Subgroups — C₃₁×#33970 #21490729

Order 7936.

528 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3396	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2250	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2211	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2210	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2206	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#559	—	2	2	no	—
11	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
17	C₃₁×#495	C₂²	1	2	yes	—
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#493	C₂²	1	2	yes	—
20	C₃₁×#463	—	2	2	no	—
21	C₃₁×#463	—	2	2	no	—
22	C₃₁×#460	—	2	2	no	—
23	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#366	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₃₁×#358	C₂²	1	2	yes	—
31	C₃₁×#357	C₂²	1	2	yes	—
32	C₃₁×#357	C₂²	1	2	yes	—
33	C₃₁×#346	C₂²	1	2	yes	—
34	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₃₁	—	2	2	no	—
37	C₂×SD₁₆×C₃₁	—	2	2	no	—
38	C₂×D₈×C₃₁	—	2	2	no	—
39	C₂×D₈×C₃₁	C₂³	1	2	yes	—
40	C₂×D₈×C₃₁	C₂³	1	2	yes	—
41	C₃₁×#127	—	2	2	no	—
42	C₃₁×#124	—	2	2	no	—
43	C₃₁×#124	—	2	2	no	—
44	C₃₁×#124	—	4	2	no	—
45	C₃₁×#123	—	4	2	no	—
46	C₃₁×#123	—	4	2	no	—
47	C₄×Q₈×C₃₁	—	2	2	no	—
48	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
49	C₄×D₄×C₃₁	—	2	2	no	—
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	—	4	2	no	—
52	C₃₁×Q₃₂	—	4	1	no	—
53	C₃₁×SD₃₂	—	2	4	no	—
54	C₃₁×SD₃₂	—	4	2	no	—
55	C₃₁×D₁₆	—	2	2	no	—
56	C₃₁×D₁₆	—	4	2	no	—
57	C₃₁×D₁₆	—	4	1	no	—
58	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
63	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₃₁×#103	—	2	2	no	—
65	C₃₁×#102	—	2	2	no	—
66	C₃₁×#102	—	2	2	no	—
67	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	2	yes	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	4	2	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	—	4	2	no	—
76	C₂²×C₄×C₃₁	—	4	2	no	—
77	Q₁₆×C₃₁	—	2	2	no	—
78	Q₁₆×C₃₁	—	2	2	no	—
79	SD₁₆×C₃₁	—	8	2	no	—
80	SD₁₆×C₃₁	—	8	2	no	—
81	D₈×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	2	2	no	—
83	D₈×C₃₁	—	2	2	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	8	2	no	—
86	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	4	2	no	—
91	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
92	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
93	C₂²⋊C₄×C₃₁	—	4	2	no	—
94	C₂²⋊C₄×C₃₁	—	4	2	no	—
95	C₂²⋊C₄×C₃₁	—	4	2	no	—
96	C₂²⋊C₄×C₃₁	—	4	2	no	—
97	C₂²⋊C₄×C₃₁	—	4	2	no	—
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₃₁	C₂×D₄	1	2	yes	—
102	C₁₆×C₃₁	—	2	2	no	—
103	C₁₆×C₃₁	—	2	1	no	—
104	C₁₆×C₃₁	#41	1	2	yes	—
105	#33970	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	4	2	no	—
107	C₂³×C₃₁	—	4	2	no	—
108	Q₈×C₃₁	—	4	1	no	—
109	Q₈×C₃₁	—	4	2	no	—
110	Q₈×C₃₁	—	4	2	no	—
111	D₄×C₃₁	—	4	2	no	—
112	D₄×C₃₁	—	4	2	no	—
113	D₄×C₃₁	—	4	2	no	—
114	D₄×C₃₁	—	4	2	no	—
115	D₄×C₃₁	—	8	4	no	—
116	D₄×C₃₁	—	16	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	16	2	no	—
120	C₂×C₄×C₃₁	—	16	2	no	—
121	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₃₁	—	2	2	no	—
127	C₈×C₃₁	—	2	1	no	—
128	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
129	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
130	#3396	C₂×C₃₁	1	2	yes	complemented
131	#2250	C₂×C₃₁	1	2	yes	complemented
132	#2249	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#2211	C₂×C₃₁	1	2	yes	complemented
134	#2210	C₂×C₃₁	1	2	yes	complemented
135	#2206	C₂×C₃₁	1	2	yes	complemented
136	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#2175	C₂×C₃₁	1	2	yes	complemented
138	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂²×C₃₁	—	8	2	no	—
140	C₂²×C₃₁	—	8	2	no	—
141	C₂²×C₃₁	—	8	2	no	—
142	C₂²×C₃₁	—	8	2	no	—
143	C₂²×C₃₁	—	8	2	no	—
144	C₂²×C₃₁	—	8	2	no	—
145	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₃₁	—	2	2	no	—
147	C₄×C₃₁	—	2	1	no	—
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	8	2	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	#550	1	1	yes	characteristic
