Subgroups — C₃₁×#14712 #21471471

Order 7936.

1854 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#389	C₂	1	6	yes	maximal
2	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×C₄⋊#95	C₂	1	6	yes	maximal
6	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
7	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
8	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#389	—	2	24	no	—
13	C₃₁×#384	—	2	4	no	—
14	C₃₁×#384	C₄	1	8	yes	complemented
15	C₂×C₃₁×#95	C₂²	1	12	yes	—
16	C₄³×C₃₁	—	2	4	no	—
17	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
18	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
22	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
28	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	24	yes	—
29	C₂×C₄²×C₃₁	—	2	6	no	—
30	C₂×C₄²×C₃₁	—	2	6	no	—
31	C₂×C₄²×C₃₁	—	2	6	no	—
32	C₂×C₄²×C₃₁	—	2	6	no	—
33	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₃₁	D₄	1	12	yes	—
35	C₂×C₄²×C₃₁	D₄	1	4	yes	—
36	C₂×C₄²×C₃₁	Q₈	1	12	yes	—
37	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
38	C₃₁×#95	—	2	24	no	—
39	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	—	2	1	no	—
41	C₂²×C₄×C₃₁	—	2	3	no	—
42	C₂²×C₄×C₃₁	—	2	3	no	—
43	C₂²×C₄×C₃₁	—	2	3	no	—
44	C₂²×C₄×C₃₁	—	2	6	no	—
45	C₂²×C₄×C₃₁	—	2	12	no	—
46	C₂²×C₄×C₃₁	—	2	12	no	—
47	C₂²×C₄×C₃₁	—	2	12	no	—
48	C₂²×C₄×C₃₁	—	2	4	no	—
49	C₂²×C₄×C₃₁	—	2	12	no	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₃₁	C₂×Q₈	1	12	yes	—
57	C₂²×C₄×C₃₁	C₂×Q₈	1	6	yes	—
58	C₂²×C₄×C₃₁	C₂×Q₈	1	6	yes	—
59	C₂²×C₄×C₃₁	#41	1	6	yes	—
60	C₂²×C₄×C₃₁	#41	1	6	yes	—
61	C₄⋊C₄×C₃₁	—	2	48	no	—
62	C₄⋊C₄×C₃₁	—	4	24	no	—
63	C₄⋊C₄×C₃₁	—	4	16	no	—
64	C₄²×C₃₁	—	2	24	no	—
65	C₄²×C₃₁	—	2	24	no	—
66	C₄²×C₃₁	C₄⋊C₄	1	16	yes	complemented
67	#14712	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
69	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
70	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
71	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
72	C₂³×C₃₁	C₂²×Q₈	1	3	yes	—
73	C₂³×C₃₁	C₂×#41	1	3	yes	—
74	C₂³×C₃₁	C₂×#41	1	3	yes	—
75	C₂×C₄×C₃₁	—	2	6	no	—
76	C₂×C₄×C₃₁	—	2	6	no	—
77	C₂×C₄×C₃₁	—	2	12	no	—
78	C₂×C₄×C₃₁	—	2	6	no	—
79	C₂×C₄×C₃₁	—	2	6	no	—
80	C₂×C₄×C₃₁	—	2	12	no	—
81	C₂×C₄×C₃₁	—	2	24	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	4	12	no	—
84	C₂×C₄×C₃₁	—	4	12	no	—
85	C₂×C₄×C₃₁	—	4	4	no	—
86	C₂×C₄×C₃₁	—	4	12	no	—
87	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	12	yes	—
88	C₂×C₄×C₃₁	C₄×D₄	1	12	yes	—
89	C₂×C₄×C₃₁	C₄×Q₈	1	12	yes	—
90	C₂×C₄×C₃₁	#121	1	12	yes	—
91	C₂×C₄×C₃₁	C₂²⋊Q₈	1	12	yes	—
92	C₂×C₄×C₃₁	#125	1	12	yes	—
93	C₂×C₄×C₃₁	#127	1	4	yes	—
94	C₂×C₄×C₃₁	C₄⋊Q₈	1	12	yes	—
95	C₂×C₄×C₃₁	C₄⋊Q₈	1	12	yes	—
96	C₂×#389	C₂×C₃₁	1	6	yes	—
97	C₂×#384	C₂×C₃₁	1	1	yes	characteristic
98	C₂×#384	C₂×C₃₁	1	1	yes	characteristic
99	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
100	C₄⋊#95	C₂×C₃₁	1	6	yes	—
101	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₃₁	C₂×C₄×D₄	1	3	yes	—
103	C₂²×C₃₁	C₂×C₄×Q₈	1	3	yes	—
104	C₂²×C₃₁	C₂×#121	1	3	yes	—
105	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	3	yes	—
106	C₂²×C₃₁	C₂×#125	1	3	yes	—
107	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
108	C₂²×C₃₁	C₂×C₄⋊Q₈	1	3	yes	—
109	C₂²×C₃₁	C₂×C₄⋊Q₈	1	3	yes	—
110	C₂²×C₃₁	#532	1	3	yes	—
111	C₂²×C₃₁	#532	1	3	yes	—
112	C₂²×C₃₁	#533	1	6	yes	—
113	C₄×C₃₁	—	2	4	no	—
114	C₄×C₃₁	—	2	12	no	—
115	C₄×C₃₁	—	4	8	no	—
116	C₄×C₃₁	—	8	4	no	—
117	C₄×C₃₁	#389	1	24	yes	complemented
118	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
119	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
120	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
121	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
122	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
123	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
124	#389	—	2	24	no	—
125	#384	—	2	4	no	—
126	#384	C₄×C₃₁	1	8	yes	complemented
127	C₂×#95	C₂²×C₃₁	1	12	yes	—
128	C₄³	—	2	4	no	—
129	C₂×C₃₁	C₂×#389	1	3	yes	—
