Subgroups — C₂×C₃₁×#1328 #21468216

Order 7936.

378 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#349	C₂	1	2	yes	maximal
3	C₃₁×#1328	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	—	2	2	no	—
8	C₃₁×#424	—	2	2	no	—
9	C₃₁×#424	C₄	1	2	yes	—
10	C₃₁×#424	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
12	C₃₁×#349	—	2	2	no	—
13	C₃₁×#349	C₂²	1	4	yes	—
14	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₃₁	—	4	1	no	—
16	C₂×M₁₆×C₃₁	—	4	1	no	—
17	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
19	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	—	2	1	no	—
21	C₂²×C₈×C₃₁	—	2	1	no	—
22	C₂²×C₈×C₃₁	—	4	1	no	—
23	C₂²×C₈×C₃₁	—	4	1	no	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₃₁	D₄	1	2	yes	—
26	C₃₁×#117	—	2	1	no	—
27	C₃₁×#117	—	2	1	no	—
28	C₃₁×#117	C₈	1	2	yes	—
29	C₃₁×#117	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
32	C₂×C₄²×C₃₁	—	2	1	no	—
33	C₃₁×M₃₂	—	2	4	no	—
34	C₃₁×M₃₂	—	4	2	no	—
35	C₂×C₁₆×C₃₁	—	4	2	no	—
36	C₃₁×C₄⋊C₈	—	2	4	no	—
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	—	2	1	no	—
40	C₂²×C₄×C₃₁	—	2	1	no	—
41	C₂²×C₄×C₃₁	—	2	1	no	—
42	C₂²×C₄×C₃₁	—	2	1	no	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₃₁	—	4	1	no	—
51	M₁₆×C₃₁	—	4	2	no	—
52	M₁₆×C₃₁	—	4	1	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	2	2	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	—	4	1	no	—
58	C₂×C₈×C₃₁	—	4	1	no	—
59	C₂×C₈×C₃₁	—	4	2	no	—
60	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	—	2	2	no	—
68	C₁₆×C₃₁	—	4	4	no	—
69	C₂×#1328	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	—	2	1	no	—
71	C₂³×C₃₁	—	2	1	no	—
72	C₂³×C₃₁	—	4	2	no	—
73	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
74	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
75	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	—	2	1	no	—
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	4	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	#99	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#99	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#101	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#101	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₃₁	—	2	2	no	—
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	4	2	no	—
99	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
100	C₂×#349	C₂×C₃₁	1	2	yes	—
101	#1328	C₂×C₃₁	1	4	yes	direct product factor
102	C₂²×C₃₁	—	2	2	no	—
103	C₂²×C₃₁	—	2	1	no	—
104	C₂²×C₃₁	—	2	1	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	—	4	1	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	4	2	no	—
109	C₂²×C₃₁	#324	1	1	yes	characteristic
110	C₂²×C₃₁	#324	1	1	yes	characteristic
111	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₄×C₃₁	—	2	1	no	—
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	4	4	no	—
115	C₄×C₃₁	#324	1	1	yes	characteristic
116	C₄×C₃₁	#324	1	1	yes	characteristic
117	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
118	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
119	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
120	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic
121	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
122	C₂×M₃₂	—	2	2	no	—
123	#424	—	2	2	no	—
124	#424	C₄×C₃₁	1	2	yes	—
125	#424	C₂²×C₃₁	1	2	yes	—
126	C₂×C₄⋊C₈	—	2	2	no	—
127	#349	—	2	2	no	—
128	#349	C₂²×C₃₁	1	4	yes	—
129	C₂×C₃₁	—	2	1	no	—
130	C₂×C₃₁	—	2	1	no	—
131	C₂×C₃₁	—	4	1	no	—
132	C₂×C₃₁	—	4	1	no	—
133	C₂×C₃₁	#1328	1	2	yes	direct product factor
134	C₂×C₃₁	C₂×#324	1	1	yes	characteristic
135	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₂×M₁₆	—	4	1	no	—
137	C₂×M₁₆	—	4	1	no	—
138	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
139	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
140	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	—	4	1	no	—
144	C₂²×C₈	—	4	1	no	—
145	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
146	C₂²×C₈	D₄×C₃₁	1	2	yes	—
147	#117	—	2	1	no	—
148	#117	—	2	1	no	—
149	#117	C₈×C₃₁	1	2	yes	—
150	#117	C₂×C₄×C₃₁	1	2	yes	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₂²⋊C₄	C₈×C₃₁	1	2	yes	—
153	C₂×C₄²	—	2	1	no	—
154	M₃₂	—	2	4	no	—
155	M₃₂	—	4	2	no	—
156	C₂×C₁₆	—	4	2	no	—
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₃₁	C₂×#1328	1	1	yes	characteristic, direct product factor, Sylow
160	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
169	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	2	no	—
174	M₁₆	—	4	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
183	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₁₆	—	4	4	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	4	2	no	—
194	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
195	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
196	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
210	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
211	C₂×C₄	C₃₁×#99	1	1	yes	characteristic
212	C₂×C₄	C₃₁×#99	1	1	yes	characteristic
213	C₂×C₄	C₃₁×#101	1	1	yes	characteristic
214	C₂×C₄	C₃₁×#101	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₃₁×#324	1	1	yes	characteristic
228	C₂²	C₃₁×#324	1	1	yes	characteristic
229	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	4	no	—
233	C₄	C₃₁×#324	1	1	yes	characteristic
234	C₄	C₃₁×#324	1	1	yes	characteristic
235	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
236	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	C₃₁×#1328	1	2	yes	direct product factor
242	C₂	C₂×C₃₁×#324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#99
commutator	C₂³	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×M₁₆	C₂³×C₃₁
socle	C₂²×C₃₁	#324
prime radical	C₂⁴×C₃₁	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1328	C₂	direct (N × H)
69. C₂×#1328	C₃₁	direct (N × H)
101. #1328	C₂×C₃₁	direct (N × H)
133. C₂×C₃₁	#1328	direct (N × H)
159. C₃₁	C₂×#1328	direct (N × H)
241. C₂	C₃₁×#1328	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×#424	2
C₂×C₃₁×#349	2
C₃₁×#1328	2
C₂×#1328	31

Subgroups — C2×C31×#1328 #21468216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1328 #21468216