Subgroups — C₃₁×#12817 #21469576

Order 7936.

430 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#360	C₂	1	2	yes	maximal
4	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
7	C₄×#41×C₃₁	—	2	1	no	—
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₃₁	C₄	1	2	yes	complemented
10	C₄×SD₁₆×C₃₁	—	2	2	no	—
11	C₄×D₈×C₃₁	C₄	1	2	yes	complemented
12	C₃₁×#427	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#107	—	2	1	no	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₃₁×#416	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#382	—	2	1	no	—
18	C₃₁×C₄⋊C₁₆	—	2	2	no	—
19	C₃₁×#360	—	2	4	no	—
20	C₂×#41×C₃₁	—	2	1	no	—
21	C₃₁×#135	—	2	1	no	—
22	C₃₁×#135	—	2	1	no	—
23	C₃₁×#135	D₄	1	2	yes	—
24	C₃₁×#135	Q₈	1	2	yes	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₄×Q₈×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	—	4	2	no	—
33	C₃₁×#117	—	2	1	no	—
34	C₃₁×#117	—	4	1	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄²×C₃₁	—	4	1	no	—
37	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
38	C₃₁×M₃₂	—	2	2	no	—
39	C₃₁×M₃₂	—	2	2	no	—
40	C₂×C₁₆×C₃₁	—	2	2	no	—
41	C₃₁×#107	—	4	2	no	—
42	C₃₁×#107	C₂×C₄	1	2	yes	—
43	C₃₁×#106	—	2	1	no	—
44	C₃₁×#106	—	2	2	no	—
45	C₃₁×#106	—	2	2	no	—
46	C₃₁×#103	—	2	2	no	—
47	C₃₁×#102	—	2	2	no	—
48	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₃₁	D₄	1	2	yes	—
50	C₃₁×#95	—	4	2	no	—
51	#41×C₃₁	—	2	2	no	—
52	#41×C₃₁	—	2	2	no	—
53	#41×C₃₁	—	4	1	no	—
54	#41×C₃₁	—	4	1	no	—
55	C₂×Q₈×C₃₁	—	2	1	no	—
56	C₂×D₄×C₃₁	—	2	1	no	—
57	C₂×D₄×C₃₁	—	4	1	no	—
58	C₂²×C₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	D₈	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₃₁	—	2	1	no	—
64	Q₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
65	SD₁₆×C₃₁	—	2	2	no	—
66	SD₁₆×C₃₁	—	2	2	no	—
67	D₈×C₃₁	—	2	1	no	—
68	D₈×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	—	2	1	no	—
71	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
77	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	1	no	—
83	C₂²⋊C₄×C₃₁	—	4	2	no	—
84	C₄²×C₃₁	—	4	2	no	—
85	C₄²×C₃₁	—	4	1	no	—
86	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
88	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
89	C₁₆×C₃₁	—	4	2	no	—
90	#12817	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	4	1	no	—
92	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	—	4	1	no	—
95	D₄×C₃₁	—	2	2	no	—
96	D₄×C₃₁	—	4	2	no	—
97	D₄×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	4	1	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	1	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	8	4	no	—
105	C₂×C₄×C₃₁	—	8	4	no	—
106	C₂×C₄×C₃₁	—	8	2	no	—
107	C₂×C₄×C₃₁	#102	1	2	yes	—
108	C₂×C₄×C₃₁	#102	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
115	C₈×C₃₁	—	2	2	no	—
116	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₈×C₃₁	#117	1	1	yes	characteristic
118	C₈×C₃₁	#121	1	1	yes	characteristic
119	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
121	#2150	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂×#360	C₂×C₃₁	1	2	yes	—
123	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#1386	C₂×C₃₁	1	2	yes	complemented
125	C₂²×C₃₁	—	2	2	no	—
126	C₂²×C₃₁	—	4	2	no	—
127	C₂²×C₃₁	—	8	2	no	—
128	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
129	C₂²×C₃₁	#459	1	1	yes	characteristic
130	C₂²×C₃₁	#474	1	1	yes	characteristic
131	C₄×C₃₁	—	2	2	no	—
132	C₄×C₃₁	—	2	2	no	—
133	C₄×C₃₁	—	4	2	no	—
134	C₄×C₃₁	—	8	2	no	—
135	C₄×C₃₁	—	16	2	no	—
136	C₄×C₃₁	#380	1	1	yes	characteristic
137	C₄×C₃₁	#417	1	1	yes	characteristic
138	C₄×C₃₁	#461	1	1	yes	characteristic