130	C₂×C₃₁	C₄×#127	1	1	yes	characteristic
131	C₂×C₃₁	C₄×C₄⋊Q₈	1	3	yes	—
132	C₂×C₃₁	#2856	1	1	yes	characteristic
133	C₂×C₃₁	#2857	1	3	yes	—
134	C₂×C₃₁	#2859	1	3	yes	—
135	C₂×C₃₁	#2868	1	1	yes	characteristic
136	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
137	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
138	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
139	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
140	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
141	C₂×C₄⋊C₄	—	2	12	no	—
142	C₂×C₄⋊C₄	—	2	12	no	—
143	C₂×C₄⋊C₄	—	2	12	no	—
144	C₂×C₄⋊C₄	—	2	12	no	—
145	C₂×C₄⋊C₄	—	2	24	no	—
146	C₂×C₄⋊C₄	—	2	12	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	24	yes	—
148	C₂×C₄²	—	2	6	no	—
149	C₂×C₄²	—	2	6	no	—
150	C₂×C₄²	—	2	6	no	—
151	C₂×C₄²	—	2	6	no	—
152	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
153	C₂×C₄²	D₄×C₃₁	1	12	yes	—
154	C₂×C₄²	D₄×C₃₁	1	4	yes	—
155	C₂×C₄²	Q₈×C₃₁	1	12	yes	—
156	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
157	#95	—	2	24	no	—
158	C₃₁	#14712	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	—	2	3	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	12	no	—
170	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
171	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
172	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
173	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
174	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
175	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
176	C₂²×C₄	C₂×Q₈×C₃₁	1	12	yes	—
177	C₂²×C₄	C₂×Q₈×C₃₁	1	6	yes	—
178	C₂²×C₄	C₂×Q₈×C₃₁	1	6	yes	—
179	C₂²×C₄	#41×C₃₁	1	6	yes	—
180	C₂²×C₄	#41×C₃₁	1	6	yes	—
181	C₄⋊C₄	—	2	48	no	—
182	C₄⋊C₄	—	4	24	no	—
183	C₄⋊C₄	—	4	16	no	—
184	C₄²	—	2	24	no	—
185	C₄²	—	2	24	no	—
186	C₄²	C₄⋊C₄×C₃₁	1	16	yes	complemented
187	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
188	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
189	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
190	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
191	C₂³	C₂²×Q₈×C₃₁	1	3	yes	—
192	C₂³	C₂×#41×C₃₁	1	3	yes	—
193	C₂³	C₂×#41×C₃₁	1	3	yes	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	12	no	—
200	C₂×C₄	—	2	24	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	12	no	—
203	C₂×C₄	—	4	12	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	12	no	—
206	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	12	yes	—
207	C₂×C₄	C₄×D₄×C₃₁	1	12	yes	—
208	C₂×C₄	C₄×Q₈×C₃₁	1	12	yes	—
209	C₂×C₄	C₃₁×#121	1	12	yes	—
210	C₂×C₄	C₃₁×C₂²⋊Q₈	1	12	yes	—
211	C₂×C₄	C₃₁×#125	1	12	yes	—
212	C₂×C₄	C₃₁×#127	1	4	yes	—
213	C₂×C₄	C₃₁×C₄⋊Q₈	1	12	yes	—
214	C₂×C₄	C₃₁×C₄⋊Q₈	1	12	yes	—
215	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂²	C₂×C₄×D₄×C₃₁	1	3	yes	—
217	C₂²	C₂×C₄×Q₈×C₃₁	1	3	yes	—
218	C₂²	C₂×C₃₁×#121	1	3	yes	—
219	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	3	yes	—
220	C₂²	C₂×C₃₁×#125	1	3	yes	—
221	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
222	C₂²	C₂×C₃₁×C₄⋊Q₈	1	3	yes	—
223	C₂²	C₂×C₃₁×C₄⋊Q₈	1	3	yes	—
224	C₂²	C₃₁×#532	1	3	yes	—
225	C₂²	C₃₁×#532	1	3	yes	—
226	C₂²	C₃₁×#533	1	6	yes	—
227	C₄	—	2	4	no	—
228	C₄	—	2	12	no	—
229	C₄	—	4	8	no	—
230	C₄	—	8	4	no	—
231	C₄	C₃₁×#389	1	24	yes	complemented
232	C₂	C₂×C₃₁×#389	1	3	yes	—
233	C₂	C₄×C₃₁×#127	1	1	yes	characteristic
234	C₂	C₄×C₃₁×C₄⋊Q₈	1	3	yes	—
235	C₂	C₃₁×#2856	1	1	yes	characteristic
236	C₂	C₃₁×#2857	1	3	yes	—
237	C₂	C₃₁×#2859	1	3	yes	—
238	C₂	C₃₁×#2868	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
14. C₃₁×#384	C₄	semidirect
66. C₄²×C₃₁	C₄⋊C₄	semidirect
67. #14712	C₃₁	direct (N × H)
117. C₄×C₃₁	#389	semidirect
126. #384	C₄×C₃₁	semidirect
158. C₃₁	#14712	direct (N × H)
186. C₄²	C₄⋊C₄×C₃₁	semidirect
231. C₄	C₃₁×#389	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#389	2
C₂×C₃₁×#384	2
C₂×C₄³×C₃₁	2
C₃₁×C₄⋊#95	2
#14712	31

Subgroups — C31×#14712 #21471471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#14712 #21471471