Subgroups — C₃₁×#32695 #21489454

Order 7936.

642 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2567	C₂	1	2	yes	maximal
2	C₃₁×#2086	C₂	1	2	yes	maximal
3	C₃₁×#2057	C₂	1	4	yes	maximal
4	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1944	C₂	1	2	yes	maximal
6	C₃₁×#1920	C₂	1	4	yes	maximal
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#107	C₂²	1	2	yes	—
10	C₂×C₃₁×#107	C₂²	1	2	yes	—
11	C₂×C₃₁×#106	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
14	C₃₁×#395	—	2	2	no	—
15	C₃₁×#395	—	2	2	no	—
16	C₃₁×#391	—	2	2	no	—
17	C₃₁×#389	C₂²	1	4	yes	—
18	C₃₁×#389	C₂²	1	4	yes	—
19	C₃₁×#384	—	2	2	no	—
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#340	C₂²	1	4	yes	—
22	C₃₁×#340	C₂²	1	4	yes	—
23	C₃₁×#340	C₂²	1	2	yes	—
24	C₃₁×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₃₁×C₄⋊Q₈	—	4	2	no	—
30	C₃₁×C₄⋊Q₈	—	4	2	no	—
31	C₄×Q₈×C₃₁	—	4	1	no	—
32	C₄×Q₈×C₃₁	—	4	1	no	—
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	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄²×C₃₁	—	2	2	no	—
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
43	C₃₁×#107	—	4	2	no	—
44	C₃₁×#107	—	4	2	no	—
45	C₃₁×#106	Q₈	1	8	yes	complemented
46	C₃₁×C₄⋊C₈	—	4	1	no	—
47	C₃₁×#97	—	4	1	no	—
48	C₃₁×#95	—	2	2	no	—
49	C₃₁×#95	—	2	4	no	—
50	C₃₁×#95	—	2	4	no	—
51	C₂×Q₈×C₃₁	—	2	4	no	—
52	C₂×Q₈×C₃₁	—	4	1	no	—
53	C₂×Q₈×C₃₁	—	4	4	no	—
54	C₂×Q₈×C₃₁	—	8	4	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	—	2	4	no	—
69	C₄⋊C₄×C₃₁	—	2	4	no	—
70	C₄⋊C₄×C₃₁	—	4	2	no	—
71	C₄⋊C₄×C₃₁	—	4	1	no	—
72	C₄⋊C₄×C₃₁	—	4	4	no	—
73	C₄⋊C₄×C₃₁	—	8	4	no	—
74	C₄⋊C₄×C₃₁	—	8	4	no	—
75	C₄⋊C₄×C₃₁	C₂×Q₈	1	8	yes	—
76	C₄²×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	—	2	1	no	—
78	C₄²×C₃₁	—	2	2	no	—
79	C₄²×C₃₁	—	2	2	no	—
80	C₄²×C₃₁	—	4	4	no	—
81	#32695	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₃₁	—	4	4	no	—
84	Q₈×C₃₁	—	8	8	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	8	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	#137	1	4	yes	—
100	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
104	C₈×C₃₁	—	2	4	no	—
105	C₈×C₃₁	—	4	2	no	—
106	#2567	C₂×C₃₁	1	2	yes	—
107	#2086	C₂×C₃₁	1	2	yes	—
108	#2057	C₂×C₃₁	1	4	yes	—
109	#1985	C₂×C₃₁	1	1	yes	characteristic
110	#1944	C₂×C₃₁	1	2	yes	—
111	#1920	C₂×C₃₁	1	4	yes	—
112	C₂²×C₃₁	#534	1	1	yes	characteristic
113	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
114	C₂²×C₃₁	C₂×#137	1	2	yes	—
115	C₂²×C₃₁	#576	1	2	yes	—
116	C₄×C₃₁	—	2	4	no	—
117	C₄×C₃₁	—	4	8	no	—
118	C₄×C₃₁	—	4	1	no	—
119	C₄×C₃₁	—	4	1	no	—
120	C₄×C₃₁	—	4	2	no	—
121	C₄×C₃₁	—	8	4	no	—
122	C₄×C₃₁	#544	1	2	yes	—
123	C₄×C₃₁	Q₈²	1	1	yes	characteristic
124	C₄×C₃₁	#563	1	1	yes	characteristic
125	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
126	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
127	C₂×#107	C₂²×C₃₁	1	2	yes	—
128	C₂×#107	C₂²×C₃₁	1	2	yes	—
129	C₂×#106	C₂²×C₃₁	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
132	#395	—	2	2	no	—
133	#395	—	2	2	no	—
134	#391	—	2	2	no	—
135	#389	C₂²×C₃₁	1	4	yes	—
136	#389	C₂²×C₃₁	1	4	yes	—
137	#384	—	2	2	no	—
138	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
139	#340	C₂²×C₃₁	1	4	yes	—
140	#340	C₂²×C₃₁	1	4	yes	—
141	#340	C₂²×C₃₁	1	2	yes	—
142	#336	C₂²×C₃₁	1	4	yes	—
143	C₂×C₃₁	#2734	1	1	yes	characteristic
144	C₂×C₃₁	#3195	1	1	yes	characteristic
145	C₂×C₃₁	#3199	1	1	yes	characteristic
146	C₂×C₃₁	#3385	1	4	yes	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
150	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
151	C₄⋊Q₈	—	4	2	no	—
152	C₄⋊Q₈	—	4	2	no	—
153	C₄×Q₈	—	4	1	no	—
154	C₄×Q₈	—	4	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
159	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
161	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
164	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
165	#107	—	4	2	no	—
166	#107	—	4	2	no	—
167	#106	Q₈×C₃₁	1	8	yes	complemented
168	C₄⋊C₈	—	4	1	no	—
169	#97	—	4	1	no	—
170	#95	—	2	2	no	—
171	#95	—	2	4	no	—
172	#95	—	2	4	no	—
173	C₃₁	#32695	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	8	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
184	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	8	4	no	—
197	C₄⋊C₄	—	8	4	no	—
198	C₄⋊C₄	C₂×Q₈×C₃₁	1	8	yes	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	4	no	—
204	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	Q₈	—	8	8	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₃₁×#137	1	4	yes	—
222	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
223	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₂²	C₃₁×#534	1	1	yes	characteristic
229	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
230	C₂²	C₂×C₃₁×#137	1	2	yes	—
231	C₂²	C₃₁×#576	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	4	8	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	—	8	4	no	—
238	C₄	C₃₁×#544	1	2	yes	—
239	C₄	Q₈²×C₃₁	1	1	yes	characteristic
240	C₄	C₃₁×#563	1	1	yes	characteristic
241	C₂	C₃₁×#2734	1	1	yes	characteristic
242	C₂	C₃₁×#3195	1	1	yes	characteristic
243	C₂	C₃₁×#3199	1	1	yes	characteristic
244	C₂	C₃₁×#3385	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
45. C₃₁×#106	Q₈	semidirect
81. #32695	C₃₁	direct (N × H)
167. #106	Q₈×C₃₁	semidirect
173. C₃₁	#32695	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×#2567	2
C₃₁×#2086	2
C₃₁×#2057	2
C₃₁×#1985	2
C₃₁×#1944	2
C₃₁×#1920	2
#32695	31

Subgroups — C31×#32695 #21489454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32695 #21489454