Subgroups — C₃₁×#3219 #16843152

Order 3968.

396 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#464	C₂	1	2	yes	maximal, complemented
8	C₃₁×#463	C₂	1	2	yes	maximal, complemented
9	C₃₁×#452	C₂	1	2	yes	maximal, complemented
10	C₃₁×#449	C₂	1	2	yes	maximal, complemented
11	C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂×D₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×#127	C₂²	1	1	yes	characteristic
19	C₃₁×#126	—	2	1	no	—
20	C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₃₁×#124	C₂²	1	2	yes	—
22	C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₃₁×#123	—	2	1	no	—
24	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×#121	—	2	1	no	—
26	C₃₁×#121	—	2	1	no	—
27	C₃₁×#121	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₄×D₄×C₃₁	—	2	1	no	—
32	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#117	—	2	1	no	—
34	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
36	C₃₁×#103	C₂²	1	2	yes	—
37	C₃₁×#103	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×#102	C₂²	1	2	yes	—
40	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₃₁×#97	C₂²	1	1	yes	characteristic
42	C₃₁×#97	C₂²	1	1	yes	characteristic
43	#41×C₃₁	—	4	2	no	—
44	#41×C₃₁	—	4	2	no	—
45	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	—	2	1	no	—
50	C₂×D₄×C₃₁	—	2	1	no	—
51	C₂×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	Q₁₆×C₃₁	—	4	2	no	—
59	SD₁₆×C₃₁	—	4	2	no	—
60	SD₁₆×C₃₁	—	4	2	no	—
61	D₈×C₃₁	—	4	2	no	—
62	C₂×C₈×C₃₁	C₂³	1	2	yes	—
63	C₂×C₈×C₃₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₄²×C₃₁	—	2	1	no	—
80	C₄²×C₃₁	D₄	1	1	yes	characteristic
81	C₄²×C₃₁	D₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂³×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₃₁	—	2	2	no	—
88	Q₈×C₃₁	—	2	2	no	—
89	Q₈×C₃₁	—	2	2	no	—
90	D₄×C₃₁	—	2	2	no	—
91	D₄×C₃₁	—	2	2	no	—
92	D₄×C₃₁	—	2	2	no	—
93	D₄×C₃₁	—	4	2	no	—
94	D₄×C₃₁	—	4	2	no	—
95	D₄×C₃₁	—	4	2	no	—
96	D₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	4	1	no	—
103	C₂×C₄×C₃₁	—	4	1	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	—	2	2	no	—
114	#3219	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	1	no	—
118	C₂²×C₃₁	—	4	2	no	—
119	C₂²×C₃₁	—	4	1	no	—
120	C₂²×C₃₁	—	4	1	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	—	4	2	no	—
123	C₂²×C₃₁	—	4	1	no	—
124	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	4	2	no	—
131	C₄×C₃₁	—	4	1	no	—
132	C₄×C₃₁	—	4	1	no	—
133	C₄×C₃₁	—	4	1	no	—
134	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
135	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#497	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#496	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#491	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#490	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#464	C₂×C₃₁	1	2	yes	complemented
142	#463	C₂×C₃₁	1	2	yes	complemented
143	#452	C₂×C₃₁	1	2	yes	complemented
144	#449	C₂×C₃₁	1	2	yes	complemented
145	#432	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂×C₃₁	—	4	1	no	—
147	C₂×C₃₁	—	8	2	no	—
148	C₂×C₃₁	—	8	1	no	—
149	C₂×C₃₁	#535	1	1	yes	characteristic
150	C₂×C₃₁	#575	1	1	yes	characteristic
151	C₂×C₃₁	#575	1	1	yes	characteristic
152	C₂×#41	C₂²×C₃₁	1	2	yes	—
153	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
154	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
155	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
156	C₂×D₈	C₂²×C₃₁	1	2	yes	—
157	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
158	#127	C₂²×C₃₁	1	1	yes	characteristic
159	#126	—	2	1	no	—
160	#125	C₂²×C₃₁	1	1	yes	characteristic
161	#124	C₂²×C₃₁	1	1	yes	characteristic
162	#124	C₂²×C₃₁	1	2	yes	—
163	#123	—	2	1	no	—
164	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
165	#121	—	2	1	no	—
166	#121	—	2	1	no	—
167	#121	C₂²×C₃₁	1	1	yes	characteristic
168	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
175	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
176	#103	C₂²×C₃₁	1	2	yes	—
177	#103	C₂²×C₃₁	1	2	yes	—
178	#102	C₂²×C₃₁	1	2	yes	—
179	#102	C₂²×C₃₁	1	2	yes	—
180	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
181	#97	C₂²×C₃₁	1	1	yes	characteristic
182	#97	C₂²×C₃₁	1	1	yes	characteristic
183	C₃₁	#3219	1	1	yes	characteristic, direct product factor, Sylow
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
187	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₂³×C₃₁	1	2	yes	—
193	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
199	Q₁₆	—	4	2	no	—
200	SD₁₆	—	4	2	no	—
201	SD₁₆	—	4	2	no	—
202	D₈	—	4	2	no	—
203	C₂×C₈	C₂³×C₃₁	1	2	yes	—
204	C₂×C₈	C₂³×C₃₁	1	2	yes	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	D₄×C₃₁	1	1	yes	characteristic
222	C₄²	D₄×C₃₁	1	1	yes	characteristic
223	C₄²	C₂³×C₃₁	1	1	yes	characteristic
224	C₄²	C₂³×C₃₁	1	1	yes	characteristic
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	2	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	—	8	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₃₁×#535	1	1	yes	characteristic
279	C₂	C₃₁×#575	1	1	yes	characteristic
280	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₃₁×#532	C₂	semidirect
2. C₃₁×#529	C₂	semidirect
3. C₃₁×#497	C₂	semidirect
4. C₃₁×#496	C₂	semidirect
5. C₃₁×#491	C₂	semidirect
6. C₃₁×#490	C₂	semidirect
7. C₃₁×#464	C₂	semidirect
8. C₃₁×#463	C₂	semidirect
9. C₃₁×#452	C₂	semidirect
10. C₃₁×#449	C₂	semidirect
11. C₃₁×#432	C₂	semidirect
114. #3219	C₃₁	direct (N × H)
135. #532	C₂×C₃₁	semidirect
136. #529	C₂×C₃₁	semidirect
137. #497	C₂×C₃₁	semidirect
138. #496	C₂×C₃₁	semidirect
139. #491	C₂×C₃₁	semidirect
140. #490	C₂×C₃₁	semidirect
141. #464	C₂×C₃₁	semidirect
142. #463	C₂×C₃₁	semidirect
143. #452	C₂×C₃₁	semidirect
144. #449	C₂×C₃₁	semidirect
145. #432	C₂×C₃₁	semidirect
183. C₃₁	#3219	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×#532	2
C₃₁×#529	2
C₃₁×#497	2
C₃₁×#496	2
C₃₁×#491	2
C₃₁×#490	2
C₃₁×#464	2
C₃₁×#463	2
C₃₁×#452	2
C₃₁×#449	2
C₃₁×#432	2
#3219	31

Subgroups — C31×#3219 #16843152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3219 #16843152