Subgroups — C₃₁×#32815 #21489574

Order 7936.

746 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2642	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2641	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2055	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2026	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2025	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2014	C₂	1	2	yes	maximal
7	C₃₁×#2013	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×#103	C₂²	1	4	yes	—
16	C₂×C₃₁×#102	C₂²	1	4	yes	—
17	C₃₁×#400	—	2	4	no	—
18	C₃₁×#394	—	2	4	no	—
19	C₃₁×#391	C₂²	1	2	yes	—
20	C₃₁×#391	C₂²	1	2	yes	—
21	C₃₁×#390	C₂²	1	2	yes	—
22	C₃₁×#390	C₂²	1	2	yes	—
23	C₃₁×#376	C₂²	1	1	yes	characteristic
24	C₃₁×#340	C₂²	1	4	yes	—
25	C₃₁×#336	C₂²	1	2	yes	—
26	C₃₁×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
28	C₂²×D₄×C₃₁	—	2	2	no	—
29	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
30	C₂×SD₁₆×C₃₁	—	4	4	no	—
31	C₂×SD₁₆×C₃₁	—	4	4	no	—
32	C₂×SD₁₆×C₃₁	—	4	4	no	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₃₁×C₄⋊Q₈	—	4	2	no	—
35	C₃₁×#127	—	4	2	no	—
36	C₃₁×#125	—	4	2	no	—
37	C₃₁×#125	—	4	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊C₈	—	4	2	no	—
49	C₃₁×#103	—	4	4	no	—
50	C₃₁×#102	—	4	4	no	—
51	C₃₁×#95	—	2	4	no	—
52	C₃₁×#95	—	2	4	no	—
53	C₃₁×#95	D₄	1	4	yes	—
54	C₂⁴×C₃₁	—	2	2	no	—
55	C₂×Q₈×C₃₁	—	2	4	no	—
56	C₂×Q₈×C₃₁	—	2	4	no	—
57	C₂×Q₈×C₃₁	—	2	2	no	—
58	C₂×Q₈×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	4	no	—
60	C₂×D₄×C₃₁	—	2	4	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	—	4	4	no	—
64	C₂×D₄×C₃₁	—	8	4	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	SD₁₆×C₃₁	—	4	16	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₄⋊C₄×C₃₁	—	4	4	no	—
79	C₄⋊C₄×C₃₁	—	4	4	no	—
80	C₄⋊C₄×C₃₁	—	4	4	no	—
81	C₂²⋊C₄×C₃₁	—	8	4	no	—
82	C₂²⋊C₄×C₃₁	—	8	4	no	—
83	C₄²×C₃₁	—	2	1	no	—
84	C₄²×C₃₁	—	2	2	no	—
85	C₄²×C₃₁	—	2	2	no	—
86	C₄²×C₃₁	—	2	1	no	—
87	#32815	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	—	4	2	no	—
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	—	4	4	no	—
92	C₂³×C₃₁	—	4	4	no	—
93	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₃₁	—	4	4	no	—
95	Q₈×C₃₁	—	4	4	no	—
96	D₄×C₃₁	—	4	4	no	—
97	D₄×C₃₁	—	4	4	no	—
98	D₄×C₃₁	—	8	8	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
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	4	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
114	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
115	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
116	C₈×C₃₁	—	4	4	no	—
117	#2642	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#2641	C₂×C₃₁	1	1	yes	characteristic
119	#2055	C₂×C₃₁	1	4	yes	complemented
120	#2026	C₂×C₃₁	1	2	yes	complemented
121	#2025	C₂×C₃₁	1	2	yes	complemented
122	#2014	C₂×C₃₁	1	2	yes	—
123	#2013	C₂×C₃₁	1	2	yes	complemented
124	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂²×C₃₁	—	8	2	no	—
126	C₂²×C₃₁	—	8	4	no	—
127	C₂²×C₃₁	—	8	2	no	—
128	C₂²×C₃₁	—	8	2	no	—
129	C₂²×C₃₁	—	8	4	no	—
130	C₂²×C₃₁	#535	1	1	yes	characteristic
131	C₂²×C₃₁	#535	1	1	yes	characteristic
132	C₂²×C₃₁	#537	1	1	yes	characteristic
