Subgroups — C₃₁×#21294 #21478053

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2923	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1974	C₂	1	2	yes	maximal
4	C₃₁×#1945	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1938	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1759	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
10	C₄×#41×C₃₁	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₃₁	C₄	1	2	yes	complemented
12	C₄×SD₁₆×C₃₁	C₄	1	4	yes	complemented
13	C₄×D₈×C₃₁	C₄	1	2	yes	complemented
14	C₃₁×#433	—	2	4	no	—
15	C₃₁×#432	—	2	4	no	—
16	C₂×C₃₁×#108	C₂²	1	2	yes	—
17	C₃₁×#427	C₂²	1	1	yes	characteristic
18	C₃₁×#424	C₂²	1	4	yes	—
19	C₂×C₃₁×#104	C₂²	1	4	yes	—
20	C₃₁×#416	C₂²	1	2	yes	—
21	C₃₁×#408	C₂²	1	4	yes	—
22	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#376	C₂²	1	1	yes	characteristic
28	C₃₁×#341	C₂²	1	2	yes	—
29	C₃₁×#339	C₂²	1	4	yes	—
30	C₂×#41×C₃₁	C₂³	1	2	yes	—
31	C₃₁×#135	—	2	4	no	—
32	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
40	C₄×D₄×C₃₁	—	2	4	no	—
41	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	C₂³	1	2	yes	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#108	—	2	4	no	—
48	C₃₁×#107	C₂×C₄	1	2	yes	—
49	C₃₁×#106	C₂×C₄	1	2	yes	—
50	C₃₁×C₄⋊C₈	—	2	8	no	—
51	C₃₁×#104	—	2	8	no	—
52	C₃₁×#103	C₂×C₄	1	4	yes	—
53	C₃₁×#102	C₂×C₄	1	4	yes	—
54	C₃₁×C₂²⋊C₈	—	2	8	no	—
55	C₃₁×#97	—	2	2	no	—
56	C₃₁×#97	—	2	4	no	—
57	C₃₁×#97	D₄	1	2	yes	—
58	C₃₁×#97	D₄	1	2	yes	—
59	C₄×C₈×C₃₁	—	2	4	no	—
60	C₄×C₈×C₃₁	—	2	2	no	—
61	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
63	C₃₁×#95	—	2	2	no	—
64	#41×C₃₁	—	2	4	no	—
65	#41×C₃₁	—	2	4	no	—
66	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	#41	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	Q₁₆×C₃₁	—	2	2	no	—
75	SD₁₆×C₃₁	—	2	4	no	—
76	D₈×C₃₁	—	2	2	no	—
77	M₁₆×C₃₁	—	2	4	no	—
78	M₁₆×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₃₁	#41	1	4	yes	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂²⋊C₄×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	4	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₃₁	#41	1	4	yes	—
99	#21294	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	2	no	—
101	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
102	Q₈×C₃₁	—	2	2	no	—
103	Q₈×C₃₁	—	2	2	no	—
104	D₄×C₃₁	—	2	2	no	—
105	D₄×C₃₁	—	2	2	no	—
106	D₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	8	4	no	—
114	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	—	2	4	no	—
122	C₈×C₃₁	—	4	4	no	—
123	C₈×C₃₁	#117	1	2	yes	—
124	C₈×C₃₁	#117	1	2	yes	—
125	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
126	#2923	C₂×C₃₁	1	2	yes	complemented
127	#1974	C₂×C₃₁	1	2	yes	—
128	#1945	C₂×C₃₁	1	2	yes	complemented
129	#1938	C₂×C₃₁	1	4	yes	complemented
130	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#1759	C₂×C₃₁	1	2	yes	complemented
132	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	4	4	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₃₁	#529	1	2	yes	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	4	4	no	—
141	C₄×C₃₁	—	4	4	no	—
142	C₄×C₃₁	—	4	2	no	—
143	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
144	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
145	C₄×C₃₁	#532	1	2	yes	—
146	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
147	C₄×#41	C₂²×C₃₁	1	2	yes	—
148	C₄×Q₁₆	C₄×C₃₁	1	2	yes	complemented
149	C₄×SD₁₆	C₄×C₃₁	1	4	yes	complemented
150	C₄×D₈	C₄×C₃₁	1	2	yes	complemented
151	#433	—	2	4	no	—
152	#432	—	2	4	no	—
153	C₂×#108	C₂²×C₃₁	1	2	yes	—