153	C₄×C₃₁	#576	1	1	yes	characteristic
154	#559	—	2	2	no	—
155	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
157	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
158	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
161	#495	C₂²×C₃₁	1	2	yes	—
162	#495	C₂²×C₃₁	1	2	yes	—
163	#493	C₂²×C₃₁	1	2	yes	—
164	#463	—	2	2	no	—
165	#463	—	2	2	no	—
166	#460	—	2	2	no	—
167	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
168	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
169	C₄×D₈	C₂²×C₃₁	1	2	yes	—
170	C₄×D₈	C₂²×C₃₁	1	2	yes	—
171	#366	C₂²×C₃₁	1	1	yes	characteristic
172	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
173	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
174	#358	C₂²×C₃₁	1	2	yes	—
175	#357	C₂²×C₃₁	1	2	yes	—
176	#357	C₂²×C₃₁	1	2	yes	—
177	#346	C₂²×C₃₁	1	2	yes	—
178	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×C₃₁	—	16	2	no	—
180	C₂×C₃₁	—	16	2	no	—
181	C₂×C₃₁	#3391	1	1	yes	characteristic
182	C₂×C₃₁	#3414	1	1	yes	characteristic
183	C₂×C₃₁	#3415	1	1	yes	characteristic
184	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
185	C₂×SD₁₆	—	2	2	no	—
186	C₂×SD₁₆	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	C₂³×C₃₁	1	2	yes	—
189	C₂×D₈	C₂³×C₃₁	1	2	yes	—
190	#127	—	2	2	no	—
191	#124	—	2	2	no	—
192	#124	—	2	2	no	—
193	#124	—	4	2	no	—
194	#123	—	4	2	no	—
195	#123	—	4	2	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	—	4	2	no	—
201	Q₃₂	—	4	1	no	—
202	SD₃₂	—	2	4	no	—
203	SD₃₂	—	4	2	no	—
204	D₁₆	—	2	2	no	—
205	D₁₆	—	4	2	no	—
206	D₁₆	—	4	1	no	—
207	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
209	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
210	#107	C₂³×C₃₁	1	2	yes	—
211	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
212	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
213	#103	—	2	2	no	—
214	#102	—	2	2	no	—
215	#102	—	2	2	no	—
216	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₃₁	1	2	yes	—
218	C₃₁	#33970	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	SD₁₆	—	8	2	no	—
230	SD₁₆	—	8	2	no	—
231	D₈	—	2	2	no	—
232	D₈	—	2	2	no	—
233	D₈	—	2	2	no	—
234	D₈	—	2	2	no	—
235	D₈	—	8	2	no	—
236	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
238	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
242	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₃₁	1	2	yes	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	2	1	no	—
254	C₁₆	#41×C₃₁	1	2	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	8	4	no	—
265	D₄	—	16	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	16	2	no	—
269	C₂×C₄	—	16	2	no	—
270	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
271	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
274	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	2	1	no	—
277	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
278	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	C₃₁×#550	1	1	yes	characteristic
293	C₄	C₃₁×#576	1	1	yes	characteristic
294	C₂	—	16	2	no	—
295	C₂	—	16	2	no	—
296	C₂	C₃₁×#3391	1	1	yes	characteristic
297	C₂	C₃₁×#3414	1	1	yes	characteristic
298	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3396	C₂	semidirect
2. C₃₁×#2250	C₂	semidirect
3. C₃₁×#2249	C₂	semidirect
4. C₃₁×#2211	C₂	semidirect
5. C₃₁×#2210	C₂	semidirect
6. C₃₁×#2206	C₂	semidirect
7. C₃₁×#2183	C₂	semidirect
8. C₃₁×#2175	C₂	semidirect
9. C₄×C₃₁×D₁₆	C₂	semidirect
105. #33970	C₃₁	direct (N × H)
130. #3396	C₂×C₃₁	semidirect
131. #2250	C₂×C₃₁	semidirect
132. #2249	C₂×C₃₁	semidirect
133. #2211	C₂×C₃₁	semidirect
134. #2210	C₂×C₃₁	semidirect
135. #2206	C₂×C₃₁	semidirect
136. #2183	C₂×C₃₁	semidirect
137. #2175	C₂×C₃₁	semidirect
138. C₄×D₁₆	C₂×C₃₁	semidirect
218. C₃₁	#33970	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×#3396	2
C₃₁×#2250	2
C₃₁×#2249	2
C₃₁×#2211	2
C₃₁×#2210	2
C₃₁×#2206	2
C₃₁×#2183	2
C₃₁×#2175	2
C₄×C₃₁×D₁₆	2
#33970	31

Subgroups — C31×#33970 #21490729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33970 #21490729