139	C₄×C₃₁	#476	1	1	yes	characteristic
140	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
141	C₄×#41	—	2	1	no	—
142	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
143	C₄×Q₁₆	C₄×C₃₁	1	2	yes	complemented
144	C₄×SD₁₆	—	2	2	no	—
145	C₄×D₈	C₄×C₃₁	1	2	yes	complemented
146	#427	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×#107	—	2	1	no	—
148	C₂×#106	C₂²×C₃₁	1	2	yes	—
149	#416	—	2	1	no	—
150	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	#382	—	2	1	no	—
152	C₄⋊C₁₆	—	2	2	no	—
153	#360	—	2	4	no	—
154	C₂×C₃₁	—	2	1	no	—
155	C₂×C₃₁	—	2	1	no	—
156	C₂×C₃₁	—	8	2	no	—
157	C₂×C₃₁	#1802	1	1	yes	characteristic
158	C₂×C₃₁	#2145	1	2	yes	—
159	C₂×#41	—	2	1	no	—
160	#135	—	2	1	no	—
161	#135	—	2	1	no	—
162	#135	D₄×C₃₁	1	2	yes	—
163	#135	Q₈×C₃₁	1	2	yes	—
164	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×SD₁₆	—	2	1	no	—
166	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
169	C₄×Q₈	—	2	2	no	—
170	C₄×D₄	—	2	2	no	—
171	C₄×D₄	—	4	2	no	—
172	#117	—	2	1	no	—
173	#117	—	4	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	4	1	no	—
176	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
177	M₃₂	—	2	2	no	—
178	M₃₂	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	#107	—	4	2	no	—
181	#107	C₂×C₄×C₃₁	1	2	yes	—
182	#106	—	2	1	no	—
183	#106	—	2	2	no	—
184	#106	—	2	2	no	—
185	#103	—	2	2	no	—
186	#102	—	2	2	no	—
187	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
188	C₄×C₈	D₄×C₃₁	1	2	yes	—
189	#95	—	4	2	no	—
190	C₃₁	#12817	1	1	yes	characteristic, direct product factor, Sylow
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	D₈×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
203	Q₁₆	—	2	1	no	—
204	Q₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
205	SD₁₆	—	2	2	no	—
206	SD₁₆	—	2	2	no	—
207	D₈	—	2	1	no	—
208	D₈	C₄⋊C₄×C₃₁	1	2	yes	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
216	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
217	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	1	no	—
226	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
227	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
228	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
229	C₁₆	—	4	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	2	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₃₁×#102	1	2	yes	—
247	C₂×C₄	C₃₁×#102	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
250	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
256	C₈	C₃₁×#117	1	1	yes	characteristic
257	C₈	C₃₁×#121	1	1	yes	characteristic
258	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
263	C₂²	C₃₁×#459	1	1	yes	characteristic
264	C₂²	C₃₁×#474	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	16	2	no	—
270	C₄	C₃₁×#380	1	1	yes	characteristic
271	C₄	C₃₁×#417	1	1	yes	characteristic
272	C₄	C₃₁×#461	1	1	yes	characteristic
273	C₄	C₃₁×#476	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	8	2	no	—
277	C₂	C₃₁×#1802	1	1	yes	characteristic
278	C₂	C₃₁×#2145	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2150	C₂	semidirect
4. C₃₁×#1846	C₂	semidirect
5. C₃₁×#1386	C₂	semidirect
9. C₄×Q₁₆×C₃₁	C₄	semidirect
11. C₄×D₈×C₃₁	C₄	semidirect
90. #12817	C₃₁	direct (N × H)
121. #2150	C₂×C₃₁	semidirect
123. #1846	C₂×C₃₁	semidirect
124. #1386	C₂×C₃₁	semidirect
143. C₄×Q₁₆	C₄×C₃₁	semidirect
145. C₄×D₈	C₄×C₃₁	semidirect
190. C₃₁	#12817	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×#2150	2
C₂×C₃₁×#360	2
C₃₁×#1846	2
C₃₁×#1386	2
#12817	31

Subgroups — C31×#12817 #21469576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12817 #21469576