154	#427	C₂²×C₃₁	1	1	yes	characteristic
155	#424	C₂²×C₃₁	1	4	yes	—
156	C₂×#104	C₂²×C₃₁	1	4	yes	—
157	#416	C₂²×C₃₁	1	2	yes	—
158	#408	C₂²×C₃₁	1	4	yes	—
159	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
160	C₂×#97	C₂²×C₃₁	1	2	yes	—
161	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
164	#376	C₂²×C₃₁	1	1	yes	characteristic
165	#341	C₂²×C₃₁	1	2	yes	—
166	#339	C₂²×C₃₁	1	4	yes	—
167	C₂×C₃₁	—	2	2	no	—
168	C₂×C₃₁	—	8	2	no	—
169	C₂×C₃₁	#2293	1	1	yes	characteristic
170	C₂×C₃₁	#2957	1	2	yes	—
171	C₂×#41	C₂³×C₃₁	1	2	yes	—
172	#135	—	2	4	no	—
173	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
177	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
179	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
180	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
181	C₄×D₄	—	2	4	no	—
182	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
183	#117	—	2	2	no	—
184	#117	C₂³×C₃₁	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
187	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
188	#108	—	2	4	no	—
189	#107	C₂×C₄×C₃₁	1	2	yes	—
190	#106	C₂×C₄×C₃₁	1	2	yes	—
191	C₄⋊C₈	—	2	8	no	—
192	#104	—	2	8	no	—
193	#103	C₂×C₄×C₃₁	1	4	yes	—
194	#102	C₂×C₄×C₃₁	1	4	yes	—
195	C₂²⋊C₈	—	2	8	no	—
196	#97	—	2	2	no	—
197	#97	—	2	4	no	—
198	#97	D₄×C₃₁	1	2	yes	—
199	#97	D₄×C₃₁	1	2	yes	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
203	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
204	#95	—	2	2	no	—
205	C₃₁	#21294	1	1	yes	characteristic, direct product factor, Sylow
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	#41×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	2	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
229	C₂×C₈	#41×C₃₁	1	4	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
238	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₄²	#41×C₃₁	1	4	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
256	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
257	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
260	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	4	no	—
264	C₈	C₃₁×#117	1	2	yes	—
265	C₈	C₃₁×#117	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	2	no	—
269	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
270	C₂²	C₃₁×#529	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	4	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	2	no	—
276	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
277	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
278	C₄	C₃₁×#532	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	8	2	no	—
281	C₂	C₃₁×#2293	1	1	yes	characteristic
282	C₂	C₃₁×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2923	C₂	semidirect
4. C₃₁×#1945	C₂	semidirect
5. C₃₁×#1938	C₂	semidirect
6. C₄×C₃₁×#108	C₂	semidirect
7. C₃₁×#1759	C₂	semidirect
8. C₃₁×#1726	C₂	semidirect
11. C₄×Q₁₆×C₃₁	C₄	semidirect
12. C₄×SD₁₆×C₃₁	C₄	semidirect
13. C₄×D₈×C₃₁	C₄	semidirect
99. #21294	C₃₁	direct (N × H)
126. #2923	C₂×C₃₁	semidirect
128. #1945	C₂×C₃₁	semidirect
129. #1938	C₂×C₃₁	semidirect
130. C₄×#108	C₂×C₃₁	semidirect
131. #1759	C₂×C₃₁	semidirect
132. #1726	C₂×C₃₁	semidirect
148. C₄×Q₁₆	C₄×C₃₁	semidirect
149. C₄×SD₁₆	C₄×C₃₁	semidirect
150. C₄×D₈	C₄×C₃₁	semidirect
205. C₃₁	#21294	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#135	2
C₃₁×#2923	2
C₃₁×#1974	2
C₃₁×#1945	2
C₃₁×#1938	2
C₄×C₃₁×#108	2
C₃₁×#1759	2
C₃₁×#1726	2
#21294	31

Subgroups — C31×#21294 #21478053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21294 #21478053