Subgroups — C₃₁×#32792 #21489551

Order 7936.

600 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×#2628	C₂	1	2	yes	maximal
2	C₃₁×#2101	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2076	C₂	1	2	yes	maximal
4	C₃₁×#2075	C₂	1	2	yes	maximal
5	C₃₁×#2057	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1818	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#107	C₂²	1	2	yes	—
10	C₂×C₃₁×#107	C₂²	1	2	yes	—
11	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
13	C₃₁×#400	—	2	2	no	—
14	C₃₁×#399	—	2	2	no	—
15	C₃₁×#395	—	2	2	no	—
16	C₃₁×#393	—	2	2	no	—
17	C₃₁×#389	C₂²	1	4	yes	—
18	C₃₁×#382	C₂²	1	2	yes	—
19	C₃₁×#380	C₂²	1	4	yes	—
20	C₃₁×#340	C₂²	1	4	yes	—
21	C₃₁×#340	C₂²	1	4	yes	—
22	C₃₁×#340	C₂²	1	2	yes	—
23	C₃₁×#340	C₂²	1	2	yes	—
24	C₃₁×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂³×C₄×C₃₁	C₂³	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	C₃₁×#117	—	4	1	no	—
32	C₃₁×#117	—	4	1	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
42	C₃₁×#107	—	4	2	no	—
43	C₃₁×#107	—	4	2	no	—
44	C₃₁×C₂²⋊C₈	—	4	1	no	—
45	C₃₁×C₂²⋊C₈	—	4	1	no	—
46	C₃₁×#95	—	2	4	no	—
47	C₃₁×#95	—	2	2	no	—
48	C₃₁×#95	—	2	2	no	—
49	C₃₁×#95	—	2	4	no	—
50	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Q₈×C₃₁	—	4	4	no	—
52	C₂×Q₈×C₃₁	—	8	4	no	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	1	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	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
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	4	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	—	2	4	no	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	—	2	8	no	—
69	C₄⋊C₄×C₃₁	—	4	1	no	—
70	C₄⋊C₄×C₃₁	—	4	1	no	—
71	C₄⋊C₄×C₃₁	—	4	4	no	—
72	C₄⋊C₄×C₃₁	—	8	4	no	—
73	C₂²⋊C₄×C₃₁	—	4	2	no	—
74	C₂²⋊C₄×C₃₁	—	8	4	no	—
75	C₄²×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	—	2	2	no	—
77	#32792	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	2	1	no	—
79	C₂³×C₃₁	—	2	2	no	—
80	C₂³×C₃₁	—	2	2	no	—
81	C₂³×C₃₁	—	2	2	no	—
82	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₃₁	—	8	8	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	4	4	no	—
93	C₂×C₄×C₃₁	—	4	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	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
101	C₈×C₃₁	—	4	2	no	—
102	C₈×C₃₁	—	4	2	no	—
103	#2628	C₂×C₃₁	1	2	yes	—
104	#2101	C₂×C₃₁	1	4	yes	complemented
105	#2076	C₂×C₃₁	1	2	yes	—
106	#2075	C₂×C₃₁	1	2	yes	—
107	#2057	C₂×C₃₁	1	4	yes	complemented
108	#1818	C₂×C₃₁	1	1	yes	characteristic
109	C₂²×C₃₁	—	4	1	no	—
110	C₂²×C₃₁	—	4	1	no	—
111	C₂²×C₃₁	—	4	1	no	—
112	C₂²×C₃₁	—	4	4	no	—
113	C₂²×C₃₁	#534	1	1	yes	characteristic
114	C₂²×C₃₁	#536	1	2	yes	—
115	C₂²×C₃₁	#576	1	2	yes	—
116	C₂²×C₃₁	#578	1	2	yes	—
117	C₄×C₃₁	—	4	1	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₄×C₃₁	—	4	2	no	—
120	C₄×C₃₁	—	8	4	no	—
121	C₄×C₃₁	—	8	4	no	—
122	C₄×C₃₁	#563	1	2	yes	—
123	C₄×C₃₁	#563	1	2	yes	—
124	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
125	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
126	C₂×#107	C₂²×C₃₁	1	2	yes	—
127	C₂×#107	C₂²×C₃₁	1	2	yes	—
128	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
129	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
130	#400	—	2	2	no	—
131	#399	—	2	2	no	—
132	#395	—	2	2	no	—
133	#393	—	2	2	no	—
134	#389	C₂²×C₃₁	1	4	yes	—
135	#382	C₂²×C₃₁	1	2	yes	—
136	#380	C₂²×C₃₁	1	4	yes	—
137	#340	C₂²×C₃₁	1	4	yes	—
138	#340	C₂²×C₃₁	1	4	yes	—
139	#340	C₂²×C₃₁	1	2	yes	—
140	#340	C₂²×C₃₁	1	2	yes	—
141	#336	C₂²×C₃₁	1	4	yes	—
142	C₂×C₃₁	—	8	1	no	—
143	C₂×C₃₁	#2817	1	1	yes	characteristic
144	C₂×C₃₁	#3199	1	1	yes	characteristic
145	C₂×C₃₁	#3202	1	1	yes	characteristic
146	C₂×C₃₁	#3243	1	4	yes	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂³×C₄	C₂³×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	#117	—	4	1	no	—
154	#117	—	4	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
161	C₂×C₂²⋊C₄	—	2	4	no	—
162	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
163	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
164	#107	—	4	2	no	—
165	#107	—	4	2	no	—
166	C₂²⋊C₈	—	4	1	no	—
167	C₂²⋊C₈	—	4	1	no	—
168	#95	—	2	4	no	—
169	#95	—	2	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	4	no	—
172	C₃₁	#32792	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	8	4	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
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	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	8	4	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	8	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
205	Q₈	—	8	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	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	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
223	C₈	—	4	2	no	—
224	C₈	—	4	2	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₃₁×#534	1	1	yes	characteristic
230	C₂²	C₃₁×#536	1	2	yes	—
231	C₂²	C₃₁×#576	1	2	yes	—
232	C₂²	C₃₁×#578	1	2	yes	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	8	4	no	—
237	C₄	—	8	4	no	—
238	C₄	C₃₁×#563	1	2	yes	—
239	C₄	C₃₁×#563	1	2	yes	—
240	C₂	—	8	1	no	—
241	C₂	C₃₁×#2817	1	1	yes	characteristic
242	C₂	C₃₁×#3199	1	1	yes	characteristic
243	C₂	C₃₁×#3202	1	1	yes	characteristic
244	C₂	C₃₁×#3243	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2101	C₂	semidirect
5. C₃₁×#2057	C₂	semidirect
77. #32792	C₃₁	direct (N × H)
104. #2101	C₂×C₃₁	semidirect
107. #2057	C₂×C₃₁	semidirect
172. C₃₁	#32792	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×#2628	2
C₃₁×#2101	2
C₃₁×#2076	2
C₃₁×#2075	2
C₃₁×#2057	2
C₃₁×#1818	2
#32792	31

Subgroups — C31×#32792 #21489551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32792 #21489551