Subgroups — C₃₁×#33659 #21490418

Order 7936.

546 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3060	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2239	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2238	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#365	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2148	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2115	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₃₁×#568	—	2	1	no	—
10	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
11	C₃₁×#537	—	2	1	no	—
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₃₁×#470	C₂²	1	2	yes	—
14	C₃₁×#469	C₂²	1	2	yes	—
15	C₃₁×#465	—	2	2	no	—
16	C₃₁×#430	—	2	1	no	—
17	C₃₁×#430	C₂²	1	4	yes	—
18	C₂×C₃₁×#108	—	2	2	no	—
19	C₂×C₃₁×#108	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₃₁×#417	—	2	1	no	—
22	C₃₁×#365	C₂²	1	8	yes	—
23	C₃₁×#362	C₂²	1	4	yes	—
24	C₃₁×#361	C₂²	1	4	yes	—
25	C₃₁×#349	C₂²	1	4	yes	—
26	C₂×#41×C₃₁	—	2	1	no	—
27	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×#135	—	2	2	no	—
29	C₃₁×#135	—	2	2	no	—
30	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×SD₁₆×C₃₁	—	2	1	no	—
32	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#131	—	2	2	no	—
34	C₃₁×#131	—	2	2	no	—
35	C₃₁×#131	—	4	4	no	—
36	C₂×M₁₆×C₃₁	—	2	4	no	—
37	C₂×M₁₆×C₃₁	—	2	2	no	—
38	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	D₄	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×C₂²⋊Q₈	—	2	2	no	—
43	C₃₁×C₂²⋊Q₈	—	4	1	no	—
44	C₃₁×#121	—	2	2	no	—
45	C₃₁×#121	—	4	1	no	—
46	C₃₁×#121	—	4	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₃₁×M₃₂	—	2	2	no	—
49	C₃₁×M₃₂	C₂³	1	4	yes	—
50	C₂×C₁₆×C₃₁	—	2	2	no	—
51	C₃₁×#108	—	2	2	no	—
52	C₃₁×#108	—	2	4	no	—
53	C₃₁×#108	C₂³	1	4	yes	—
54	C₃₁×#106	—	2	1	no	—
55	C₃₁×#106	C₂³	1	2	yes	—
56	C₃₁×#103	—	2	2	no	—
57	C₃₁×#102	—	2	2	no	—
58	#41×C₃₁	—	2	2	no	—
59	#41×C₃₁	—	2	2	no	—
60	#41×C₃₁	—	4	2	no	—
61	#41×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	2	1	no	—
63	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
64	C₂×D₄×C₃₁	—	2	1	no	—
65	C₂×D₄×C₃₁	—	4	1	no	—
66	C₂×D₄×C₃₁	—	4	2	no	—
67	C₂×D₄×C₃₁	SD₁₆	1	2	yes	—
68	C₂×D₄×C₃₁	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	—	4	1	no	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₃₁	—	2	2	no	—
74	SD₁₆×C₃₁	—	4	1	no	—
75	SD₁₆×C₃₁	—	4	1	no	—
76	D₈×C₃₁	—	2	2	no	—
77	M₁₆×C₃₁	—	2	2	no	—
78	M₁₆×C₃₁	—	2	4	no	—
79	M₁₆×C₃₁	—	2	4	no	—
80	M₁₆×C₃₁	—	4	4	no	—
81	M₁₆×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	4	4	no	—
85	C₂×C₈×C₃₁	—	4	2	no	—
86	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	#41	1	2	yes	—
89	C₂×C₈×C₃₁	#41	1	2	yes	—
90	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	4	1	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₂²⋊C₄×C₃₁	—	4	2	no	—
95	C₂²⋊C₄×C₃₁	—	4	2	no	—
96	C₁₆×C₃₁	—	2	4	no	—
97	#33659	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	4	1	no	—
99	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
100	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
101	Q₈×C₃₁	—	2	2	no	—
102	Q₈×C₃₁	—	4	1	no	—
103	Q₈×C₃₁	—	4	1	no	—
104	D₄×C₃₁	—	2	4	no	—
105	D₄×C₃₁	—	2	2	no	—
106	D₄×C₃₁	—	4	1	no	—
107	D₄×C₃₁	—	4	1	no	—
108	D₄×C₃₁	—	8	2	no	—
109	D₄×C₃₁	—	16	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	1	no	—
114	C₂×C₄×C₃₁	—	8	2	no	—
115	C₂×C₄×C₃₁	—	16	1	no	—
116	C₂×C₄×C₃₁	#123	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₃₁	—	2	2	no	—
122	C₈×C₃₁	—	4	4	no	—
123	C₈×C₃₁	#123	1	1	yes	characteristic
124	C₈×C₃₁	#123	1	1	yes	characteristic
125	C₈×C₃₁	C₂×#41	1	2	yes	—
126	#3060	C₂×C₃₁	1	1	yes	characteristic
127	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#2239	C₂×C₃₁	1	4	yes	complemented
129	#2238	C₂×C₃₁	1	4	yes	complemented
130	C₂×#365	C₂×C₃₁	1	2	yes	complemented
131	#2148	C₂×C₃₁	1	2	yes	complemented
132	#2115	C₂×C₃₁	1	1	yes	characteristic, complemented
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	4	1	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	—	8	1	no	—
137	C₂²×C₃₁	—	8	1	no	—
138	C₂²×C₃₁	—	8	1	no	—
139	C₂²×C₃₁	#481	1	2	yes	—
