Subgroups — C₂×C₃₁×#2166 #21490184

Order 7936.

538 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2166	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃₁×#364	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#349	C₂	1	2	yes	maximal
5	C₂×C₃₁×#347	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic
8	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
9	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
10	C₃₁×#405	C₄	1	4	yes	—
11	C₃₁×#405	C₂²	1	4	yes	—
12	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
13	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#364	C₂²	1	8	yes	complemented
17	C₃₁×#349	C₂²	1	8	yes	—
18	C₃₁×#347	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×#117	—	2	2	no	—
32	C₃₁×#117	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₃₁×M₃₂	—	2	4	no	—
38	C₃₁×M₃₂	C₂³	1	8	yes	—
39	C₂×C₁₆×C₃₁	—	2	4	no	—
40	C₃₁×#97	C₂×C₄	1	4	yes	—
41	C₃₁×#97	C₂×C₄	1	2	yes	—
42	C₃₁×#97	C₂³	1	2	yes	—
43	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂⁴×C₃₁	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₃₁	—	2	4	no	—
58	M₁₆×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	2	1	no	—
60	C₂×C₈×C₃₁	—	2	1	no	—
61	C₂×C₈×C₃₁	—	2	1	no	—
62	C₂×C₈×C₃₁	—	2	1	no	—
63	C₂×C₈×C₃₁	—	2	8	no	—
64	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
65	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
66	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₃₁	#41	1	2	yes	—
71	C₂×C₈×C₃₁	#41	1	2	yes	—
72	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₂²⋊C₄×C₃₁	—	2	4	no	—
75	C₄²×C₃₁	—	2	1	no	—
76	C₄²×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
78	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
79	C₁₆×C₃₁	—	2	8	no	—
80	C₂×#2166	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	—	2	1	no	—
82	C₂³×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	—	4	2	no	—
84	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
97	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₃₁	—	2	2	no	—
104	C₈×C₃₁	—	2	2	no	—
105	C₈×C₃₁	#117	1	2	yes	—
106	C₈×C₃₁	C₂×M₁₆	1	4	yes	—
107	C₈×C₃₁	C₂×#41	1	2	yes	—
108	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
109	#2166	C₂×C₃₁	1	8	yes	direct product factor
110	C₂×#364	C₂×C₃₁	1	2	yes	complemented
111	C₂×#349	C₂×C₃₁	1	2	yes	—
112	C₂×#347	C₂×C₃₁	1	2	yes	complemented
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	—	2	2	no	—
115	C₂²×C₃₁	—	4	2	no	—
116	C₂²×C₃₁	—	4	2	no	—
117	C₂²×C₃₁	—	4	2	no	—
118	C₂²×C₃₁	#432	1	1	yes	characteristic
119	C₂²×C₃₁	#432	1	1	yes	characteristic
120	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
121	C₄×C₃₁	—	2	2	no	—
122	C₄×C₃₁	—	4	2	no	—
123	C₄×C₃₁	—	4	2	no	—
124	C₄×C₃₁	#432	1	1	yes	characteristic
125	C₄×C₃₁	#432	1	1	yes	characteristic
126	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
127	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
128	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
129	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic
130	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
131	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
132	#405	C₄×C₃₁	1	4	yes	—
133	#405	C₂²×C₃₁	1	4	yes	—
134	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
135	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
138	#364	C₂²×C₃₁	1	8	yes	complemented
139	#349	C₂²×C₃₁	1	8	yes	—
140	#347	C₂²×C₃₁	1	8	yes	complemented
141	C₂×C₃₁	—	2	1	no	—
142	C₂×C₃₁	—	2	1	no	—
143	C₂×C₃₁	—	4	2	no	—
144	C₂×C₃₁	#2166	1	2	yes	direct product factor
145	C₂×C₃₁	C₂×#432	1	1	yes	characteristic
146	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
150	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
154	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
155	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
158	#117	—	2	2	no	—
159	#117	C₂×C₄×C₃₁	1	4	yes	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₂²⋊C₄	—	2	1	no	—
162	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
164	M₃₂	—	2	4	no	—
165	M₃₂	C₂³×C₃₁	1	8	yes	—
166	C₂×C₁₆	—	2	4	no	—
167	#97	C₂×C₄×C₃₁	1	4	yes	—
168	#97	C₂×C₄×C₃₁	1	2	yes	—
169	#97	C₂³×C₃₁	1	2	yes	—
170	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
171	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
172	C₄×C₈	C₂³×C₃₁	1	2	yes	—
173	C₃₁	C₂×#2166	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁴	M₁₆×C₃₁	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
182	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
185	M₁₆	—	2	4	no	—
186	M₁₆	—	2	4	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
193	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
194	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
195	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
196	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
197	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
198	C₂×C₈	#41×C₃₁	1	2	yes	—
199	C₂×C₈	#41×C₃₁	1	2	yes	—
200	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
201	C₄⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
206	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
207	C₁₆	—	2	8	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
212	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
224	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
225	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
226	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
227	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	2	2	no	—
232	C₈	C₃₁×#117	1	2	yes	—
233	C₈	C₂×M₁₆×C₃₁	1	4	yes	—
234	C₈	C₂×#41×C₃₁	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₃₁×#432	1	1	yes	characteristic
241	C₂²	C₃₁×#432	1	1	yes	characteristic
242	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	C₃₁×#432	1	1	yes	characteristic
247	C₄	C₃₁×#432	1	1	yes	characteristic
248	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
249	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	C₃₁×#2166	1	2	yes	direct product factor
254	C₂	C₂×C₃₁×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	#432
prime radical	C₂⁴×C₃₁	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2166	C₂	direct (N × H)
3. C₂×C₃₁×#364	C₂	semidirect
5. C₂×C₃₁×#347	C₂	semidirect
16. C₃₁×#364	C₂²	semidirect
18. C₃₁×#347	C₂²	semidirect
80. C₂×#2166	C₃₁	direct (N × H)
109. #2166	C₂×C₃₁	direct (N × H)
110. C₂×#364	C₂×C₃₁	semidirect
112. C₂×#347	C₂×C₃₁	semidirect
138. #364	C₂²×C₃₁	semidirect
140. #347	C₂²×C₃₁	semidirect
144. C₂×C₃₁	#2166	direct (N × H)
173. C₃₁	C₂×#2166	direct (N × H)
253. C₂	C₃₁×#2166	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×#405	2
C₃₁×#2166	2
C₂×C₃₁×#364	2
C₂×C₃₁×#349	2
C₂×C₃₁×#347	2
C₂×#2166	31

Subgroups — C2×C31×#2166 #21490184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2166 #21490184