Subgroups — C₃₁×#21955 #21478714

Order 7936.

914 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	C₂×C₃₁×#432	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1940	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1797	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#432	—	2	8	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
11	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃₁×#97	C₂²	1	4	yes	—
13	C₄×C₄⋊C₄×C₃₁	C₄	1	4	yes	—
14	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
15	C₂×C₃₁×#95	C₄	1	1	yes	characteristic
16	C₂×C₃₁×#95	C₄	1	1	yes	characteristic
17	C₃₁×#336	C₂²	1	8	yes	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
24	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
25	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₃₁	—	4	4	no	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	D₄	1	4	yes	—
30	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
31	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
33	C₃₁×C₄⋊C₈	—	2	8	no	—
34	C₃₁×C₄⋊C₈	—	4	8	no	—
35	C₃₁×C₂²⋊C₈	—	2	8	no	—
36	C₃₁×#97	—	2	8	no	—
37	C₃₁×#95	C₂×C₄	1	4	yes	—
38	C₃₁×#95	C₂×C₄	1	4	yes	—
39	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	—	2	4	no	—
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	—	2	2	no	—
43	C₂²×C₄×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	—	4	2	no	—
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	—	4	2	no	—
47	C₂²×C₄×C₃₁	—	4	2	no	—
48	C₂²×C₄×C₃₁	—	4	2	no	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₃₁	M₁₆	1	8	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₃₁	—	2	8	no	—
61	C₂×C₈×C₃₁	—	2	8	no	—
62	C₂×C₈×C₃₁	—	2	8	no	—
63	C₄⋊C₄×C₃₁	—	2	16	no	—
64	C₄²×C₃₁	—	2	8	no	—
65	C₄²×C₃₁	—	4	4	no	—
66	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
67	C₄²×C₃₁	C₂×D₄	1	4	yes	—
68	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
69	#21955	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	—	2	1	no	—
71	C₂³×C₃₁	—	2	1	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
75	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	—	2	4	no	—
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	4	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	8	no	—
83	C₂×C₄×C₃₁	—	2	4	no	—
84	C₂×C₄×C₃₁	—	2	4	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	4	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	4	8	no	—
90	C₂×C₄×C₃₁	—	4	8	no	—
91	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₃₁	C₂×M₁₆	1	8	yes	—
95	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
98	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
99	C₈×C₃₁	—	4	8	no	—
100	C₂×#432	C₂×C₃₁	1	2	yes	—
101	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
102	#1940	C₂×C₃₁	1	8	yes	complemented
103	#1797	C₂×C₃₁	1	4	yes	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	2	2	no	—
106	C₂²×C₃₁	—	2	4	no	—
107	C₂²×C₃₁	—	2	4	no	—
108	C₂²×C₃₁	#423	1	4	yes	—
109	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃₁	#518	1	1	yes	characteristic
111	C₂²×C₃₁	#519	1	1	yes	characteristic
112	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
113	C₂²×C₃₁	#568	1	2	yes	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	4	no	—
116	C₄×C₃₁	—	2	4	no	—
117	C₄×C₃₁	—	2	8	no	—
118	C₄×C₃₁	—	4	4	no	—
119	C₄×C₃₁	#537	1	2	yes	—
120	C₄×C₃₁	#544	1	2	yes	—
121	C₂²×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
122	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
123	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
124	#432	—	2	8	no	—
125	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
126	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
127	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
128	C₂×#97	C₂²×C₃₁	1	4	yes	—
129	C₄×C₄⋊C₄	C₄×C₃₁	1	4	yes	—
130	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
131	C₂×#95	C₄×C₃₁	1	1	yes	characteristic
132	C₂×#95	C₄×C₃₁	1	1	yes	characteristic
133	#336	C₂²×C₃₁	1	8	yes	—
134	C₂×C₃₁	—	2	4	no	—
135	C₂×C₃₁	#2316	1	1	yes	characteristic
136	C₂×C₃₁	C₂×#423	1	1	yes	characteristic
137	C₂×C₃₁	#2905	1	1	yes	characteristic
138	C₂×C₃₁	#2973	1	2	yes	—
139	C₂×C₃₁	#2983	1	2	yes	—
140	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
141	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
142	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
143	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
144	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
146	C₂×C₄⋊C₄	—	2	8	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
148	C₂×C₄²	—	4	4	no	—
149	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
150	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
151	C₂×C₄²	D₄×C₃₁	1	4	yes	—
152	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
153	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
154	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
155	C₄⋊C₈	—	2	8	no	—
156	C₄⋊C₈	—	4	8	no	—
157	C₂²⋊C₈	—	2	8	no	—
158	#97	—	2	8	no	—
159	#95	C₂×C₄×C₃₁	1	4	yes	—
160	#95	C₂×C₄×C₃₁	1	4	yes	—
161	C₃₁	#21955	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
173	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
174	C₂²×C₄	M₁₆×C₃₁	1	8	yes	—
175	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
180	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
181	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
182	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	8	no	—
186	C₄⋊C₄	—	2	16	no	—
187	C₄²	—	2	8	no	—
188	C₄²	—	4	4	no	—
189	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
190	C₄²	C₂×D₄×C₃₁	1	4	yes	—
191	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
196	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
197	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
214	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
215	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
216	C₂×C₄	C₂×M₁₆×C₃₁	1	8	yes	—
217	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
220	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
221	C₈	—	4	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	C₃₁×#423	1	4	yes	—
227	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂²	C₃₁×#518	1	1	yes	characteristic
229	C₂²	C₃₁×#519	1	1	yes	characteristic
230	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
231	C₂²	C₃₁×#568	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	4	4	no	—
237	C₄	C₃₁×#537	1	2	yes	—
238	C₄	C₃₁×#544	1	2	yes	—
239	C₂	—	2	4	no	—
240	C₂	C₃₁×#2316	1	1	yes	characteristic
241	C₂	C₂×C₃₁×#423	1	1	yes	characteristic
242	C₂	C₃₁×#2905	1	1	yes	characteristic
243	C₂	C₃₁×#2973	1	2	yes	—
244	C₂	C₃₁×#2983	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1940	C₂	semidirect
69. #21955	C₃₁	direct (N × H)
102. #1940	C₂×C₃₁	semidirect
161. C₃₁	#21955	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#432	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1940	2
C₃₁×#1797	2
#21955	31

Subgroups — C31×#21955 #21478714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21955 #21478714