Subgroups — C₃₁×#10607 #21467366

Order 7936.

374 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
9	C₃₁×#431	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#108	—	2	1	no	—
11	C₃₁×#423	C₂²	1	1	yes	characteristic
12	C₃₁×#423	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#100	—	2	1	no	—
16	C₂×C₃₁×#100	—	2	1	no	—
17	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#341	—	2	1	no	—
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#334	—	2	1	no	—
21	C₃₁×#334	—	2	2	no	—
22	C₃₁×#331	—	2	1	no	—
23	C₃₁×#331	—	2	1	no	—
24	C₃₁×#331	—	2	2	no	—
25	C₂²×D₄×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	Q₈	1	2	yes	—
28	C₂×M₁₆×C₃₁	—	2	1	no	—
29	C₂×M₁₆×C₃₁	—	2	1	no	—
30	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₃₁×#127	—	2	1	no	—
35	C₃₁×#127	—	2	1	no	—
36	C₃₁×#127	—	2	1	no	—
37	C₃₁×#127	—	2	1	no	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#108	—	4	2	no	—
40	C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₃₁×C₄⋊C₈	—	2	1	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
48	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
50	C₃₁×#100	—	4	2	no	—
51	C₃₁×#100	—	4	2	no	—
52	C₃₁×C₂²⋊C₈	—	2	1	no	—
53	C₃₁×#97	—	2	1	no	—
54	C₄×C₈×C₃₁	—	2	1	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₂⁴×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	4	1	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂×D₄×C₃₁	—	4	1	no	—
62	C₂×D₄×C₃₁	—	4	1	no	—
63	C₂×D₄×C₃₁	—	4	1	no	—
64	C₂×D₄×C₃₁	—	4	1	no	—
65	C₂×D₄×C₃₁	—	4	1	no	—
66	C₂×D₄×C₃₁	—	4	1	no	—
67	C₂×D₄×C₃₁	—	8	1	no	—
68	C₂×D₄×C₃₁	—	8	1	no	—
69	C₂×D₄×C₃₁	—	8	2	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	M₁₆×C₃₁	—	2	1	no	—
74	M₁₆×C₃₁	—	2	1	no	—
75	M₁₆×C₃₁	—	2	1	no	—
76	M₁₆×C₃₁	—	4	1	no	—
77	M₁₆×C₃₁	—	4	1	no	—
78	M₁₆×C₃₁	—	4	1	no	—
79	M₁₆×C₃₁	—	4	1	no	—
80	M₁₆×C₃₁	SD₁₆	1	2	yes	—
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	2	1	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	4	1	no	—
87	C₂×C₈×C₃₁	—	4	1	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	1	no	—
90	C₂×C₈×C₃₁	—	4	1	no	—
91	C₂×C₈×C₃₁	—	8	1	no	—
92	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₃₁	#41	1	1	yes	characteristic
95	C₄²×C₃₁	#41	1	1	yes	characteristic
96	#10607	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	4	2	no	—
98	C₂³×C₃₁	—	4	2	no	—
99	C₂³×C₃₁	—	4	2	no	—
100	C₂³×C₃₁	—	8	2	no	—
101	C₂³×C₃₁	—	8	2	no	—
102	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
103	D₄×C₃₁	—	8	2	no	—
104	D₄×C₃₁	—	8	2	no	—
105	D₄×C₃₁	—	8	2	no	—
106	D₄×C₃₁	—	8	1	no	—
107	D₄×C₃₁	—	8	1	no	—
108	D₄×C₃₁	—	8	1	no	—
109	D₄×C₃₁	—	8	1	no	—
110	D₄×C₃₁	—	8	1	no	—
111	D₄×C₃₁	—	8	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#126	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
123	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₈×C₃₁	—	2	2	no	—
126	C₈×C₃₁	—	4	2	no	—
127	C₈×C₃₁	—	4	1	no	—
128	C₈×C₃₁	—	4	1	no	—
129	C₈×C₃₁	—	4	1	no	—
130	C₈×C₃₁	—	8	1	no	—
131	C₈×C₃₁	—	8	1	no	—
132	C₈×C₃₁	—	8	1	no	—
133	#1965	C₂×C₃₁	1	1	yes	characteristic
134	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#1545	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#331	C₂×C₃₁	1	1	yes	characteristic
139	#1292	C₂×C₃₁	1	1	yes	characteristic, complemented
140	C₂²×C₃₁	—	2	1	no	—
141	C₂²×C₃₁	—	2	1	no	—
142	C₂²×C₃₁	—	8	2	no	—
143	C₂²×C₃₁	—	8	2	no	—
144	C₂²×C₃₁	—	8	2	no	—
145	C₂²×C₃₁	—	16	2	no	—
146	C₂²×C₃₁	—	16	2	no	—
147	C₂²×C₃₁	#396	1	1	yes	characteristic
148	C₂²×C₃₁	#476	1	1	yes	characteristic
149	C₂²×C₃₁	#478	1	1	yes	characteristic
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	#460	1	1	yes	characteristic
154	C₄×C₃₁	#463	1	1	yes	characteristic
155	C₄×C₃₁	#465	1	1	yes	characteristic
156	C₄×C₃₁	#469	1	1	yes	characteristic
