Subgroups — C₃₁×#9844 #21466603

Order 7936.

554 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1962	C₂	1	2	yes	maximal, complemented
3	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1492	C₂	1	2	yes	maximal, complemented
6	C₃₁×#532	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	1	yes	characteristic, complemented
8	C₃₁×#493	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#486	C₄	1	2	yes	complemented
10	C₃₁×#466	—	2	2	no	—
11	C₄×D₈×C₃₁	—	2	2	no	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₂×C₃₁×#104	—	2	4	no	—
14	C₂×C₄×C₈×C₃₁	—	2	2	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#335	—	2	2	no	—
19	C₃₁×D₄⋊C₈	—	2	4	no	—
20	C₂×#41×C₃₁	—	2	2	no	—
21	C₂×D₈×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	—	2	1	no	—
24	C₂²×C₈×C₃₁	D₄	1	2	yes	—
25	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	C₃₁×#127	C₂×C₄	1	2	yes	—
27	C₃₁×#124	—	2	2	no	—
28	C₃₁×#121	—	2	2	no	—
29	C₃₁×#121	—	2	2	no	—
30	C₄×D₄×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₂×C₄²×C₃₁	—	2	1	no	—
33	C₂×C₄²×C₃₁	—	2	1	no	—
34	C₂×C₄²×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×#107	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×#104	—	4	8	no	—
40	C₃₁×#102	—	2	4	no	—
41	C₄×C₈×C₃₁	—	2	8	no	—
42	C₄×C₈×C₃₁	—	2	1	no	—
43	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	2	yes	—
46	C₄×C₈×C₃₁	D₄	1	2	yes	—
47	#41×C₃₁	—	4	4	no	—
48	#41×C₃₁	—	4	4	no	—
49	C₂×Q₈×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	2	2	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
59	D₈×C₃₁	—	4	4	no	—
60	M₁₆×C₃₁	—	4	4	no	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂²⋊C₄×C₃₁	—	4	2	no	—
70	C₂²⋊C₄×C₃₁	—	4	2	no	—
71	C₄²×C₃₁	—	2	4	no	—
72	C₄²×C₃₁	—	2	4	no	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₄²×C₃₁	D₈	1	4	yes	complemented
75	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₃₁	#41	1	1	yes	characteristic
79	#9844	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	4	2	no	—
81	C₂³×C₃₁	#121	1	1	yes	characteristic
82	Q₈×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	—	8	4	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	8	2	no	—
94	C₂×C₄×C₃₁	—	8	2	no	—
95	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₃₁	#135	1	2	yes	—
103	C₈×C₃₁	—	2	4	no	—
104	C₈×C₃₁	—	2	2	no	—
105	C₈×C₃₁	—	8	2	no	—
106	C₈×C₃₁	#104	1	4	yes	complemented
107	#3169	C₂×C₃₁	1	1	yes	characteristic
108	#1962	C₂×C₃₁	1	2	yes	complemented
109	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
110	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
111	#1492	C₂×C₃₁	1	2	yes	complemented
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	8	2	no	—
115	C₂²×C₃₁	—	8	2	no	—
116	C₂²×C₃₁	—	8	2	no	—
117	C₂²×C₃₁	#390	1	1	yes	characteristic
118	C₂²×C₃₁	#466	1	2	yes	—
119	C₄×C₃₁	—	2	8	no	—
120	C₄×C₃₁	—	2	1	no	—
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	8	2	no	—
123	C₄×C₃₁	C₂×#104	1	2	yes	—
124	C₄×C₃₁	C₄×D₈	1	2	yes	—
125	C₄×C₃₁	#443	1	2	yes	—
126	C₄×C₃₁	#493	1	1	yes	characteristic
127	C₄×C₃₁	#495	1	1	yes	characteristic
128	#532	C₂²×C₃₁	1	2	yes	—
129	#500	C₄×C₃₁	1	1	yes	characteristic, complemented
130	#493	C₄×C₃₁	1	1	yes	characteristic, complemented
131	#486	C₄×C₃₁	1	2	yes	complemented
132	#466	—	2	2	no	—
133	C₄×D₈	—	2	2	no	—
134	#423	C₂²×C₃₁	1	2	yes	—
135	C₂×#104	—	2	4	no	—
136	C₂×C₄×C₈	—	2	2	no	—
