Subgroups — C₃₁×#8592 #21465351

Order 7936.

644 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#331	C₂	1	2	yes	maximal
4	C₃₁×#1296	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
6	C₂×C₄×D₄×C₃₁	—	2	2	no	—
7	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
9	C₃₁×#390	—	2	2	no	—
10	C₃₁×#390	C₄	1	2	yes	—
11	C₃₁×#384	C₂²	1	1	yes	characteristic
12	C₄³×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#336	—	2	2	no	—
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#331	—	2	4	no	—
16	C₂²×D₄×C₃₁	—	2	2	no	—
17	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	—	2	4	no	—
20	C₃₁×#127	—	2	2	no	—
21	C₃₁×#127	D₄	1	2	yes	—
22	C₃₁×#127	Q₈	1	2	yes	—
23	C₄×D₄×C₃₁	—	4	8	no	—
24	C₄×D₄×C₃₁	—	4	4	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	2	4	no	—
35	C₃₁×C₄⋊C₈	—	2	4	no	—
36	C₂⁴×C₃₁	—	2	2	no	—
37	C₂×D₄×C₃₁	—	2	2	no	—
38	C₂×D₄×C₃₁	—	2	4	no	—
39	C₂×D₄×C₃₁	—	2	4	no	—
40	C₂×D₄×C₃₁	—	4	2	no	—
41	C₂×D₄×C₃₁	—	4	2	no	—
42	C₂×D₄×C₃₁	—	4	4	no	—
43	C₂×D₄×C₃₁	—	4	4	no	—
44	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	—	4	8	no	—
47	C₂²×C₄×C₃₁	—	4	4	no	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₃₁	—	4	4	no	—
54	C₂×C₈×C₃₁	—	4	4	no	—
55	C₂×C₈×C₃₁	—	4	4	no	—
56	C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₄⋊C₄×C₃₁	—	4	4	no	—
58	C₂²⋊C₄×C₃₁	—	4	4	no	—
59	C₂²⋊C₄×C₃₁	—	4	8	no	—
60	C₄²×C₃₁	—	2	2	no	—
61	C₄²×C₃₁	—	2	4	no	—
62	C₄²×C₃₁	—	2	4	no	—
63	C₄²×C₃₁	—	4	1	no	—
64	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₃₁	#41	1	2	yes	—
67	#8592	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	—	2	4	no	—
69	C₂³×C₃₁	—	4	2	no	—
70	C₂³×C₃₁	—	4	2	no	—
71	C₂³×C₃₁	—	4	4	no	—
72	C₂³×C₃₁	—	4	4	no	—
73	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	D₄×C₃₁	—	4	4	no	—
75	D₄×C₃₁	—	4	4	no	—
76	D₄×C₃₁	—	4	8	no	—
77	D₄×C₃₁	—	4	8	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	4	2	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	—	8	8	no	—
89	C₂×C₄×C₃₁	#100	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#121	1	2	yes	—
93	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₃₁	#135	1	4	yes	—
95	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
96	C₈×C₃₁	—	8	4	no	—
97	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
98	#1831	C₂×C₃₁	1	2	yes	complemented
99	C₂×#331	C₂×C₃₁	1	2	yes	—
100	#1296	C₂×C₃₁	1	2	yes	complemented
101	C₂²×C₃₁	—	4	4	no	—
102	C₂²×C₃₁	—	4	8	no	—
103	C₂²×C₃₁	—	8	2	no	—
104	C₂²×C₃₁	—	8	4	no	—
105	C₂²×C₃₁	—	8	2	no	—
106	C₂²×C₃₁	#380	1	1	yes	characteristic
107	C₂²×C₃₁	C₂×#100	1	1	yes	characteristic
108	C₂²×C₃₁	#413	1	1	yes	characteristic
109	C₂²×C₃₁	#416	1	2	yes	—
110	C₂²×C₃₁	#418	1	2	yes	—
111	C₄×C₃₁	—	2	1	no	—
112	C₄×C₃₁	—	2	1	no	—
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	—	4	1	no	—
115	C₄×C₃₁	—	4	2	no	—
116	C₄×C₃₁	—	8	2	no	—
117	C₄×C₃₁	#463	1	4	yes	—
118	C₄×C₃₁	#478	1	4	yes	—
119	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
120	C₂×C₄×D₄	—	2	2	no	—
121	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
123	#390	—	2	2	no	—
124	#390	C₄×C₃₁	1	2	yes	—
125	#384	C₂²×C₃₁	1	1	yes	characteristic
126	C₄³	C₂²×C₃₁	1	1	yes	characteristic
127	#336	—	2	2	no	—
128	#336	—	2	2	no	—
129	#331	—	2	4	no	—
130	C₂×C₃₁	—	8	4	no	—
131	C₂×C₃₁	#1543	1	1	yes	characteristic
132	C₂×C₃₁	#1553	1	2	yes	—
133	C₂×C₃₁	#1555	1	1	yes	characteristic
134	C₂×C₃₁	#1800	1	1	yes	characteristic
135	C₂×C₃₁	#1804	1	2	yes	—
136	C₂²×D₄	—	2	2	no	—
137	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
138	C₂³×C₄	—	2	2	no	—
139	C₂²×C₈	—	2	4	no	—
140	#127	—	2	2	no	—
141	#127	D₄×C₃₁	1	2	yes	—
142	#127	Q₈×C₃₁	1	2	yes	—
143	C₄×D₄	—	4	8	no	—
144	C₄×D₄	—	4	4	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₂²⋊C₄	—	2	4	no	—
148	C₂×C₂²⋊C₄	—	2	2	no	—
149	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄²	D₄×C₃₁	1	2	yes	—
152	C₂×C₄²	D₄×C₃₁	1	2	yes	—
153	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	2	4	no	—
156	C₃₁	#8592	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁴	—	2	2	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	8	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	4	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₂²⋊C₄	—	4	4	no	—
180	C₂²⋊C₄	—	4	8	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	4	1	no	—
185	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
187	C₄²	#41×C₃₁	1	2	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	—	4	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
194	D₄	—	4	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	8	no	—
197	D₄	—	4	8	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	C₃₁×#100	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
212	C₂×C₄	C₃₁×#121	1	2	yes	—
213	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
214	C₂×C₄	C₃₁×#135	1	4	yes	—
215	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
216	C₈	—	8	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	8	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	4	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₃₁×#380	1	1	yes	characteristic
223	C₂²	C₂×C₃₁×#100	1	1	yes	characteristic
224	C₂²	C₃₁×#413	1	1	yes	characteristic
225	C₂²	C₃₁×#416	1	2	yes	—
226	C₂²	C₃₁×#418	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	2	no	—
233	C₄	C₃₁×#463	1	4	yes	—
234	C₄	C₃₁×#478	1	4	yes	—
235	C₂	—	8	4	no	—
236	C₂	C₃₁×#1543	1	1	yes	characteristic
237	C₂	C₃₁×#1553	1	2	yes	—
238	C₂	C₃₁×#1555	1	1	yes	characteristic
239	C₂	C₃₁×#1800	1	1	yes	characteristic
240	C₂	C₃₁×#1804	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1831	C₂	semidirect
4. C₃₁×#1296	C₂	semidirect
67. #8592	C₃₁	direct (N × H)
98. #1831	C₂×C₃₁	semidirect
100. #1296	C₂×C₃₁	semidirect
156. C₃₁	#8592	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₃₁×#127	2
C₃₁×#1831	2
C₂×C₃₁×#331	2
C₃₁×#1296	2
#8592	31

Subgroups — C31×#8592 #21465351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8592 #21465351