Subgroups — C₃₁×#14709 #21471468

Order 7936.

1662 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	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1436	C₂	1	8	yes	maximal
5	C₃₁×C₄⋊#95	C₂	1	4	yes	maximal
6	C₂×C₃₁×#125	—	2	4	no	—
7	C₂×C₃₁×#125	—	2	4	no	—
8	C₂×C₃₁×#125	C₄	1	8	yes	complemented
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
13	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#382	—	2	4	no	—
16	C₂×C₃₁×#95	C₂²	1	16	yes	—
17	C₂×C₃₁×#95	C₂²	1	4	yes	—
18	C₄³×C₃₁	—	2	4	no	—
19	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₃₁×#125	—	2	32	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₃₁	—	2	8	no	—
34	C₂×C₄²×C₃₁	—	2	4	no	—
35	C₂×C₄²×C₃₁	—	2	16	no	—
36	C₂×C₄²×C₃₁	—	2	4	no	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₃₁	D₄	1	4	yes	—
39	C₂×C₄²×C₃₁	D₄	1	4	yes	—
40	C₂×C₄²×C₃₁	Q₈	1	8	yes	—
41	C₃₁×#95	—	2	8	no	—
42	C₃₁×#95	—	2	32	no	—
43	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	—	2	8	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	—	2	8	no	—
48	C₂²×C₄×C₃₁	—	2	4	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	16	no	—
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	2	16	no	—
53	C₂²×C₄×C₃₁	—	2	8	no	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	—
62	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₃₁	#41	1	16	yes	—
64	C₂²×C₄×C₃₁	#41	1	4	yes	—
65	C₄⋊C₄×C₃₁	—	2	32	no	—
66	C₄⋊C₄×C₃₁	—	2	16	no	—
67	C₄⋊C₄×C₃₁	—	4	8	no	—
68	C₄²×C₃₁	—	2	32	no	—
69	C₄²×C₃₁	—	2	16	no	—
70	C₄²×C₃₁	C₂²⋊C₄	1	16	yes	—
71	#14709	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
73	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
74	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
75	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
76	C₂³×C₃₁	C₂×#41	1	2	yes	—
77	C₂³×C₃₁	C₂×#41	1	8	yes	—
78	C₂×C₄×C₃₁	—	2	8	no	—
79	C₂×C₄×C₃₁	—	2	8	no	—
80	C₂×C₄×C₃₁	—	2	8	no	—
81	C₂×C₄×C₃₁	—	2	16	no	—
82	C₂×C₄×C₃₁	—	2	16	no	—
83	C₂×C₄×C₃₁	—	2	8	no	—
84	C₂×C₄×C₃₁	—	2	8	no	—
85	C₂×C₄×C₃₁	—	2	8	no	—
86	C₂×C₄×C₃₁	—	2	16	no	—
87	C₂×C₄×C₃₁	—	4	8	no	—
88	C₂×C₄×C₃₁	—	4	8	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	4	16	no	—
91	C₂×C₄×C₃₁	—	4	8	no	—
92	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
93	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
95	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
96	C₂×C₄×C₃₁	#124	1	4	yes	—
97	C₂×C₄×C₃₁	#125	1	16	yes	complemented
98	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
99	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic
100	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
101	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
102	#1436	C₂×C₃₁	1	8	yes	—
103	C₄⋊#95	C₂×C₃₁	1	4	yes	—
104	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₃₁	C₂×C₄×Q₈	1	2	yes	—
106	C₂²×C₃₁	C₄×#41	1	4	yes	—
107	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₃₁	#525	1	4	yes	—
109	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
110	C₂²×C₃₁	C₂×#125	1	4	yes	—
111	C₂²×C₃₁	#529	1	8	yes	—
112	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
113	C₂²×C₃₁	#532	1	4	yes	—
114	C₂²×C₃₁	#533	1	4	yes	—
115	C₄×C₃₁	—	2	16	no	—
116	C₄×C₃₁	—	2	8	no	—
117	C₄×C₃₁	—	4	8	no	—
118	C₄×C₃₁	—	8	4	no	—
119	C₄×C₃₁	#391	1	8	yes	—
120	C₂×#125	—	2	4	no	—
121	C₂×#125	—	2	4	no	—
122	C₂×#125	C₄×C₃₁	1	8	yes	complemented
123	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
124	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
126	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
127	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
128	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
129	#382	—	2	4	no	—
130	C₂×#95	C₂²×C₃₁	1	16	yes	—
131	C₂×#95	C₂²×C₃₁	1	4	yes	—
132	C₄³	—	2	4	no	—
