Subgroups — C₃₁×#20774 #21477533

Order 7936.

818 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1840	C₂	1	4	yes	maximal, complemented
3	C₃₁×#1839	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1729	C₂	1	2	yes	maximal
5	C₃₁×#1729	C₂	1	2	yes	maximal
6	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
15	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
16	C₃₁×#376	C₂²	1	2	yes	—
17	C₄³×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#336	C₂²	1	2	yes	—
19	C₃₁×#336	C₂²	1	2	yes	—
20	C₃₁×#336	C₂²	1	8	yes	—
21	C₃₁×#336	C₂²	1	8	yes	—
22	C₃₁×#336	C₂²	1	2	yes	—
23	C₃₁×#336	C₂²	1	2	yes	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	2	4	no	—
27	C₂×M₁₆×C₃₁	—	2	4	no	—
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂²×C₈×C₃₁	—	2	4	no	—
30	C₂²×C₈×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	—	2	4	no	—
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	—	2	4	no	—
36	C₃₁×#117	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	—	2	4	no	—
43	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#95	C₂×C₄	1	4	yes	—
51	C₃₁×#95	C₂×C₄	1	4	yes	—
52	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	#41	1	4	yes	—
63	C₂²×C₄×C₃₁	#41	1	2	yes	—
64	C₂²×C₄×C₃₁	#41	1	4	yes	—
65	C₂²×C₄×C₃₁	#41	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₃₁	—	4	8	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	4	8	no	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₂²⋊C₄×C₃₁	—	2	4	no	—
79	C₂²⋊C₄×C₃₁	—	2	4	no	—
80	C₄²×C₃₁	—	2	4	no	—
81	C₄²×C₃₁	—	2	8	no	—
82	C₄²×C₃₁	—	2	2	no	—
83	C₄²×C₃₁	—	2	2	no	—
84	C₄²×C₃₁	—	2	4	no	—
85	C₄²×C₃₁	—	2	2	no	—
86	C₄²×C₃₁	—	2	2	no	—
87	C₄²×C₃₁	—	4	4	no	—
88	#20774	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	2	2	no	—
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	#117	1	2	yes	—
92	C₂³×C₃₁	#117	1	2	yes	—
93	C₂³×C₃₁	#117	1	2	yes	—
94	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	4	8	no	—
108	C₂×C₄×C₃₁	#117	1	2	yes	—
109	C₂×C₄×C₃₁	#117	1	2	yes	—
110	C₂×C₄×C₃₁	#117	1	2	yes	—
111	C₂×C₄×C₃₁	#126	1	8	yes	—
112	C₂×C₄×C₃₁	#126	1	8	yes	—
113	C₂×C₄×C₃₁	#131	1	4	yes	—
114	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
118	C₈×C₃₁	—	4	4	no	—
119	C₈×C₃₁	—	4	4	no	—
120	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
121	#1840	C₂×C₃₁	1	4	yes	complemented
122	#1839	C₂×C₃₁	1	4	yes	complemented
123	#1729	C₂×C₃₁	1	2	yes	—
124	#1729	C₂×C₃₁	1	2	yes	—
125	#1728	C₂×C₃₁	1	1	yes	characteristic
126	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	4	no	—
130	C₂²×C₃₁	—	2	4	no	—
131	C₂²×C₃₁	#383	1	4	yes	—
132	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
135	C₂²×C₃₁	C₂×#131	1	2	yes	—
136	C₂²×C₃₁	#568	1	2	yes	—
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	4	2	no	—
139	C₄×C₃₁	—	4	2	no	—
140	C₄×C₃₁	—	4	2	no	—
141	C₄×C₃₁	—	4	4	no	—
142	C₄×C₃₁	#383	1	4	yes	—
143	C₄×C₃₁	C₂×#126	1	2	yes	—
144	C₄×C₃₁	C₂×#126	1	2	yes	—
145	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
146	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
147	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
149	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
151	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
152	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
153	#376	C₂²×C₃₁	1	2	yes	—
154	C₄³	C₂²×C₃₁	1	2	yes	—
155	#336	C₂²×C₃₁	1	2	yes	—
156	#336	C₂²×C₃₁	1	2	yes	—
157	#336	C₂²×C₃₁	1	8	yes	—
158	#336	C₂²×C₃₁	1	8	yes	—
159	#336	C₂²×C₃₁	1	2	yes	—
160	#336	C₂²×C₃₁	1	2	yes	—
161	C₂×C₃₁	—	2	4	no	—
162	C₂×C₃₁	C₂×#383	1	1	yes	characteristic
163	C₂×C₃₁	#2922	1	1	yes	characteristic
164	C₂×C₃₁	#2923	1	1	yes	characteristic
165	C₂×C₃₁	#2924	1	2	yes	—
166	C₂×C₃₁	#2925	1	2	yes	—
167	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
168	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
176	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
177	#117	—	2	2	no	—
178	#117	—	2	4	no	—
179	#117	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
187	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
188	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
189	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
191	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
193	#95	C₂×C₄×C₃₁	1	4	yes	—
194	#95	C₂×C₄×C₃₁	1	4	yes	—
195	C₃₁	#20774	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	#41×C₃₁	1	4	yes	—
207	C₂²×C₄	#41×C₃₁	1	2	yes	—
208	C₂²×C₄	#41×C₃₁	1	4	yes	—
209	C₂²×C₄	#41×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
211	M₁₆	—	4	8	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	8	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	8	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	4	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₃₁×#117	1	2	yes	—
235	C₂³	C₃₁×#117	1	2	yes	—
236	C₂³	C₃₁×#117	1	2	yes	—
237	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	C₃₁×#117	1	2	yes	—
252	C₂×C₄	C₃₁×#117	1	2	yes	—
253	C₂×C₄	C₃₁×#117	1	2	yes	—
254	C₂×C₄	C₃₁×#126	1	8	yes	—
255	C₂×C₄	C₃₁×#126	1	8	yes	—
256	C₂×C₄	C₃₁×#131	1	4	yes	—
257	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
258	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
259	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
261	C₈	—	4	4	no	—
262	C₈	—	4	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	C₃₁×#383	1	4	yes	—
268	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
269	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
270	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
271	C₂²	C₂×C₃₁×#131	1	2	yes	—
272	C₂²	C₃₁×#568	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	4	no	—
278	C₄	C₃₁×#383	1	4	yes	—
279	C₄	C₂×C₃₁×#126	1	2	yes	—
280	C₄	C₂×C₃₁×#126	1	2	yes	—
281	C₂	—	2	4	no	—
282	C₂	C₂×C₃₁×#383	1	1	yes	characteristic
283	C₂	C₃₁×#2922	1	1	yes	characteristic
284	C₂	C₃₁×#2923	1	1	yes	characteristic
285	C₂	C₃₁×#2924	1	2	yes	—
286	C₂	C₃₁×#2925	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1840	C₂	semidirect
3. C₃₁×#1839	C₂	semidirect
88. #20774	C₃₁	direct (N × H)
121. #1840	C₂×C₃₁	semidirect
122. #1839	C₂×C₃₁	semidirect
195. C₃₁	#20774	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#117	2
C₃₁×#1840	2
C₃₁×#1839	2
C₃₁×#1729	2
C₃₁×#1728	2
C₂×C₃₁×#336	2
#20774	31

Subgroups — C31×#20774 #21477533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20774 #21477533