Subgroups — C₃₁×#19063 #21475822

Order 7936.

806 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3235	C₂	1	2	yes	maximal
3	C₃₁×#3135	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1663	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1658	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1537	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1531	C₂	1	2	yes	maximal, complemented
8	C₃₁×#564	C₂²	1	4	yes	—
9	C₃₁×#557	—	2	2	no	—
10	C₃₁×#557	—	2	2	no	—
11	C₃₁×#557	—	2	2	no	—
12	C₃₁×#541	—	2	2	no	—
13	C₃₁×#541	—	2	4	no	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₃₁×#533	C₂²	1	2	yes	—
16	C₃₁×#532	C₂²	1	1	yes	characteristic
17	C₃₁×#501	C₂²	1	4	yes	—
18	C₃₁×#491	C₂²	1	2	yes	—
19	C₃₁×#490	C₂²	1	2	yes	—
20	C₃₁×#490	C₂²	1	4	yes	—
21	C₃₁×#489	—	2	2	no	—
22	C₃₁×#482	—	2	4	no	—
23	C₃₁×#476	—	2	2	no	—
24	C₃₁×#474	—	2	2	no	—
25	C₃₁×#432	C₂²	1	2	yes	—
26	C₃₁×#416	—	2	2	no	—
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₃₁×#330	C₂²	1	8	yes	—
29	C₃₁×#330	C₂²	1	4	yes	—
30	C₂×#41×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₃₁×C₄⋊Q₈	—	2	4	no	—
33	C₃₁×C₄⋊Q₈	—	2	2	no	—
34	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
35	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
36	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
37	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
38	C₃₁×#127	C₂³	1	1	yes	characteristic
39	C₃₁×#126	—	2	8	no	—
40	C₃₁×#125	—	2	4	no	—
41	C₃₁×#125	—	2	2	no	—
42	C₃₁×#125	—	2	8	no	—
43	C₃₁×#125	—	2	8	no	—
44	C₃₁×#125	—	2	8	no	—
45	C₃₁×#125	C₂³	1	2	yes	—
46	C₃₁×#125	C₂³	1	4	yes	—
47	C₃₁×#124	—	2	1	no	—
48	C₃₁×C₂²⋊Q₈	—	2	2	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	4	no	—
53	C₄×Q₈×C₃₁	—	2	4	no	—
54	C₄×Q₈×C₃₁	—	2	4	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₃₁×#117	D₄	1	2	yes	—
60	C₃₁×#117	D₄	1	4	yes	—
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#107	—	2	4	no	—
63	C₃₁×#106	—	2	4	no	—
64	C₃₁×C₄⋊C₈	—	2	2	no	—
65	C₃₁×#103	—	2	4	no	—
66	C₃₁×#102	—	2	4	no	—
67	C₃₁×#102	—	2	4	no	—
68	C₃₁×#102	—	2	4	no	—
69	C₃₁×C₂²⋊C₈	—	2	2	no	—
70	C₃₁×#97	—	2	1	no	—
71	C₃₁×#97	C₂³	1	2	yes	—
72	C₃₁×#97	C₂³	1	4	yes	—
73	#41×C₃₁	—	4	4	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	2	1	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	1	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	4	2	no	—
87	C₄⋊C₄×C₃₁	—	2	8	no	—
88	C₄⋊C₄×C₃₁	—	2	8	no	—
89	C₄⋊C₄×C₃₁	—	2	8	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
98	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
99	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
100	C₂²⋊C₄×C₃₁	—	2	2	no	—
101	C₂²⋊C₄×C₃₁	—	2	4	no	—
102	C₂²⋊C₄×C₃₁	—	4	2	no	—
103	C₄²×C₃₁	—	2	8	no	—
104	C₄²×C₃₁	—	2	4	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	2	yes	—
108	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
109	#19063	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	—	4	1	no	—
111	C₂³×C₃₁	#120	1	1	yes	characteristic
112	Q₈×C₃₁	—	4	4	no	—
113	Q₈×C₃₁	—	4	4	no	—
114	Q₈×C₃₁	—	4	2	no	—
115	Q₈×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	4	4	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	8	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	2	4	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	1	no	—
125	C₂×C₄×C₃₁	—	8	2	no	—
126	C₂×C₄×C₃₁	#120	1	2	yes	—
127	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₃₁	—	4	4	no	—
131	C₈×C₃₁	—	4	2	no	—
132	#3556	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#3235	C₂×C₃₁	1	2	yes	—
134	#3135	C₂×C₃₁	1	1	yes	characteristic
135	#1663	C₂×C₃₁	1	4	yes	complemented
136	#1658	C₂×C₃₁	1	4	yes	complemented
137	#1537	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#1531	C₂×C₃₁	1	2	yes	complemented