140	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	8	2	no	—
143	C₄×C₃₁	—	8	1	no	—
144	C₄×C₃₁	#481	1	2	yes	—
145	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
146	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
148	#568	—	2	1	no	—
149	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
150	#537	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
152	#470	C₂²×C₃₁	1	2	yes	—
153	#469	C₂²×C₃₁	1	2	yes	—
154	#465	—	2	2	no	—
155	#430	—	2	1	no	—
156	#430	C₂²×C₃₁	1	4	yes	—
157	C₂×#108	—	2	2	no	—
158	C₂×#108	C₂²×C₃₁	1	2	yes	—
159	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
160	#417	—	2	1	no	—
161	#365	C₂²×C₃₁	1	8	yes	—
162	#362	C₂²×C₃₁	1	4	yes	—
163	#361	C₂²×C₃₁	1	4	yes	—
164	#349	C₂²×C₃₁	1	4	yes	—
165	C₂×C₃₁	—	2	1	no	—
166	C₂×C₃₁	—	2	2	no	—
167	C₂×C₃₁	—	4	2	no	—
168	C₂×C₃₁	—	16	1	no	—
169	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
170	C₂×#41	—	2	1	no	—
171	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
172	#135	—	2	2	no	—
173	#135	—	2	2	no	—
174	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
175	C₂×SD₁₆	—	2	1	no	—
176	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
177	#131	—	2	2	no	—
178	#131	—	2	2	no	—
179	#131	—	4	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
183	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
184	C₂²×C₈	D₄×C₃₁	1	2	yes	—
185	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
186	C₂²⋊Q₈	—	2	2	no	—
187	C₂²⋊Q₈	—	4	1	no	—
188	#121	—	2	2	no	—
189	#121	—	4	1	no	—
190	#121	—	4	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	C₂³×C₃₁	1	4	yes	—
194	C₂×C₁₆	—	2	2	no	—
195	#108	—	2	2	no	—
196	#108	—	2	4	no	—
197	#108	C₂³×C₃₁	1	4	yes	—
198	#106	—	2	1	no	—
199	#106	C₂³×C₃₁	1	2	yes	—
200	#103	—	2	2	no	—
201	#102	—	2	2	no	—
202	C₃₁	#33659	1	1	yes	characteristic, direct product factor, Sylow
203	#41	—	2	2	no	—
204	#41	—	2	2	no	—
205	#41	—	4	2	no	—
206	#41	—	4	2	no	—
207	C₂×Q₈	—	2	1	no	—
208	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	SD₁₆×C₃₁	1	2	yes	—
213	C₂×D₄	SD₁₆×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	Q₁₆	—	2	2	no	—
219	SD₁₆	—	4	1	no	—
220	SD₁₆	—	4	1	no	—
221	D₈	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	4	no	—
226	M₁₆	—	4	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₈	#41×C₃₁	1	2	yes	—
234	C₂×C₈	#41×C₃₁	1	2	yes	—
235	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₁₆	—	2	4	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
244	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	2	4	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	1	no	—
252	D₄	—	8	2	no	—
253	D₄	—	16	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	16	1	no	—
260	C₂×C₄	C₃₁×#123	1	2	yes	—
261	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
262	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
264	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	4	4	no	—
267	C₈	C₃₁×#123	1	1	yes	characteristic
268	C₈	C₃₁×#123	1	1	yes	characteristic
269	C₈	C₂×#41×C₃₁	1	2	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	8	1	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	1	no	—
276	C₂²	C₃₁×#481	1	2	yes	—
277	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	1	no	—
281	C₄	C₃₁×#481	1	2	yes	—
282	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
283	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	16	1	no	—
288	C₂	C₂×C₃₁×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3060	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2916	C₂	semidirect
3. C₃₁×#2239	C₂	semidirect
4. C₃₁×#2238	C₂	semidirect
5. C₂×C₃₁×#365	C₂	semidirect
6. C₃₁×#2148	C₂	semidirect
7. C₃₁×#2115	C₂	semidirect
97. #33659	C₃₁	direct (N × H)
127. #2916	C₂×C₃₁	semidirect
128. #2239	C₂×C₃₁	semidirect
129. #2238	C₂×C₃₁	semidirect
130. C₂×#365	C₂×C₃₁	semidirect
131. #2148	C₂×C₃₁	semidirect
132. #2115	C₂×C₃₁	semidirect
202. C₃₁	#33659	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×#3060	2
C₃₁×#2916	2
C₃₁×#2239	2
C₃₁×#2238	2
C₂×C₃₁×#365	2
C₃₁×#2148	2
C₃₁×#2115	2
#33659	31

Subgroups — C31×#33659 #21490418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33659 #21490418