Subgroups — C₃₁×#13268 #21470027

Order 7936.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1377	C₂	1	4	yes	maximal
5	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
7	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
8	C₃₁×#446	—	2	1	no	—
9	C₃₁×#446	—	2	1	no	—
10	C₃₁×#446	—	2	1	no	—
11	C₃₁×#446	—	2	1	no	—
12	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₃₁×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	—	2	1	no	—
17	C₃₁×C₄⋊C₁₆	—	2	1	no	—
18	C₃₁×#346	—	2	1	no	—
19	C₄×C₁₆×C₃₁	—	2	1	no	—
20	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
24	C₄×Q₈×C₃₁	—	2	1	no	—
25	C₄×Q₈×C₃₁	—	2	1	no	—
26	C₄×Q₈×C₃₁	—	2	1	no	—
27	C₄×Q₈×C₃₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₃₁	C₈	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₃₁×M₃₂	—	2	2	no	—
33	C₃₁×M₃₂	—	2	4	no	—
34	C₃₁×M₃₂	Q₈	1	4	yes	complemented
35	C₂×C₁₆×C₃₁	—	2	2	no	—
36	C₂×C₁₆×C₃₁	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	1	no	—
40	C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₃₁×#97	—	2	2	no	—
42	C₃₁×#97	—	2	2	no	—
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₃₁	—	2	1	no	—
48	C₂×Q₈×C₃₁	—	2	1	no	—
49	C₂×Q₈×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₃₁	#41	1	2	yes	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
73	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
74	C₁₆×C₃₁	—	2	4	no	—
75	C₁₆×C₃₁	—	4	2	no	—
76	#13268	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
78	Q₈×C₃₁	—	4	2	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	—	4	1	no	—
84	C₂×C₄×C₃₁	—	4	2	no	—
85	C₂×C₄×C₃₁	—	4	2	no	—
86	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#131	1	2	yes	—
94	C₈×C₃₁	—	2	1	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₈×C₃₁	—	4	2	no	—
97	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
98	C₈×C₃₁	#124	1	1	yes	characteristic
99	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×#446	C₂×C₃₁	1	1	yes	characteristic
101	#2150	C₂×C₃₁	1	1	yes	characteristic
102	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
103	#1377	C₂×C₃₁	1	4	yes	—
104	C₂²×C₃₁	—	2	1	no	—
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₂²×C₃₁	C₈×Q₈	1	2	yes	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	—	4	2	no	—
113	C₄×C₃₁	#349	1	2	yes	—
114	C₄×C₃₁	#391	1	1	yes	characteristic
115	C₄×C₃₁	#408	1	1	yes	characteristic
116	C₄×C₃₁	#446	1	2	yes	—
117	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
118	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
119	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
120	#446	—	2	1	no	—
121	#446	—	2	1	no	—
122	#446	—	2	1	no	—
123	#446	—	2	1	no	—
124	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
125	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
126	C₂×#97	C₂²×C₃₁	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	1	no	—
129	C₄⋊C₁₆	—	2	1	no	—
130	#346	—	2	1	no	—
131	C₄×C₁₆	—	2	1	no	—
132	C₂×C₃₁	—	2	2	no	—
133	C₂×C₃₁	#1984	1	1	yes	characteristic
134	C₂×C₃₁	C₂×#349	1	1	yes	characteristic
135	C₂×C₃₁	#2115	1	1	yes	characteristic
136	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
138	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	—	2	1	no	—
144	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₈×C₃₁	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	4	no	—
150	M₃₂	Q₈×C₃₁	1	4	yes	complemented
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	1	no	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
160	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
161	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
162	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
163	C₃₁	#13268	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
170	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
179	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
180	C₂×C₈	#41×C₃₁	1	2	yes	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
189	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
190	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
191	C₁₆	—	2	4	no	—
192	C₁₆	—	4	2	no	—
193	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
204	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
205	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
206	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
208	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
209	C₂×C₄	C₃₁×#131	1	2	yes	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
214	C₈	C₃₁×#124	1	1	yes	characteristic
215	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
219	C₂²	C₈×Q₈×C₃₁	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₃₁×#349	1	2	yes	—
226	C₄	C₃₁×#391	1	1	yes	characteristic
227	C₄	C₃₁×#408	1	1	yes	characteristic
228	C₄	C₃₁×#446	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₃₁×#1984	1	1	yes	characteristic
231	C₂	C₂×C₃₁×#349	1	1	yes	characteristic
232	C₂	C₃₁×#2115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₃₁	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₃₁×M₃₂	Q₈	semidirect
76. #13268	C₃₁	direct (N × H)
150. M₃₂	Q₈×C₃₁	semidirect
163. C₃₁	#13268	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#446	2
C₃₁×#2150	2
C₄×C₃₁×M₃₂	2
C₃₁×#1377	2
#13268	31

Subgroups — C31×#13268 #21470027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13268 #21470027