137	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
139	C₄³	C₂²×C₃₁	1	1	yes	characteristic
140	#335	—	2	2	no	—
141	D₄⋊C₈	—	2	4	no	—
142	C₂×C₃₁	—	2	2	no	—
143	C₂×C₃₁	—	16	2	no	—
144	C₂×C₃₁	#1962	1	1	yes	characteristic
145	C₂×C₃₁	#1970	1	1	yes	characteristic
146	C₂×C₃₁	#1974	1	1	yes	characteristic
147	C₂×#41	—	2	2	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	D₄×C₃₁	1	2	yes	—
152	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
153	#127	C₂×C₄×C₃₁	1	2	yes	—
154	#124	—	2	2	no	—
155	#121	—	2	2	no	—
156	#121	—	2	2	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	D₄×C₃₁	1	2	yes	—
162	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
163	#107	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	#104	—	4	8	no	—
167	#102	—	2	4	no	—
168	C₄×C₈	—	2	8	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
172	C₄×C₈	D₄×C₃₁	1	2	yes	—
173	C₄×C₈	D₄×C₃₁	1	2	yes	—
174	C₃₁	#9844	1	1	yes	characteristic, direct product factor, Sylow
175	#41	—	4	4	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
187	D₈	—	4	4	no	—
188	M₁₆	—	4	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
194	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	2	no	—
202	C₄²	D₈×C₃₁	1	4	yes	complemented
203	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
204	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₄²	#41×C₃₁	1	1	yes	characteristic
207	C₂³	—	4	2	no	—
208	C₂³	C₃₁×#121	1	1	yes	characteristic
209	Q₈	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	8	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
226	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
227	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
228	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
229	C₂×C₄	C₃₁×#135	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	2	2	no	—
232	C₈	—	8	2	no	—
233	C₈	C₃₁×#104	1	4	yes	complemented
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	8	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	C₃₁×#390	1	1	yes	characteristic
240	C₂²	C₃₁×#466	1	2	yes	—
241	C₄	—	2	8	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	8	2	no	—
245	C₄	C₂×C₃₁×#104	1	2	yes	—
246	C₄	C₄×D₈×C₃₁	1	2	yes	—
247	C₄	C₃₁×#443	1	2	yes	—
248	C₄	C₃₁×#493	1	1	yes	characteristic
249	C₄	C₃₁×#495	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	16	2	no	—
252	C₂	C₃₁×#1962	1	1	yes	characteristic
253	C₂	C₃₁×#1970	1	1	yes	characteristic
254	C₂	C₃₁×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#390
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1962	C₂	semidirect
3. C₄²×C₈×C₃₁	C₂	semidirect
4. C₃₁×#1567	C₂	semidirect
5. C₃₁×#1492	C₂	semidirect
7. C₃₁×#500	C₄	semidirect
8. C₃₁×#493	C₄	semidirect
9. C₃₁×#486	C₄	semidirect
74. C₄²×C₃₁	D₈	semidirect
79. #9844	C₃₁	direct (N × H)
106. C₈×C₃₁	#104	semidirect
108. #1962	C₂×C₃₁	semidirect
109. C₄²×C₈	C₂×C₃₁	semidirect
110. #1567	C₂×C₃₁	semidirect
111. #1492	C₂×C₃₁	semidirect
129. #500	C₄×C₃₁	semidirect
130. #493	C₄×C₃₁	semidirect
131. #486	C₄×C₃₁	semidirect
174. C₃₁	#9844	direct (N × H)
202. C₄²	D₈×C₃₁	semidirect
233. C₈	C₃₁×#104	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3169	2
C₃₁×#1962	2
C₄²×C₈×C₃₁	2
C₃₁×#1567	2
C₃₁×#1492	2
#9844	31

Subgroups — C31×#9844 #21466603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9844 #21466603