Subgroups — C₃₁×#33860 #21490619

Order 7936.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3417	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3293	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2222	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2214	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2195	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#578	—	2	2	no	—
10	C₃₁×#578	C₂²	1	4	yes	—
11	C₃₁×#510	—	2	4	no	—
12	C₃₁×#510	C₂²	1	4	yes	—
13	C₃₁×#508	—	2	2	no	—
14	C₃₁×#508	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×Q₃₂	—	2	2	no	—
16	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
17	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
18	C₃₁×#504	C₂²	1	2	yes	—
19	C₃₁×#494	C₂²	1	2	yes	—
20	C₃₁×#473	—	2	2	no	—
21	C₃₁×#473	C₂²	1	4	yes	—
22	C₃₁×#456	—	2	4	no	—
23	C₃₁×#443	C₂²	1	2	yes	—
24	C₃₁×#443	C₂²	1	1	yes	characteristic
25	C₃₁×#368	C₂²	1	1	yes	characteristic
26	C₃₁×#364	C₂²	1	1	yes	characteristic
27	C₃₁×#362	C₂²	1	4	yes	—
28	C₃₁×#359	C₂²	1	2	yes	—
29	C₃₁×#350	C₂²	1	2	yes	—
30	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×2₋¹⁺⁴	—	2	4	no	—
32	C₃₁×#137	—	2	4	no	—
33	C₃₁×#137	—	2	4	no	—
34	C₃₁×#137	—	4	4	no	—
35	C₃₁×#137	—	4	4	no	—
36	C₃₁×#135	—	2	4	no	—
37	C₃₁×#135	—	4	2	no	—
38	C₃₁×#135	D₄	1	1	yes	characteristic
39	C₃₁×#135	C₂³	1	2	yes	—
40	C₂×Q₁₆×C₃₁	—	2	4	no	—
41	C₂×Q₁₆×C₃₁	—	2	2	no	—
42	C₂×Q₁₆×C₃₁	—	4	2	no	—
43	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
44	C₃₁×#131	—	2	2	no	—
45	C₃₁×#131	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×Q₃₂	—	2	4	no	—
48	C₃₁×Q₃₂	—	2	2	no	—
49	C₃₁×Q₃₂	—	2	2	no	—
50	C₃₁×Q₃₂	—	4	2	no	—
51	C₃₁×Q₃₂	D₄	1	1	yes	characteristic
52	C₃₁×SD₃₂	—	2	4	no	—
53	C₃₁×SD₃₂	D₄	1	2	yes	—
54	C₃₁×D₁₆	D₄	1	1	yes	characteristic
55	C₃₁×M₃₂	—	2	2	no	—
56	C₃₁×M₃₂	C₂³	1	2	yes	—
57	C₂×C₁₆×C₃₁	—	2	2	no	—
58	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#108	D₄	1	1	yes	characteristic
61	C₃₁×#108	C₂³	1	2	yes	—
62	C₃₁×#104	—	2	2	no	—
63	C₃₁×#104	D₄	1	2	yes	—
64	C₃₁×#101	—	2	4	no	—
65	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	#41×C₃₁	—	2	2	no	—
67	#41×C₃₁	—	4	4	no	—
68	#41×C₃₁	—	4	4	no	—
69	#41×C₃₁	—	8	4	no	—
70	#41×C₃₁	C₂×D₄	1	2	yes	—
71	C₂×Q₈×C₃₁	—	2	4	no	—
72	C₂×Q₈×C₃₁	—	4	4	no	—
73	C₂×Q₈×C₃₁	—	4	4	no	—
74	Q₁₆×C₃₁	—	2	4	no	—
75	Q₁₆×C₃₁	—	2	4	no	—
76	Q₁₆×C₃₁	—	4	4	no	—
77	Q₁₆×C₃₁	—	4	4	no	—
78	Q₁₆×C₃₁	—	4	2	no	—
79	Q₁₆×C₃₁	—	4	2	no	—
80	Q₁₆×C₃₁	D₈	1	1	yes	characteristic
81	Q₁₆×C₃₁	C₂×D₄	1	2	yes	—
82	SD₁₆×C₃₁	—	2	2	no	—
83	SD₁₆×C₃₁	—	4	4	no	—
84	SD₁₆×C₃₁	—	4	4	no	—
85	SD₁₆×C₃₁	D₈	1	2	yes	—
86	D₈×C₃₁	D₈	1	1	yes	characteristic
87	D₈×C₃₁	C₂×D₄	1	2	yes	—
88	M₁₆×C₃₁	—	2	2	no	—
89	M₁₆×C₃₁	—	2	2	no	—
90	M₁₆×C₃₁	—	4	2	no	—
91	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	—	8	2	no	—
97	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₁₆×C₃₁	—	2	2	no	—
99	C₁₆×C₃₁	—	2	1	no	—
100	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
102	#33860	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	Q₈×C₃₁	—	2	2	no	—
104	Q₈×C₃₁	—	4	4	no	—
105	Q₈×C₃₁	—	4	4	no	—
106	Q₈×C₃₁	—	8	4	no	—
107	Q₈×C₃₁	C₂×D₈	1	2	yes	—
108	D₄×C₃₁	—	2	2	no	—
109	D₄×C₃₁	—	2	2	no	—
110	D₄×C₃₁	—	4	2	no	—
111	D₄×C₃₁	—	8	4	no	—
112	D₄×C₃₁	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	8	4	no	—
118	C₂×C₄×C₃₁	—	8	4	no	—
119	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
123	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
124	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
126	#3417	C₂×C₃₁	1	2	yes	complemented
127	#3293	C₂×C₃₁	1	2	yes	complemented
128	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#2222	C₂×C₃₁	1	2	yes	complemented
130	#2214	C₂×C₃₁	1	2	yes	complemented
131	#2195	C₂×C₃₁	1	4	yes	complemented
132	#2178	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#2170	C₂×C₃₁	1	1	yes	characteristic, complemented
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	#576	1	1	yes	characteristic
137	C₄×C₃₁	—	2	2	no	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	4	2	no	—
140	C₄×C₃₁	—	8	4	no	—