157	C₄×C₃₁	#471	1	1	yes	characteristic
158	C₄×C₃₁	#472	1	1	yes	characteristic
159	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
160	#431	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#108	—	2	1	no	—
162	#423	C₂²×C₃₁	1	1	yes	characteristic
163	#423	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#100	—	2	1	no	—
167	C₂×#100	—	2	1	no	—
168	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
169	#341	—	2	1	no	—
170	#335	—	2	1	no	—
171	#334	—	2	1	no	—
172	#334	—	2	2	no	—
173	#331	—	2	2	no	—
174	#331	—	2	1	no	—
175	#331	—	2	1	no	—
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	—	2	1	no	—
178	C₂×C₃₁	—	16	2	no	—
179	C₂×C₃₁	#2063	1	1	yes	characteristic
180	C₂×C₃₁	#2065	1	1	yes	characteristic
181	C₂×C₃₁	#2069	1	1	yes	characteristic
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	—	2	1	no	—
184	C₂²×D₄	Q₈×C₃₁	1	2	yes	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
188	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	#127	—	2	1	no	—
192	#127	—	2	1	no	—
193	#127	—	2	1	no	—
194	#127	—	2	1	no	—
195	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
196	#108	—	4	2	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
205	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
206	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
207	#100	—	4	2	no	—
208	#100	—	4	2	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	#97	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₃₁	#10607	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	8	1	no	—
226	C₂×D₄	—	8	1	no	—
227	C₂×D₄	—	8	2	no	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	4	1	no	—
237	M₁₆	—	4	1	no	—
238	M₁₆	SD₁₆×C₃₁	1	2	yes	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	8	1	no	—
250	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₄²	#41×C₃₁	1	1	yes	characteristic
253	C₄²	#41×C₃₁	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	2	no	—
259	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
260	D₄	—	8	2	no	—
261	D₄	—	8	2	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	1	no	—
264	D₄	—	8	1	no	—
265	D₄	—	8	1	no	—
266	D₄	—	8	1	no	—
267	D₄	—	8	1	no	—
268	D₄	—	8	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
274	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
275	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
277	C₂×C₄	C₃₁×#126	1	1	yes	characteristic
278	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
280	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	8	1	no	—
288	C₈	—	8	1	no	—
289	C₈	—	8	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	2	no	—
296	C₂²	—	16	2	no	—
297	C₂²	C₃₁×#396	1	1	yes	characteristic
298	C₂²	C₃₁×#476	1	1	yes	characteristic
299	C₂²	C₃₁×#478	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	C₃₁×#460	1	1	yes	characteristic
304	C₄	C₃₁×#463	1	1	yes	characteristic
305	C₄	C₃₁×#465	1	1	yes	characteristic
306	C₄	C₃₁×#469	1	1	yes	characteristic
307	C₄	C₃₁×#471	1	1	yes	characteristic
308	C₄	C₃₁×#472	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	16	2	no	—
312	C₂	C₃₁×#2063	1	1	yes	characteristic
313	C₂	C₃₁×#2065	1	1	yes	characteristic
314	C₂	C₃₁×#2069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1950	C₂	semidirect
3. C₃₁×#1570	C₂	semidirect
4. C₃₁×#1565	C₂	semidirect
7. C₃₁×#1292	C₂	semidirect
96. #10607	C₃₁	direct (N × H)
134. #1950	C₂×C₃₁	semidirect
135. #1570	C₂×C₃₁	semidirect
136. #1565	C₂×C₃₁	semidirect
139. #1292	C₂×C₃₁	semidirect
213. C₃₁	#10607	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×#1965	2
C₃₁×#1950	2
C₃₁×#1570	2
C₃₁×#1565	2
C₃₁×#1545	2
C₂×C₃₁×#331	2
C₃₁×#1292	2
#10607	31

Subgroups — C31×#10607 #21467366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10607 #21467366