Subgroups — C₃₁×#8585 #21465344

Order 7936.

684 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#1842	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	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
11	C₃₁×#390	—	2	2	no	—
12	C₃₁×#390	C₄	1	2	yes	—
13	C₃₁×#384	C₂²	1	1	yes	characteristic
14	C₄³×C₃₁	C₂²	1	1	yes	characteristic
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	4	no	—
24	C₄×D₄×C₃₁	—	4	4	no	—
25	C₄×D₄×C₃₁	—	4	4	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
29	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×C₄⋊C₈	—	2	4	no	—
36	C₃₁×C₄⋊C₈	—	2	4	no	—
37	C₃₁×C₄⋊C₈	—	4	4	no	—
38	C₃₁×C₄⋊C₈	—	4	4	no	—
39	C₂⁴×C₃₁	—	2	2	no	—
40	C₂×D₄×C₃₁	—	2	2	no	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	2	4	no	—
43	C₂×D₄×C₃₁	—	4	2	no	—
44	C₂×D₄×C₃₁	—	4	2	no	—
45	C₂×D₄×C₃₁	—	4	2	no	—
46	C₂×D₄×C₃₁	—	4	2	no	—
47	C₂×D₄×C₃₁	—	4	2	no	—
48	C₂×D₄×C₃₁	—	4	2	no	—
49	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	4	4	no	—
52	C₂²×C₄×C₃₁	—	4	8	no	—
53	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₃₁	—	4	4	no	—
59	C₂×C₈×C₃₁	—	4	4	no	—
60	C₂×C₈×C₃₁	—	4	4	no	—
61	C₄⋊C₄×C₃₁	—	4	2	no	—
62	C₄⋊C₄×C₃₁	—	4	2	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₂²⋊C₄×C₃₁	—	4	4	no	—
65	C₂²⋊C₄×C₃₁	—	4	8	no	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	—	4	1	no	—
70	C₄²×C₃₁	D₈	1	4	yes	—
71	C₄²×C₃₁	SD₁₆	1	4	yes	—
72	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₃₁	#41	1	2	yes	—
75	#8585	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	—	2	4	no	—
77	C₂³×C₃₁	—	4	2	no	—
78	C₂³×C₃₁	—	4	2	no	—
79	C₂³×C₃₁	—	4	4	no	—
80	C₂³×C₃₁	—	4	4	no	—
81	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	D₄×C₃₁	—	4	4	no	—
83	D₄×C₃₁	—	4	4	no	—
84	D₄×C₃₁	—	4	4	no	—
85	D₄×C₃₁	—	4	4	no	—
86	D₄×C₃₁	—	4	4	no	—
87	D₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	8	8	no	—
97	C₂×C₄×C₃₁	#100	1	2	yes	—
98	C₂×C₄×C₃₁	#102	1	4	yes	—
99	C₂×C₄×C₃₁	#102	1	4	yes	—
100	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#121	1	2	yes	—
103	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₃₁	#137	1	2	yes	—
108	C₈×C₃₁	—	8	4	no	—
109	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
110	#1842	C₂×C₃₁	1	2	yes	complemented
111	C₂×#331	C₂×C₃₁	1	2	yes	—
112	#1296	C₂×C₃₁	1	2	yes	complemented
113	C₂²×C₃₁	—	4	4	no	—
114	C₂²×C₃₁	—	4	8	no	—
115	C₂²×C₃₁	—	8	2	no	—
116	C₂²×C₃₁	—	8	4	no	—
117	C₂²×C₃₁	—	8	2	no	—
118	C₂²×C₃₁	#380	1	1	yes	characteristic
119	C₂²×C₃₁	C₂×#100	1	1	yes	characteristic
120	C₂²×C₃₁	#413	1	1	yes	characteristic
121	C₂²×C₃₁	C₂×#102	1	2	yes	—
122	C₂²×C₃₁	#417	1	2	yes	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	4	no	—
126	C₄×C₃₁	—	4	1	no	—
127	C₄×C₃₁	—	8	2	no	—
128	C₄×C₃₁	#331	1	4	yes	complemented
129	C₄×C₃₁	#459	1	2	yes	—
130	C₄×C₃₁	#461	1	2	yes	—
131	C₄×C₃₁	#474	1	2	yes	—
132	C₄×C₃₁	#476	1	2	yes	—
133	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×C₄×D₄	—	2	2	no	—
