Subgroups — C₃₁×#33768 #21490527

Order 7936.

552 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3059	C₂	1	2	yes	maximal
2	C₃₁×#2237	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2219	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2216	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2188	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2113	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
9	C₃₁×#536	—	2	2	no	—
10	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
11	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#470	—	2	2	no	—
15	C₃₁×#470	—	2	2	no	—
16	C₃₁×#467	C₂²	1	4	yes	—
17	C₃₁×#467	C₂²	1	4	yes	—
18	C₃₁×#427	C₂²	1	2	yes	—
19	C₃₁×#419	—	2	2	no	—
20	C₃₁×#367	C₂²	1	2	yes	—
21	C₃₁×#366	C₂²	1	2	yes	—
22	C₃₁×#358	C₂²	1	4	yes	—
23	C₃₁×#358	C₂²	1	4	yes	—
24	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂×Q₁₆×C₃₁	—	2	2	no	—
28	C₂×Q₁₆×C₃₁	—	4	2	no	—
29	C₂×Q₁₆×C₃₁	—	4	2	no	—
30	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
31	C₃₁×#131	—	4	1	no	—
32	C₃₁×#131	—	4	1	no	—
33	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
34	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊Q₈	—	4	2	no	—
38	C₃₁×#124	—	4	2	no	—
39	C₃₁×#123	—	4	2	no	—
40	C₃₁×#123	—	4	2	no	—
41	C₃₁×C₂²⋊Q₈	—	2	4	no	—
42	C₃₁×C₂²⋊Q₈	—	2	4	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×Q₃₂	—	4	2	no	—
45	C₃₁×Q₃₂	—	4	2	no	—
46	C₃₁×M₃₂	—	4	1	no	—
47	C₂×C₁₆×C₃₁	—	4	1	no	—
48	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#107	C₂³	1	4	yes	—
51	C₃₁×#106	—	2	2	no	—
52	C₃₁×#103	—	2	4	no	—
53	C₃₁×#103	—	2	4	no	—
54	#41×C₃₁	—	4	1	no	—
55	#41×C₃₁	—	4	1	no	—
56	C₂×Q₈×C₃₁	—	2	4	no	—
57	C₂×Q₈×C₃₁	—	4	4	no	—
58	C₂×Q₈×C₃₁	—	4	4	no	—
59	C₂×Q₈×C₃₁	—	4	2	no	—
60	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
61	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
62	C₂×D₄×C₃₁	D₈	1	2	yes	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₃₁	—	2	4	no	—
67	Q₁₆×C₃₁	—	2	4	no	—
68	Q₁₆×C₃₁	—	4	4	no	—
69	M₁₆×C₃₁	—	4	2	no	—
70	M₁₆×C₃₁	—	4	1	no	—
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	—	2	1	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	4	4	no	—
80	C₄⋊C₄×C₃₁	—	4	4	no	—
81	C₂²⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	4	no	—
83	C₂²⋊C₄×C₃₁	—	4	4	no	—
84	C₄²×C₃₁	—	4	2	no	—
85	C₁₆×C₃₁	—	2	2	no	—
86	C₁₆×C₃₁	—	2	2	no	—
87	#33768	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
89	C₂³×C₃₁	C₂×D₈	1	2	yes	—
90	Q₈×C₃₁	—	4	1	no	—
91	Q₈×C₃₁	—	4	4	no	—
92	Q₈×C₃₁	—	4	4	no	—
93	Q₈×C₃₁	—	8	4	no	—
94	D₄×C₃₁	—	4	1	no	—
95	D₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	1	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	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	2	no	—
108	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
109	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
110	#3059	C₂×C₃₁	1	2	yes	—
111	#2237	C₂×C₃₁	1	4	yes	complemented
112	#2219	C₂×C₃₁	1	2	yes	complemented
113	#2216	C₂×C₃₁	1	2	yes	complemented
114	#2188	C₂×C₃₁	1	4	yes	complemented
115	#2113	C₂×C₃₁	1	1	yes	characteristic
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	4	no	—
118	C₂²×C₃₁	—	2	2	no	—
119	C₂²×C₃₁	—	2	1	no	—
120	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	2	2	no	—
123	C₄×C₃₁	—	8	4	no	—
124	C₄×C₃₁	—	8	4	no	—
125	C₄×C₃₁	#534	1	1	yes	characteristic
126	C₄×C₃₁	#575	1	1	yes	characteristic
