Subgroups — C₃₁×#22270 #21479029

Order 7936.

886 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	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₃₁×#1956	C₂	1	2	yes	maximal
4	C₃₁×#1878	C₂	1	8	yes	maximal
5	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
10	C₃₁×#494	C₄	1	8	yes	complemented
11	C₂×C₃₁×#103	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#391	C₂²	1	4	yes	—
15	C₃₁×#389	—	2	2	no	—
16	C₃₁×#389	—	2	2	no	—
17	C₃₁×#384	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
19	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#340	C₂²	1	4	yes	—
22	C₃₁×#336	C₂²	1	4	yes	—
23	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
24	C₂×Q₁₆×C₃₁	—	4	8	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	16	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₃₁×C₄⋊Q₈	—	2	4	no	—
29	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
30	C₄×Q₈×C₃₁	—	4	8	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₂×C₄²×C₃₁	—	2	4	no	—
38	C₂×C₄²×C₃₁	D₄	1	2	yes	—
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#103	—	4	8	no	—
44	C₄×C₈×C₃₁	—	2	2	no	—
45	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
46	C₃₁×#95	—	2	4	no	—
47	C₃₁×#95	—	2	4	no	—
48	C₂×Q₈×C₃₁	—	2	4	no	—
49	C₂×Q₈×C₃₁	—	2	8	no	—
50	C₂×Q₈×C₃₁	—	2	8	no	—
51	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₃₁	—	4	16	no	—
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	2	4	no	—
63	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
64	C₄⋊C₄×C₃₁	—	2	8	no	—
65	C₄⋊C₄×C₃₁	—	2	8	no	—
66	C₄⋊C₄×C₃₁	—	2	8	no	—
67	C₄⋊C₄×C₃₁	—	4	8	no	—
68	C₄⋊C₄×C₃₁	—	4	4	no	—
69	C₄⋊C₄×C₃₁	—	4	4	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	2	4	no	—
72	C₄²×C₃₁	—	2	4	no	—
73	C₄²×C₃₁	—	4	8	no	—
74	C₄²×C₃₁	Q₁₆	1	4	yes	complemented
75	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₃₁	#41	1	2	yes	—
77	#22270	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
79	Q₈×C₃₁	—	2	16	no	—
80	Q₈×C₃₁	—	4	8	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	8	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	4	4	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
95	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₃₁	#135	1	2	yes	—
97	C₂×C₄×C₃₁	#137	1	4	yes	—
98	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
101	C₈×C₃₁	—	2	4	no	—
102	C₈×C₃₁	—	4	2	no	—
103	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
104	C₄×C₄⋊Q₈	C₂×C₃₁	1	2	yes	—
105	#1956	C₂×C₃₁	1	2	yes	—
106	#1878	C₂×C₃₁	1	8	yes	—
107	#1834	C₂×C₃₁	1	1	yes	characteristic
108	#1744	C₂×C₃₁	1	1	yes	characteristic
109	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂²×C₃₁	#535	1	2	yes	—
111	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
112	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×#137	1	2	yes	—
114	C₄×C₃₁	—	2	4	no	—
115	C₄×C₃₁	—	4	8	no	—
116	C₄×C₃₁	—	4	2	no	—
117	C₄×C₃₁	—	8	4	no	—
118	C₄×C₃₁	C₄×Q₁₆	1	4	yes	—
119	C₄×C₃₁	#441	1	4	yes	—
120	C₄×C₃₁	#518	1	2	yes	—
121	C₄×C₃₁	#540	1	2	yes	—
122	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
123	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
125	#494	C₄×C₃₁	1	8	yes	complemented
126	C₂×#103	C₂²×C₃₁	1	8	yes	—
127	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
129	#391	C₂²×C₃₁	1	4	yes	—
130	#389	—	2	2	no	—
131	#389	—	2	2	no	—
132	#384	C₂²×C₃₁	1	2	yes	—
133	C₄×C₄⋊C₄	—	2	2	no	—
134	C₄×C₄⋊C₄	—	2	2	no	—
135	C₄³	C₂²×C₃₁	1	1	yes	characteristic
136	#340	C₂²×C₃₁	1	4	yes	—
137	#336	C₂²×C₃₁	1	4	yes	—
138	C₂×C₃₁	#2328	1	1	yes	characteristic
139	C₂×C₃₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
140	C₂×C₃₁	C₂×#441	1	1	yes	characteristic
141	C₂×C₃₁	#3209	1	2	yes	—
142	C₂×C₃₁	#3210	1	2	yes	—
143	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
144	C₂×Q₁₆	—	4	8	no	—
145	C₂×Q₁₆	C₂×C₄×C₃₁	1	16	yes	—
146	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
147	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
148	C₄⋊Q₈	—	2	4	no	—
149	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
150	C₄×Q₈	—	4	8	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	D₄×C₃₁	1	2	yes	—
159	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
160	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
162	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
163	#103	—	4	8	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
166	#95	—	2	4	no	—
167	#95	—	2	4	no	—
168	C₃₁	#22270	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	8	no	—
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
180	Q₁₆	—	4	16	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	4	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	4	8	no	—
195	C₄²	Q₁₆×C₃₁	1	4	yes	complemented
196	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
197	C₄²	#41×C₃₁	1	2	yes	—
198	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
199	Q₈	—	2	16	no	—
200	Q₈	—	4	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
213	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
214	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
215	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
216	C₂×C₄	C₃₁×#135	1	2	yes	—
217	C₂×C₄	C₃₁×#137	1	4	yes	—
218	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
220	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
224	C₂²	C₃₁×#535	1	2	yes	—
225	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
226	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
227	C₂²	C₂×C₃₁×#137	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	4	8	no	—
230	C₄	—	4	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₄×Q₁₆×C₃₁	1	4	yes	—
233	C₄	C₃₁×#441	1	4	yes	—
234	C₄	C₃₁×#518	1	2	yes	—
235	C₄	C₃₁×#540	1	2	yes	—
236	C₂	C₃₁×#2328	1	1	yes	characteristic
237	C₂	C₂×C₄×Q₁₆×C₃₁	1	1	yes	characteristic
238	C₂	C₂×C₃₁×#441	1	1	yes	characteristic
239	C₂	C₃₁×#3209	1	2	yes	—
240	C₂	C₃₁×#3210	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₃₁×#494	C₄	semidirect
74. C₄²×C₃₁	Q₁₆	semidirect
77. #22270	C₃₁	direct (N × H)
125. #494	C₄×C₃₁	semidirect
168. C₃₁	#22270	direct (N × H)
195. C₄²	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#494	2
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#1956	2
C₃₁×#1878	2
C₃₁×#1834	2
C₃₁×#1744	2
#22270	31

Subgroups — C31×#22270 #21479029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22270 #21479029