	—	4	8	no	—
91	D₄×C₃₁	—	8	4	no	—
92	D₄×C₃₁	—	8	4	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	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	#124	1	2	yes	—
105	C₂×C₄×C₃₁	#124	1	2	yes	—
106	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
108	C₂×C₄×C₃₁	#137	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
111	C₈×C₃₁	—	2	8	no	—
112	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
113	#2858	C₂×C₃₁	1	1	yes	characteristic
114	#1990	C₂×C₃₁	1	2	yes	complemented
115	#1957	C₂×C₃₁	1	2	yes	—
116	#1936	C₂×C₃₁	1	8	yes	complemented
117	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
118	C₂²×C₃₁	—	8	2	no	—
119	C₂²×C₃₁	—	8	2	no	—
120	C₂²×C₃₁	—	8	2	no	—
121	C₂²×C₃₁	—	8	4	no	—
122	C₂²×C₃₁	—	8	4	no	—
123	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
124	C₂²×C₃₁	#532	1	2	yes	—
125	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
127	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
128	C₄×C₃₁	—	4	2	no	—
129	C₄×C₃₁	—	4	2	no	—
130	C₄×C₃₁	—	8	2	no	—
131	C₄×C₃₁	—	8	4	no	—
132	C₄×C₃₁	#489	1	4	yes	—
133	C₄×C₃₁	#490	1	4	yes	—
134	C₄×C₃₁	#529	1	4	yes	—
135	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×#125	—	2	1	no	—
138	C₂×#124	—	2	2	no	—
139	C₂×#102	C₂²×C₃₁	1	8	yes	—
140	C₂×#97	C₂²×C₃₁	1	4	yes	—
141	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
142	#390	—	2	1	no	—
143	#390	—	2	1	no	—
144	#390	C₂²×C₃₁	1	4	yes	—
145	#389	—	2	2	no	—
146	#389	C₂²×C₃₁	1	4	yes	—
147	#384	C₂²×C₃₁	1	2	yes	—
148	#382	—	2	1	no	—
149	C₄³	C₂²×C₃₁	1	1	yes	characteristic
150	#340	C₂²×C₃₁	1	8	yes	—
151	C₂×C₃₁	—	16	2	no	—
152	C₂×C₃₁	#2861	1	1	yes	characteristic
153	C₂×C₃₁	C₂×#489	1	1	yes	characteristic
154	C₂×C₃₁	C₂×#490	1	1	yes	characteristic
155	C₂×C₃₁	#3182	1	2	yes	—
156	C₂×C₃₁	#3184	1	2	yes	—
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×D₄	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
161	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
162	C₄⋊Q₈	—	2	4	no	—
163	#127	—	2	4	no	—
164	#125	—	4	4	no	—
165	#124	—	4	8	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
171	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	4	no	—
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₄²	D₄×C₃₁	1	2	yes	—
175	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
178	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
179	#102	—	2	16	no	—
180	#97	—	2	8	no	—
181	C₄×C₈	—	2	4	no	—
182	#95	—	2	4	no	—
183	C₃₁	#33286	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	—	2	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×D₄	—	2	8	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	#41×C₃₁	1	8	yes	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	8	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	8	no	—
211	C₂²⋊C₄	—	4	8	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	#41×C₃₁	1	4	yes	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
223	Q₈	—	8	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	8	4	no	—
226	D₄	—	8	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₃₁×#124	1	2	yes	—
239	C₂×C₄	C₃₁×#124	1	2	yes	—
240	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
241	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
242	C₂×C₄	C₃₁×#137	1	2	yes	—
243	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
244	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
245	C₈	—	2	8	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
252	C₂²	C₃₁×#532	1	2	yes	—
253	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
255	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	8	2	no	—
259	C₄	—	8	4	no	—
260	C₄	C₃₁×#489	1	4	yes	—
261	C₄	C₃₁×#490	1	4	yes	—
262	C₄	C₃₁×#529	1	4	yes	—
263	C₂	—	16	2	no	—
264	C₂	C₃₁×#2861	1	1	yes	characteristic
265	C₂	C₂×C₃₁×#489	1	1	yes	characteristic
266	C₂	C₂×C₃₁×#490	1	1	yes	characteristic
267	C₂	C₃₁×#3182	1	2	yes	—
268	C₂	C₃₁×#3184	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2859	C₂	semidirect
3. C₃₁×#1990	C₂	semidirect
5. C₃₁×#1936	C₂	semidirect
6. C₄×C₃₁×#97	C₂	semidirect
82. #33286	C₃₁	direct (N × H)
112. #2859	C₂×C₃₁	semidirect
114. #1990	C₂×C₃₁	semidirect
116. #1936	C₂×C₃₁	semidirect
117. C₄×#97	C₂×C₃₁	semidirect
183. C₃₁	#33286	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×#2859	2
C₃₁×#2858	2
C₃₁×#1990	2
C₃₁×#1957	2
C₃₁×#1936	2
C₄×C₃₁×#97	2
#33286	31

Subgroups — C31×#33286 #21490045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33286 #21490045