133	C₂²×C₃₁	#577	1	4	yes	—
134	C₄×C₃₁	—	4	2	no	—
135	C₄×C₃₁	—	4	2	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	8	2	no	—
138	C₄×C₃₁	—	8	2	no	—
139	C₄×C₃₁	—	8	2	no	—
140	C₄×C₃₁	H(3,𝔽₄)	1	2	yes	—
141	C₄×C₃₁	#562	1	2	yes	—
142	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
143	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
148	C₂×#103	C₂²×C₃₁	1	4	yes	—
149	C₂×#102	C₂²×C₃₁	1	4	yes	—
150	#400	—	2	4	no	—
151	#394	—	2	4	no	—
152	#391	C₂²×C₃₁	1	2	yes	—
153	#391	C₂²×C₃₁	1	2	yes	—
154	#390	C₂²×C₃₁	1	2	yes	—
155	#390	C₂²×C₃₁	1	2	yes	—
156	#376	C₂²×C₃₁	1	1	yes	characteristic
157	#340	C₂²×C₃₁	1	4	yes	—
158	#336	C₂²×C₃₁	1	2	yes	—
159	#336	C₂²×C₃₁	1	2	yes	—
160	C₂×C₃₁	—	16	2	no	—
161	C₂×C₃₁	#2819	1	1	yes	characteristic
162	C₂×C₃₁	#3221	1	2	yes	—
163	C₂×C₃₁	#3222	1	2	yes	—
164	C₂×C₃₁	#3260	1	2	yes	—
165	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
166	C₂²×D₄	—	2	2	no	—
167	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
168	C₂×SD₁₆	—	4	4	no	—
169	C₂×SD₁₆	—	4	4	no	—
170	C₂×SD₁₆	—	4	4	no	—
171	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
172	C₄⋊Q₈	—	4	2	no	—
173	#127	—	4	2	no	—
174	#125	—	4	2	no	—
175	#125	—	4	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
181	C₂×C₂²⋊C₄	—	2	4	no	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
186	C₄⋊C₈	—	4	2	no	—
187	#103	—	4	4	no	—
188	#102	—	4	4	no	—
189	#95	—	2	4	no	—
190	#95	—	2	4	no	—
191	#95	D₄×C₃₁	1	4	yes	—
192	C₃₁	#32815	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	8	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
211	SD₁₆	—	4	16	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₂²⋊C₄	—	8	4	no	—
221	C₂²⋊C₄	—	8	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
232	Q₈	—	4	4	no	—
233	Q₈	—	4	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	8	8	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
252	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
253	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
254	C₈	—	4	4	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₃₁×#535	1	1	yes	characteristic
261	C₂²	C₃₁×#535	1	1	yes	characteristic
262	C₂²	C₃₁×#537	1	1	yes	characteristic
263	C₂²	C₃₁×#577	1	4	yes	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	C₃₁×H(3,𝔽₄)	1	2	yes	—
271	C₄	C₃₁×#562	1	2	yes	—
272	C₂	—	16	2	no	—
273	C₂	C₃₁×#2819	1	1	yes	characteristic
274	C₂	C₃₁×#3221	1	2	yes	—
275	C₂	C₃₁×#3222	1	2	yes	—
276	C₂	C₃₁×#3260	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2642	C₂	semidirect
3. C₃₁×#2055	C₂	semidirect
4. C₃₁×#2026	C₂	semidirect
5. C₃₁×#2025	C₂	semidirect
7. C₃₁×#2013	C₂	semidirect
8. C₃₁×#1832	C₂	semidirect
87. #32815	C₃₁	direct (N × H)
117. #2642	C₂×C₃₁	semidirect
119. #2055	C₂×C₃₁	semidirect
120. #2026	C₂×C₃₁	semidirect
121. #2025	C₂×C₃₁	semidirect
123. #2013	C₂×C₃₁	semidirect
124. #1832	C₂×C₃₁	semidirect
192. C₃₁	#32815	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×#2642	2
C₃₁×#2641	2
C₃₁×#2055	2
C₃₁×#2026	2
C₃₁×#2025	2
C₃₁×#2014	2
C₃₁×#2013	2
C₃₁×#1832	2
#32815	31

Subgroups — C31×#32815 #21489574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32815 #21489574