Subgroups — C₃₁×#33067 #21489826

Order 7936.

750 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1948	C₂	1	2	yes	maximal
4	C₃₁×#1919	C₂	1	4	yes	maximal
5	C₃₁×#1865	C₂	1	4	yes	maximal
6	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1803	C₂	1	2	yes	maximal
8	C₂×C₃₁×#125	—	2	1	no	—
9	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#106	C₂²	1	4	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×#103	C₂²	1	4	yes	—
14	C₃₁×#389	—	2	2	no	—
15	C₃₁×#389	C₂²	1	4	yes	—
16	C₃₁×#384	C₂²	1	2	yes	—
17	C₃₁×#382	—	2	1	no	—
18	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
19	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₄³×C₃₁	—	2	1	no	—
23	C₄³×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#340	C₂²	1	4	yes	—
25	C₃₁×#340	C₂²	1	4	yes	—
26	C₃₁×#336	C₂²	1	4	yes	—
27	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
29	C₃₁×#125	—	4	4	no	—
30	C₄×Q₈×C₃₁	—	2	4	no	—
31	C₄×Q₈×C₃₁	—	2	4	no	—
32	C₄×Q₈×C₃₁	—	2	2	no	—
33	C₄×Q₈×C₃₁	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	2	yes	—
48	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#106	—	2	8	no	—
52	C₃₁×C₄⋊C₈	—	2	4	no	—
53	C₃₁×#103	—	2	8	no	—
54	C₃₁×#95	—	2	2	no	—
55	C₃₁×#95	—	2	1	no	—
56	C₃₁×#95	—	2	4	no	—
57	C₂×Q₈×C₃₁	—	2	1	no	—
58	C₂×Q₈×C₃₁	—	2	2	no	—
59	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
60	C₂×Q₈×C₃₁	#41	1	4	yes	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₄⋊C₄×C₃₁	—	2	8	no	—
74	C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	4	4	no	—
80	C₄⋊C₄×C₃₁	—	4	8	no	—
81	C₄⋊C₄×C₃₁	—	4	4	no	—
82	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
83	C₄⋊C₄×C₃₁	#41	1	4	yes	—
84	C₄²×C₃₁	—	2	8	no	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	2	2	no	—
87	C₄²×C₃₁	—	2	2	no	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	—	4	2	no	—
90	C₄²×C₃₁	—	4	4	no	—
91	C₄²×C₃₁	—	4	2	no	—
92	C₄²×C₃₁	SD₁₆	1	4	yes	—
93	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
94	C₄²×C₃₁	#41	1	2	yes	—
95	#33067	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₃₁	—	2	4	no	—
98	Q₈×C₃₁	—	2	2	no	—
99	Q₈×C₃₁	#125	1	4	yes	complemented
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	4	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
119	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₃₁	#137	1	2	yes	—
121	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
125	C₈×C₃₁	—	4	4	no	—
126	#2859	C₂×C₃₁	1	1	yes	characteristic
127	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
128	#1948	C₂×C₃₁	1	2	yes	—
129	#1919	C₂×C₃₁	1	4	yes	—
130	#1865	C₂×C₃₁	1	4	yes	—
131	#1831	C₂×C₃₁	1	1	yes	characteristic
132	#1803	C₂×C₃₁	1	2	yes	—
133	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
134	C₂²×C₃₁	#525	1	2	yes	—
135	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
137	C₂²×C₃₁	#575	1	2	yes	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	2	4	no	—
140	C₄×C₃₁	—	2	4	no	—
141	C₄×C₃₁	—	4	2	no	—
142	C₄×C₃₁	—	4	2	no	—
143	C₄×C₃₁	—	4	4	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	#475	1	4	yes	—
146	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
147	C₄×C₃₁	#529	1	2	yes	—
148	C₄×C₃₁	#533	1	1	yes	characteristic
149	C₂×#125	—	2	1	no	—
150	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
151	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×#106	C₂²×C₃₁	1	4	yes	—
153	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
154	C₂×#103	C₂²×C₃₁	1	4	yes	—
155	#389	—	2	2	no	—
156	#389	C₂²×C₃₁	1	4	yes	—
157	#384	C₂²×C₃₁	1	2	yes	—
158	#382	—	2	1	no	—
159	C₄×C₄⋊C₄	—	2	1	no	—
160	C₄×C₄⋊C₄	—	2	2	no	—
161	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
162	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
163	C₄³	—	2	1	no	—
164	C₄³	C₂²×C₃₁	1	1	yes	characteristic
165	#340	C₂²×C₃₁	1	4	yes	—
166	#340	C₂²×C₃₁	1	4	yes	—
167	#336	C₂²×C₃₁	1	4	yes	—
168	C₂×C₃₁	#2851	1	1	yes	characteristic
169	C₂×C₃₁	C₂×#475	1	1	yes	characteristic
170	C₂×C₃₁	#3080	1	1	yes	characteristic
171	C₂×C₃₁	#3103	1	2	yes	—
172	C₂×C₃₁	#3116	1	2	yes	—
173	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
175	#125	—	4	4	no	—
176	C₄×Q₈	—	2	4	no	—
177	C₄×Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	2	no	—
179	C₄×Q₈	Q₈×C₃₁	1	4	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₃₁	1	2	yes	—
194	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
196	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
197	#106	—	2	8	no	—
198	C₄⋊C₈	—	2	4	no	—
199	#103	—	2	8	no	—
200	#95	—	2	2	no	—
201	#95	—	2	1	no	—
202	#95	—	2	4	no	—
203	C₃₁	#33067	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×Q₈	—	2	1	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
207	C₂×Q₈	#41×C₃₁	1	4	yes	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	8	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
230	C₄⋊C₄	#41×C₃₁	1	4	yes	—
231	C₄²	—	2	8	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	4	no	—
238	C₄²	—	4	2	no	—
239	C₄²	SD₁₆×C₃₁	1	4	yes	—
240	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
241	C₄²	#41×C₃₁	1	2	yes	—
242	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	2	2	no	—
245	Q₈	C₃₁×#125	1	4	yes	complemented
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
263	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
264	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
265	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
266	C₂×C₄	C₃₁×#137	1	2	yes	—
267	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
268	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
269	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
273	C₂²	C₃₁×#525	1	2	yes	—
274	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
275	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
276	C₂²	C₃₁×#575	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	8	4	no	—
284	C₄	C₃₁×#475	1	4	yes	—
285	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
286	C₄	C₃₁×#529	1	2	yes	—
287	C₄	C₃₁×#533	1	1	yes	characteristic
288	C₂	C₃₁×#2851	1	1	yes	characteristic
289	C₂	C₂×C₃₁×#475	1	1	yes	characteristic
290	C₂	C₃₁×#3080	1	1	yes	characteristic
291	C₂	C₃₁×#3103	1	2	yes	—
292	C₂	C₃₁×#3116	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
95. #33067	C₃₁	direct (N × H)
99. Q₈×C₃₁	#125	semidirect
203. C₃₁	#33067	direct (N × H)
245. Q₈	C₃₁×#125	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#2859	2
C₄²×Q₈×C₃₁	2
C₃₁×#1948	2
C₃₁×#1919	2
C₃₁×#1865	2
C₃₁×#1831	2
C₃₁×#1803	2
#33067	31

Subgroups — C31×#33067 #21489826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33067 #21489826