139	C₂²×C₃₁	—	2	1	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	8	1	no	—
142	C₂²×C₃₁	—	8	2	no	—
143	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	2	2	no	—
146	C₄×C₃₁	—	2	2	no	—
147	C₄×C₃₁	—	4	4	no	—
148	C₄×C₃₁	—	4	4	no	—
149	C₄×C₃₁	—	4	4	no	—
150	C₄×C₃₁	—	8	1	no	—
151	C₄×C₃₁	#454	1	2	yes	—
152	#564	C₂²×C₃₁	1	4	yes	—
153	#557	—	2	2	no	—
154	#557	—	2	2	no	—
155	#557	—	2	2	no	—
156	#541	—	2	2	no	—
157	#541	—	2	4	no	—
158	#533	C₂²×C₃₁	1	1	yes	characteristic
159	#533	C₂²×C₃₁	1	2	yes	—
160	#532	C₂²×C₃₁	1	1	yes	characteristic
161	#501	C₂²×C₃₁	1	4	yes	—
162	#491	C₂²×C₃₁	1	2	yes	—
163	#490	C₂²×C₃₁	1	2	yes	—
164	#490	C₂²×C₃₁	1	4	yes	—
165	#489	—	2	2	no	—
166	#482	—	2	4	no	—
167	#476	—	2	2	no	—
168	#474	—	2	2	no	—
169	#432	C₂²×C₃₁	1	2	yes	—
170	#416	—	2	2	no	—
171	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
172	#330	C₂²×C₃₁	1	4	yes	—
173	#330	C₂²×C₃₁	1	8	yes	—
174	C₂×C₃₁	—	4	1	no	—
175	C₂×C₃₁	—	16	1	no	—
176	C₂×C₃₁	C₂×#454	1	1	yes	characteristic
177	C₂×C₃₁	#3020	1	2	yes	—
178	C₂×#41	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₄⋊Q₈	—	2	4	no	—
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
183	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
184	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
185	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
186	#127	C₂³×C₃₁	1	1	yes	characteristic
187	#126	—	2	8	no	—
188	#125	—	2	4	no	—
189	#125	—	2	2	no	—
190	#125	—	2	8	no	—
191	#125	—	2	8	no	—
192	#125	—	2	8	no	—
193	#125	C₂³×C₃₁	1	2	yes	—
194	#125	C₂³×C₃₁	1	4	yes	—
195	#124	—	2	1	no	—
196	C₂²⋊Q₈	—	2	2	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#121	—	2	2	no	—
200	C₄×Q₈	—	2	4	no	—
201	C₄×Q₈	—	2	4	no	—
202	C₄×Q₈	—	2	4	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×D₄	—	2	2	no	—
207	#117	D₄×C₃₁	1	2	yes	—
208	#117	D₄×C₃₁	1	4	yes	—
209	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	C₄⋊C₈	—	2	2	no	—
213	#103	—	2	4	no	—
214	#102	—	2	4	no	—
215	#102	—	2	4	no	—
216	#102	—	2	4	no	—
217	C₂²⋊C₈	—	2	2	no	—
218	#97	—	2	1	no	—
219	#97	C₂³×C₃₁	1	2	yes	—
220	#97	C₂³×C₃₁	1	4	yes	—
221	C₃₁	#19063	1	1	yes	characteristic, direct product factor, Sylow
222	#41	—	4	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
247	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
248	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₄²	—	2	8	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₄²	C₂×D₄×C₃₁	1	2	yes	—
257	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
258	C₂³	—	4	1	no	—
259	C₂³	C₃₁×#120	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	Q₈	—	4	4	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	2	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	C₃₁×#120	1	2	yes	—
275	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	C₈	—	4	4	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	8	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	1	no	—
292	C₄	C₃₁×#454	1	2	yes	—
293	C₂	—	4	1	no	—
294	C₂	—	16	1	no	—
295	C₂	C₂×C₃₁×#454	1	1	yes	characteristic
296	C₂	C₃₁×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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×#3556	C₂	semidirect
4. C₃₁×#1663	C₂	semidirect
5. C₃₁×#1658	C₂	semidirect
6. C₃₁×#1537	C₂	semidirect
7. C₃₁×#1531	C₂	semidirect
109. #19063	C₃₁	direct (N × H)
132. #3556	C₂×C₃₁	semidirect
135. #1663	C₂×C₃₁	semidirect
136. #1658	C₂×C₃₁	semidirect
137. #1537	C₂×C₃₁	semidirect
138. #1531	C₂×C₃₁	semidirect
221. C₃₁	#19063	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×#3556	2
C₃₁×#3235	2
C₃₁×#3135	2
C₃₁×#1663	2
C₃₁×#1658	2
C₃₁×#1537	2
C₃₁×#1531	2
#19063	31

Subgroups — C31×#19063 #21475822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19063 #21475822