Subgroups — C₃₁×#9726 #21466485

Order 7936.

436 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	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1525	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1484	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#532	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	1	yes	characteristic
8	C₃₁×#493	C₄	1	1	yes	characteristic
9	C₃₁×#486	C₄	1	2	yes	—
10	C₃₁×#466	—	2	1	no	—
11	C₃₁×#466	—	2	1	no	—
12	C₄×D₈×C₃₁	—	2	1	no	—
13	C₄×D₈×C₃₁	—	2	1	no	—
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	—	2	1	no	—
16	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
17	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#341	—	2	1	no	—
20	C₃₁×#329	—	2	4	no	—
21	C₃₁×D₄⋊C₈	—	2	4	no	—
22	C₃₁×#322	—	2	1	no	—
23	C₃₁×#322	—	2	1	no	—
24	C₂×#41×C₃₁	—	2	2	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	1	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₃₁×#127	C₂×C₄	1	2	yes	—
33	C₃₁×#124	—	2	2	no	—
34	C₃₁×#121	—	2	2	no	—
35	C₃₁×#121	—	2	2	no	—
36	C₄×D₄×C₃₁	—	2	2	no	—
37	C₄×D₄×C₃₁	—	2	2	no	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#108	—	2	4	no	—
40	C₃₁×#107	—	2	1	no	—
41	C₃₁×#107	—	2	1	no	—
42	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
43	C₃₁×#102	—	2	2	no	—
44	C₃₁×#102	—	2	2	no	—
45	C₃₁×C₂²⋊C₈	—	2	1	no	—
46	C₃₁×#97	—	2	2	no	—
47	C₃₁×#97	D₄	1	2	yes	—
48	C₄×C₈×C₃₁	—	2	1	no	—
49	C₄×C₈×C₃₁	—	2	1	no	—
50	C₄×C₈×C₃₁	—	2	2	no	—
51	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	#41×C₃₁	—	8	4	no	—
56	C₂×Q₈×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
65	D₈×C₃₁	—	4	2	no	—
66	D₈×C₃₁	—	4	2	no	—
67	M₁₆×C₃₁	—	2	4	no	—
68	M₁₆×C₃₁	—	4	2	no	—
69	M₁₆×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	4	1	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₂²⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	2	no	—
83	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
87	#9726	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	#124	1	1	yes	characteristic
90	Q₈×C₃₁	—	8	2	no	—
91	D₄×C₃₁	—	4	2	no	—
92	D₄×C₃₁	—	4	2	no	—
93	D₄×C₃₁	—	4	4	no	—
94	D₄×C₃₁	—	8	2	no	—
95	D₄×C₃₁	—	8	4	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	8	4	no	—
102	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
107	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	4	2	no	—
112	C₈×C₃₁	—	4	4	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₈×C₃₁	#104	1	2	yes	—
115	#3169	C₂×C₃₁	1	1	yes	characteristic
116	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#1525	C₂×C₃₁	1	2	yes	complemented
118	#1484	C₂×C₃₁	1	2	yes	complemented
119	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	8	2	no	—
124	C₂²×C₃₁	—	8	2	no	—
125	C₂²×C₃₁	—	8	2	no	—
126	C₂²×C₃₁	#390	1	1	yes	characteristic
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	8	2	no	—
131	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
132	C₄×C₃₁	#421	1	1	yes	characteristic
133	C₄×C₃₁	#444	1	2	yes	—
134	C₄×C₃₁	#472	1	2	yes	—
135	#532	C₂²×C₃₁	1	2	yes	—
136	#500	C₄×C₃₁	1	1	yes	characteristic
137	#493	C₄×C₃₁	1	1	yes	characteristic
138	#486	C₄×C₃₁	1	2	yes	—
139	#466	—	2	1	no	—
140	#466	—	2	1	no	—
141	C₄×D₈	—	2	1	no	—
142	C₄×D₈	—	2	1	no	—
143	#431	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×#108	—	2	1	no	—
145	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
146	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	#341	—	2	1	no	—
149	#329	—	2	4	no	—
150	D₄⋊C₈	—	2	4	no	—
151	#322	—	2	1	no	—
152	#322	—	2	1	no	—
153	C₂×C₃₁	—	4	1	no	—
154	C₂×C₃₁	—	16	2	no	—
155	C₂×C₃₁	#1975	1	1	yes	characteristic
156	C₂×C₃₁	#1978	1	1	yes	characteristic
157	C₂×C₃₁	#1980	1	1	yes	characteristic
158	C₂×#41	—	2	2	no	—
159	C₂×D₈	—	2	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
165	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
166	#127	C₂×C₄×C₃₁	1	2	yes	—
167	#124	—	2	2	no	—
168	#121	—	2	2	no	—
169	#121	—	2	2	no	—
170	C₄×D₄	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
173	#108	—	2	4	no	—
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
177	#102	—	2	2	no	—
178	#102	—	2	2	no	—
179	C₂²⋊C₈	—	2	1	no	—
180	#97	—	2	2	no	—
181	#97	D₄×C₃₁	1	2	yes	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
188	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
189	C₃₁	#9726	1	1	yes	characteristic, direct product factor, Sylow
190	#41	—	8	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	4	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂³	—	4	2	no	—
223	C₂³	C₃₁×#124	1	1	yes	characteristic
224	Q₈	—	8	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	4	no	—
228	D₄	—	8	2	no	—
229	D₄	—	8	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
240	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
241	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	—	4	4	no	—
247	C₈	—	4	2	no	—
248	C₈	C₃₁×#104	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	C₃₁×#390	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	8	2	no	—
260	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
261	C₄	C₃₁×#421	1	1	yes	characteristic
262	C₄	C₃₁×#444	1	2	yes	—
263	C₄	C₃₁×#472	1	2	yes	—
264	C₂	—	4	1	no	—
265	C₂	—	16	2	no	—
266	C₂	C₃₁×#1975	1	1	yes	characteristic
267	C₂	C₃₁×#1978	1	1	yes	characteristic
268	C₂	C₃₁×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
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. C₄×C₃₁×#108	C₂	semidirect
3. C₃₁×#1525	C₂	semidirect
4. C₃₁×#1484	C₂	semidirect
5. C₃₁×#1451	C₂	semidirect
87. #9726	C₃₁	direct (N × H)
116. C₄×#108	C₂×C₃₁	semidirect
117. #1525	C₂×C₃₁	semidirect
118. #1484	C₂×C₃₁	semidirect
119. #1451	C₂×C₃₁	semidirect
189. C₃₁	#9726	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3169	2
C₄×C₃₁×#108	2
C₃₁×#1525	2
C₃₁×#1484	2
C₃₁×#1451	2
#9726	31

Subgroups — C31×#9726 #21466485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9726 #21466485