127	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
128	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
129	#536	—	2	2	no	—
130	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
131	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
132	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
133	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
134	#470	—	2	2	no	—
135	#470	—	2	2	no	—
136	#467	C₂²×C₃₁	1	4	yes	—
137	#467	C₂²×C₃₁	1	4	yes	—
138	#427	C₂²×C₃₁	1	2	yes	—
139	#419	—	2	2	no	—
140	#367	C₂²×C₃₁	1	2	yes	—
141	#366	C₂²×C₃₁	1	2	yes	—
142	#358	C₂²×C₃₁	1	4	yes	—
143	#358	C₂²×C₃₁	1	4	yes	—
144	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
145	C₂×C₃₁	—	4	2	no	—
146	C₂×C₃₁	—	4	1	no	—
147	C₂×C₃₁	#3186	1	1	yes	characteristic
148	C₂×C₃₁	#3417	1	2	yes	—
149	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
150	C₂²×Q₈	—	2	2	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	4	2	no	—
153	C₂×Q₁₆	—	4	2	no	—
154	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
155	#131	—	4	1	no	—
156	#131	—	4	1	no	—
157	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
158	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
160	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
161	C₄⋊Q₈	—	4	2	no	—
162	#124	—	4	2	no	—
163	#123	—	4	2	no	—
164	#123	—	4	2	no	—
165	C₂²⋊Q₈	—	2	4	no	—
166	C₂²⋊Q₈	—	2	4	no	—
167	#117	—	2	2	no	—
168	Q₃₂	—	4	2	no	—
169	Q₃₂	—	4	2	no	—
170	M₃₂	—	4	1	no	—
171	C₂×C₁₆	—	4	1	no	—
172	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
173	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
174	#107	C₂³×C₃₁	1	4	yes	—
175	#106	—	2	2	no	—
176	#103	—	2	4	no	—
177	#103	—	2	4	no	—
178	C₃₁	#33768	1	1	yes	characteristic, direct product factor, Sylow
179	#41	—	4	1	no	—
180	#41	—	4	1	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
186	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
187	C₂×D₄	D₈×C₃₁	1	2	yes	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
191	Q₁₆	—	2	4	no	—
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	4	4	no	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
202	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₄²	—	4	2	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	—	2	2	no	—
212	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
213	C₂³	C₂×D₈×C₃₁	1	2	yes	—
214	Q₈	—	4	1	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	4	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	2	no	—
232	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
233	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	1	no	—
241	C₄	—	8	4	no	—
242	C₄	—	8	4	no	—
243	C₄	C₃₁×#534	1	1	yes	characteristic
244	C₄	C₃₁×#575	1	1	yes	characteristic
245	C₂	—	4	2	no	—
246	C₂	—	4	1	no	—
247	C₂	C₃₁×#3186	1	1	yes	characteristic
248	C₂	C₃₁×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2237	C₂	semidirect
3. C₃₁×#2219	C₂	semidirect
4. C₃₁×#2216	C₂	semidirect
5. C₃₁×#2188	C₂	semidirect
87. #33768	C₃₁	direct (N × H)
111. #2237	C₂×C₃₁	semidirect
112. #2219	C₂×C₃₁	semidirect
113. #2216	C₂×C₃₁	semidirect
114. #2188	C₂×C₃₁	semidirect
178. C₃₁	#33768	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×#3059	2
C₃₁×#2237	2
C₃₁×#2219	2
C₃₁×#2216	2
C₃₁×#2188	2
C₃₁×#2113	2
#33768	31

Subgroups — C31×#33768 #21490527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33768 #21490527