141	C₄×C₃₁	D₄²	1	1	yes	characteristic
142	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
143	#578	—	2	2	no	—
144	#578	C₂²×C₃₁	1	4	yes	—
145	#510	—	2	4	no	—
146	#510	C₂²×C₃₁	1	4	yes	—
147	#508	—	2	2	no	—
148	#508	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×Q₃₂	—	2	2	no	—
150	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
151	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
152	#504	C₂²×C₃₁	1	2	yes	—
153	#494	C₂²×C₃₁	1	2	yes	—
154	#473	—	2	2	no	—
155	#473	C₂²×C₃₁	1	4	yes	—
156	#456	—	2	4	no	—
157	#443	C₂²×C₃₁	1	2	yes	—
158	#443	C₂²×C₃₁	1	1	yes	characteristic
159	#368	C₂²×C₃₁	1	1	yes	characteristic
160	#364	C₂²×C₃₁	1	1	yes	characteristic
161	#362	C₂²×C₃₁	1	4	yes	—
162	#359	C₂²×C₃₁	1	2	yes	—
163	#350	C₂²×C₃₁	1	2	yes	—
164	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₃₁	—	2	1	no	—
166	C₂×C₃₁	—	4	2	no	—
167	C₂×C₃₁	—	8	2	no	—
168	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
169	2₋¹⁺⁴	—	2	4	no	—
170	#137	—	2	4	no	—
171	#137	—	2	4	no	—
172	#137	—	4	4	no	—
173	#137	—	4	4	no	—
174	#135	—	2	4	no	—
175	#135	—	4	2	no	—
176	#135	D₄×C₃₁	1	1	yes	characteristic
177	#135	C₂³×C₃₁	1	2	yes	—
178	C₂×Q₁₆	—	2	4	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	4	2	no	—
181	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
182	#131	—	2	2	no	—
183	#131	C₂³×C₃₁	1	2	yes	—
184	C₄⋊Q₈	—	2	2	no	—
185	Q₃₂	—	2	4	no	—
186	Q₃₂	—	2	2	no	—
187	Q₃₂	—	2	2	no	—
188	Q₃₂	—	4	2	no	—
189	Q₃₂	D₄×C₃₁	1	1	yes	characteristic
190	SD₃₂	—	2	4	no	—
191	SD₃₂	D₄×C₃₁	1	2	yes	—
192	D₁₆	D₄×C₃₁	1	1	yes	characteristic
193	M₃₂	—	2	2	no	—
194	M₃₂	C₂³×C₃₁	1	2	yes	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
197	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
198	#108	D₄×C₃₁	1	1	yes	characteristic
199	#108	C₂³×C₃₁	1	2	yes	—
200	#104	—	2	2	no	—
201	#104	D₄×C₃₁	1	2	yes	—
202	#101	—	2	4	no	—
203	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
204	C₃₁	#33860	1	1	yes	characteristic, direct product factor, Sylow
205	#41	—	2	2	no	—
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	#41	—	8	4	no	—
209	#41	C₂×D₄×C₃₁	1	2	yes	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	4	no	—
216	Q₁₆	—	4	4	no	—
217	Q₁₆	—	4	2	no	—
218	Q₁₆	—	4	2	no	—
219	Q₁₆	D₈×C₃₁	1	1	yes	characteristic
220	Q₁₆	C₂×D₄×C₃₁	1	2	yes	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	4	4	no	—
223	SD₁₆	—	4	4	no	—
224	SD₁₆	D₈×C₃₁	1	2	yes	—
225	D₈	D₈×C₃₁	1	1	yes	characteristic
226	D₈	C₂×D₄×C₃₁	1	2	yes	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
235	C₄⋊C₄	—	8	2	no	—
236	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₁₆	—	2	2	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	Q₈	C₂×D₈×C₃₁	1	2	yes	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	4	no	—
250	D₄	C₂×D₈×C₃₁	1	2	yes	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
261	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
262	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
263	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₃₁×#576	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	4	no	—
271	C₄	D₄²×C₃₁	1	1	yes	characteristic
272	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	4	2	no	—
275	C₂	—	8	2	no	—
276	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3417	C₂	semidirect
2. C₃₁×#3293	C₂	semidirect
3. C₃₁×#2247	C₂	semidirect
4. C₃₁×#2222	C₂	semidirect
5. C₃₁×#2214	C₂	semidirect
6. C₃₁×#2195	C₂	semidirect
7. C₃₁×#2178	C₂	semidirect
8. C₃₁×#2170	C₂	semidirect
102. #33860	C₃₁	direct (N × H)
126. #3417	C₂×C₃₁	semidirect
127. #3293	C₂×C₃₁	semidirect
128. #2247	C₂×C₃₁	semidirect
129. #2222	C₂×C₃₁	semidirect
130. #2214	C₂×C₃₁	semidirect
131. #2195	C₂×C₃₁	semidirect
132. #2178	C₂×C₃₁	semidirect
133. #2170	C₂×C₃₁	semidirect
204. C₃₁	#33860	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×#3417	2
C₃₁×#3293	2
C₃₁×#2247	2
C₃₁×#2222	2
C₃₁×#2214	2
C₃₁×#2195	2
C₃₁×#2178	2
C₃₁×#2170	2
#33860	31

Subgroups — C31×#33860 #21490619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33860 #21490619