135	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
136	C₂×C₄⋊C₈	—	2	2	no	—
137	C₂×C₄⋊C₈	—	2	2	no	—
138	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
139	#390	—	2	2	no	—
140	#390	C₄×C₃₁	1	2	yes	—
141	#384	C₂²×C₃₁	1	1	yes	characteristic
142	C₄³	C₂²×C₃₁	1	1	yes	characteristic
143	#331	—	2	4	no	—
144	C₂×C₃₁	—	8	4	no	—
145	C₂×C₃₁	C₂×#331	1	1	yes	characteristic
146	C₂×C₃₁	#1547	1	2	yes	—
147	C₂×C₃₁	#1560	1	1	yes	characteristic
148	C₂×C₃₁	#1800	1	1	yes	characteristic
149	C₂×C₃₁	#1802	1	2	yes	—
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
152	C₂³×C₄	—	2	2	no	—
153	C₂²×C₈	—	2	4	no	—
154	#127	—	2	2	no	—
155	#127	D₄×C₃₁	1	2	yes	—
156	#127	Q₈×C₃₁	1	2	yes	—
157	C₄×D₄	—	4	4	no	—
158	C₄×D₄	—	4	4	no	—
159	C₄×D₄	—	4	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
162	C₂×C₂²⋊C₄	—	2	4	no	—
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₃₁	1	2	yes	—
167	C₂×C₄²	D₄×C₃₁	1	2	yes	—
168	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	4	4	no	—
172	C₄⋊C₈	—	4	4	no	—
173	C₃₁	#8585	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁴	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	4	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	D₈×C₃₁	1	4	yes	—
206	C₄²	SD₁₆×C₃₁	1	4	yes	—
207	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
209	C₄²	#41×C₃₁	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	D₄	—	4	4	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	C₃₁×#100	1	2	yes	—
232	C₂×C₄	C₃₁×#102	1	4	yes	—
233	C₂×C₄	C₃₁×#102	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
236	C₂×C₄	C₃₁×#121	1	2	yes	—
237	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
239	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
240	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
241	C₂×C₄	C₃₁×#137	1	2	yes	—
242	C₈	—	8	4	no	—
243	C₂²	—	4	4	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₃₁×#380	1	1	yes	characteristic
249	C₂²	C₂×C₃₁×#100	1	1	yes	characteristic
250	C₂²	C₃₁×#413	1	1	yes	characteristic
251	C₂²	C₂×C₃₁×#102	1	2	yes	—
252	C₂²	C₃₁×#417	1	2	yes	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	4	no	—
256	C₄	—	4	1	no	—
257	C₄	—	8	2	no	—
258	C₄	C₃₁×#331	1	4	yes	complemented
259	C₄	C₃₁×#459	1	2	yes	—
260	C₄	C₃₁×#461	1	2	yes	—
261	C₄	C₃₁×#474	1	2	yes	—
262	C₄	C₃₁×#476	1	2	yes	—
263	C₂	—	8	4	no	—
264	C₂	C₂×C₃₁×#331	1	1	yes	characteristic
265	C₂	C₃₁×#1547	1	2	yes	—
266	C₂	C₃₁×#1560	1	1	yes	characteristic
267	C₂	C₃₁×#1800	1	1	yes	characteristic
268	C₂	C₃₁×#1802	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×#1842	C₂	semidirect
4. C₃₁×#1296	C₂	semidirect
75. #8585	C₃₁	direct (N × H)
110. #1842	C₂×C₃₁	semidirect
112. #1296	C₂×C₃₁	semidirect
128. C₄×C₃₁	#331	semidirect
173. C₃₁	#8585	direct (N × H)
258. C₄	C₃₁×#331	semidirect

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×#1842	2
C₂×C₃₁×#331	2
C₃₁×#1296	2
#8585	31

Subgroups — C31×#8585 #21465344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8585 #21465344