133	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
134	C₂×C₃₁	#2297	1	2	yes	—
135	C₂×C₃₁	C₄×#125	1	4	yes	—
136	C₂×C₃₁	#2851	1	4	yes	—
137	C₂×C₃₁	#2859	1	2	yes	—
138	C₂×C₃₁	#2861	1	2	yes	—
139	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
140	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
141	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
142	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
143	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
144	#125	—	2	32	no	—
145	C₂×C₄⋊C₄	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	8	no	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	4	no	—
150	C₂×C₄⋊C₄	—	2	8	no	—
151	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	16	yes	—
152	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
153	C₂×C₄²	—	2	8	no	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	—	2	16	no	—
156	C₂×C₄²	—	2	4	no	—
157	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
158	C₂×C₄²	D₄×C₃₁	1	4	yes	—
159	C₂×C₄²	D₄×C₃₁	1	4	yes	—
160	C₂×C₄²	Q₈×C₃₁	1	8	yes	—
161	#95	—	2	8	no	—
162	#95	—	2	32	no	—
163	C₃₁	#14709	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	16	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
176	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
178	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
179	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
180	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
182	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	—
183	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
184	C₂²×C₄	#41×C₃₁	1	16	yes	—
185	C₂²×C₄	#41×C₃₁	1	4	yes	—
186	C₄⋊C₄	—	2	32	no	—
187	C₄⋊C₄	—	2	16	no	—
188	C₄⋊C₄	—	4	8	no	—
189	C₄²	—	2	32	no	—
190	C₄²	—	2	16	no	—
191	C₄²	C₂²⋊C₄×C₃₁	1	16	yes	—
192	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
193	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
194	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
195	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
196	C₂³	C₂×#41×C₃₁	1	2	yes	—
197	C₂³	C₂×#41×C₃₁	1	8	yes	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	16	no	—
202	C₂×C₄	—	2	16	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	16	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
214	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
215	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
216	C₂×C₄	C₃₁×#124	1	4	yes	—
217	C₂×C₄	C₃₁×#125	1	16	yes	complemented
218	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
219	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
220	C₂²	C₂×C₄×Q₈×C₃₁	1	2	yes	—
221	C₂²	C₄×#41×C₃₁	1	4	yes	—
222	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
223	C₂²	C₃₁×#525	1	4	yes	—
224	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
225	C₂²	C₂×C₃₁×#125	1	4	yes	—
226	C₂²	C₃₁×#529	1	8	yes	—
227	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
228	C₂²	C₃₁×#532	1	4	yes	—
229	C₂²	C₃₁×#533	1	4	yes	—
230	C₄	—	2	16	no	—
231	C₄	—	2	8	no	—
232	C₄	—	4	8	no	—
233	C₄	—	8	4	no	—
234	C₄	C₃₁×#391	1	8	yes	—
235	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
236	C₂	C₃₁×#2297	1	2	yes	—
237	C₂	C₄×C₃₁×#125	1	4	yes	—
238	C₂	C₃₁×#2851	1	4	yes	—
239	C₂	C₃₁×#2859	1	2	yes	—
240	C₂	C₃₁×#2861	1	2	yes	—

	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
8. C₂×C₃₁×#125	C₄	semidirect
71. #14709	C₃₁	direct (N × H)
97. C₂×C₄×C₃₁	#125	semidirect
122. C₂×#125	C₄×C₃₁	semidirect
163. C₃₁	#14709	direct (N × H)
217. C₂×C₄	C₃₁×#125	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#125	2
C₂×C₃₁×#382	2
C₂×C₄³×C₃₁	2
C₃₁×#1436	2
C₃₁×C₄⋊#95	2
#14709	31

Subgroups — C31×#14709 #21471468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#14709 #21471468