Subgroups — C₃₁×#8176 #21464935

Order 7936.

542 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₃₁×#1279	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₁×#97	—	2	1	no	—
9	C₂×C₄×C₈×C₃₁	—	2	2	no	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#391	C₄	1	2	yes	—
12	C₃₁×#389	C₄	1	2	yes	—
13	C₃₁×#387	—	2	2	no	—
14	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#376	—	2	1	no	—
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#336	—	2	1	no	—
19	C₃₁×Q₈⋊C₈	C₄	1	8	yes	complemented
20	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₄×Q₈×C₃₁	—	2	2	no	—
24	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	—	2	1	no	—
30	C₂×C₄²×C₃₁	—	2	1	no	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	—	2	2	no	—
37	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
38	C₃₁×#97	—	4	2	no	—
39	C₄×C₈×C₃₁	—	2	8	no	—
40	C₄×C₈×C₃₁	—	2	2	no	—
41	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
42	C₃₁×#95	—	2	2	no	—
43	C₃₁×#95	—	2	2	no	—
44	C₃₁×#95	—	2	2	no	—
45	C₃₁×#95	—	2	1	no	—
46	C₂×Q₈×C₃₁	—	2	1	no	—
47	C₂×Q₈×C₃₁	—	2	4	no	—
48	C₂×Q₈×C₃₁	C₄²	1	2	yes	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	—	2	8	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	—	4	2	no	—
61	C₂×C₈×C₃₁	—	4	2	no	—
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
67	C₄²×C₃₁	—	2	4	no	—
68	C₄²×C₃₁	—	2	4	no	—
69	C₄²×C₃₁	—	2	4	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	4	2	no	—
72	C₄²×C₃₁	SD₁₆	1	2	yes	—
73	C₄²×C₃₁	Q₁₆	1	2	yes	—
74	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₃₁	#41	1	2	yes	—
76	#8176	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₃₁	—	2	4	no	—
79	Q₈×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	4	2	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	#103	1	4	yes	—
93	C₂×C₄×C₃₁	#104	1	4	yes	—
94	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	#131	1	4	yes	—
99	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#135	1	2	yes	—
102	C₈×C₃₁	—	2	8	no	—
103	C₈×C₃₁	—	4	4	no	—
104	#2295	C₂×C₃₁	1	1	yes	characteristic
105	#1767	C₂×C₃₁	1	1	yes	characteristic
106	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
107	C₂×Q₈⋊C₈	C₂×C₃₁	1	2	yes	—
108	#1279	C₂×C₃₁	1	2	yes	—
109	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₃₁	#408	1	2	yes	—
111	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
112	C₂²×C₃₁	#416	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×#104	1	2	yes	—
114	C₄×C₃₁	—	2	4	no	—
115	C₄×C₃₁	—	2	4	no	—
116	C₄×C₃₁	—	2	2	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₄×C₃₁	—	8	2	no	—
120	C₄×C₃₁	#405	1	2	yes	—
121	C₄×C₃₁	#433	1	2	yes	—
122	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
123	C₄×C₃₁	C₄×Q₁₆	1	2	yes	—
124	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
126	C₂×#97	—	2	1	no	—
127	C₂×C₄×C₈	—	2	2	no	—
128	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
129	#391	C₄×C₃₁	1	2	yes	—
130	#389	C₄×C₃₁	1	2	yes	—
131	#387	—	2	2	no	—
132	C₄×C₄⋊C₄	—	2	1	no	—
133	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
134	#376	—	2	1	no	—
135	C₄³	C₂²×C₃₁	1	1	yes	characteristic
136	#336	—	2	1	no	—
137	Q₈⋊C₈	C₄×C₃₁	1	8	yes	complemented
138	C₂×C₃₁	#1493	1	2	yes	—
139	C₂×C₃₁	#1494	1	2	yes	—
140	C₂×C₃₁	#1750	1	1	yes	characteristic
141	C₂×C₃₁	C₄×#104	1	1	yes	characteristic
142	C₂×C₃₁	C₄×#103	1	1	yes	characteristic
143	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
155	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
156	C₂×C₄²	D₄×C₃₁	1	2	yes	—
157	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
161	#97	—	4	2	no	—
162	C₄×C₈	—	2	8	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
165	#95	—	2	2	no	—
166	#95	—	2	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	1	no	—
169	C₃₁	#8176	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₄²×C₃₁	1	2	yes	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	SD₁₆×C₃₁	1	2	yes	—
197	C₄²	Q₁₆×C₃₁	1	2	yes	—
198	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
199	C₄²	#41×C₃₁	1	2	yes	—
200	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₃₁×#103	1	4	yes	—
216	C₂×C₄	C₃₁×#104	1	4	yes	—
217	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
219	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
220	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
221	C₂×C₄	C₃₁×#131	1	4	yes	—
222	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
224	C₂×C₄	C₃₁×#135	1	2	yes	—
225	C₈	—	2	8	no	—
226	C₈	—	4	4	no	—
227	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂²	C₃₁×#408	1	2	yes	—
229	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
230	C₂²	C₃₁×#416	1	1	yes	characteristic
231	C₂²	C₂×C₃₁×#104	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	8	2	no	—
238	C₄	C₃₁×#405	1	2	yes	—
239	C₄	C₃₁×#433	1	2	yes	—
240	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
241	C₄	C₄×Q₁₆×C₃₁	1	2	yes	—
242	C₂	C₃₁×#1493	1	2	yes	—
243	C₂	C₃₁×#1494	1	2	yes	—
244	C₂	C₃₁×#1750	1	1	yes	characteristic
245	C₂	C₄×C₃₁×#104	1	1	yes	characteristic
246	C₂	C₄×C₃₁×#103	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
19. C₃₁×Q₈⋊C₈	C₄	semidirect
76. #8176	C₃₁	direct (N × H)
137. Q₈⋊C₈	C₄×C₃₁	semidirect
169. C₃₁	#8176	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×#2295	2
C₃₁×#1767	2
C₄²×C₈×C₃₁	2
C₂×C₃₁×Q₈⋊C₈	2
C₃₁×#1279	2
#8176	31

Subgroups — C31×#8176 #21464935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8176 #21464935