Subgroups — C₂×C₃₁×#2171 #21490199

Order 7936.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#504	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2171	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#364	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#350	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#135	C₄	1	1	yes	characteristic
8	C₂×C₃₁×#131	C₂²	1	2	yes	—
9	C₃₁×#504	—	2	4	no	—
10	C₃₁×#504	C₂²	1	8	yes	complemented
11	C₂×C₃₁×M₃₂	—	2	2	no	—
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₃₁	—	2	2	no	—
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#443	C₄	1	4	yes	—
17	C₃₁×#443	C₂²	1	4	yes	—
18	C₂×C₃₁×#108	C₄	1	1	yes	characteristic
19	C₂×C₃₁×#104	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#364	C₂²	1	4	yes	complemented
22	C₃₁×#350	C₂²	1	8	yes	complemented
23	C₃₁×#346	C₂²	1	4	yes	complemented
24	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
25	C₃₁×#135	C₈	1	4	yes	—
26	C₃₁×#135	C₂×C₄	1	4	yes	—
27	C₂×Q₁₆×C₃₁	C₈	1	1	yes	characteristic
28	C₂×SD₁₆×C₃₁	C₈	1	2	yes	—
29	C₂×D₈×C₃₁	C₈	1	1	yes	characteristic
30	C₃₁×#131	—	2	2	no	—
31	C₃₁×#131	—	2	2	no	—
32	C₃₁×#131	C₂×C₄	1	4	yes	—
33	C₃₁×#131	C₂³	1	4	yes	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₃₁×M₃₂	—	2	2	no	—
41	C₃₁×M₃₂	—	2	8	no	—
42	C₃₁×M₃₂	C₂³	1	4	yes	—
43	C₂×C₁₆×C₃₁	—	2	2	no	—
44	C₂×C₁₆×C₃₁	—	2	1	no	—
45	C₂×C₁₆×C₃₁	—	2	1	no	—
46	C₂×C₁₆×C₃₁	—	2	8	no	—
47	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
48	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#108	C₂×C₄	1	4	yes	—
52	C₃₁×#104	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₃₁	C₂³	1	2	yes	—
55	#41×C₃₁	—	2	2	no	—
56	#41×C₃₁	—	2	2	no	—
57	#41×C₃₁	C₂×C₈	1	4	yes	—
58	#41×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	1	no	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	Q₁₆×C₃₁	C₂×C₈	1	4	yes	—
66	SD₁₆×C₃₁	C₂×C₈	1	8	yes	—
67	D₈×C₃₁	C₂×C₈	1	4	yes	—
68	M₁₆×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₄²×C₃₁	—	2	1	no	—
86	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
87	C₁₆×C₃₁	—	2	4	no	—
88	C₁₆×C₃₁	—	2	2	no	—
89	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
90	C₂×#2171	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	#131	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	C₂²×C₈	1	4	yes	—
95	D₄×C₃₁	—	2	4	no	—
96	D₄×C₃₁	—	2	4	no	—
97	D₄×C₃₁	—	2	2	no	—
98	D₄×C₃₁	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₃₁	—	2	4	no	—
111	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
112	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
113	C₈×C₃₁	C₂²×C₈	1	2	yes	—
114	C₈×C₃₁	C₂²×C₈	1	2	yes	—
115	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₂×#504	C₂×C₃₁	1	2	yes	complemented
118	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
119	#2171	C₂×C₃₁	1	8	yes	direct product factor
120	C₂×#364	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂×#350	C₂×C₃₁	1	2	yes	complemented
122	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	2	4	no	—
125	C₂²×C₃₁	—	4	4	no	—
126	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
127	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
128	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
129	C₄×C₃₁	—	2	4	no	—
130	C₄×C₃₁	—	4	2	no	—
131	C₄×C₃₁	C₈×D₄	1	1	yes	characteristic
132	C₄×C₃₁	C₈×D₄	1	1	yes	characteristic
133	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
134	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
135	C₂×#135	C₄×C₃₁	1	1	yes	characteristic
136	C₂×#131	C₂²×C₃₁	1	2	yes	—
137	#504	—	2	4	no	—
138	#504	C₂²×C₃₁	1	8	yes	complemented
139	C₂×M₃₂	—	2	2	no	—
140	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
141	C₂²×C₁₆	—	2	2	no	—
142	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
144	#443	C₄×C₃₁	1	4	yes	—
145	#443	C₂²×C₃₁	1	4	yes	—
146	C₂×#108	C₄×C₃₁	1	1	yes	characteristic
147	C₂×#104	C₄×C₃₁	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	#364	C₂²×C₃₁	1	4	yes	complemented
150	#350	C₂²×C₃₁	1	8	yes	complemented
151	#346	C₂²×C₃₁	1	4	yes	complemented
152	C₂×C₃₁	—	2	1	no	—
153	C₂×C₃₁	—	2	1	no	—
154	C₂×C₃₁	—	4	4	no	—
155	C₂×C₃₁	#2171	1	2	yes	direct product factor
156	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
157	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
158	#135	C₈×C₃₁	1	4	yes	—
159	#135	C₂×C₄×C₃₁	1	4	yes	—
160	C₂×Q₁₆	C₈×C₃₁	1	1	yes	characteristic
161	C₂×SD₁₆	C₈×C₃₁	1	2	yes	—
162	C₂×D₈	C₈×C₃₁	1	1	yes	characteristic
163	#131	—	2	2	no	—
164	#131	—	2	2	no	—
165	#131	C₂×C₄×C₃₁	1	4	yes	—
166	#131	C₂³×C₃₁	1	4	yes	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
173	M₃₂	—	2	2	no	—
174	M₃₂	—	2	8	no	—
175	M₃₂	C₂³×C₃₁	1	4	yes	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	8	no	—
180	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
181	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
182	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
183	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
184	#108	C₂×C₄×C₃₁	1	4	yes	—
185	#104	C₂×C₄×C₃₁	1	8	yes	—
186	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
187	C₄×C₈	C₂³×C₃₁	1	2	yes	—
188	C₃₁	C₂×#2171	1	1	yes	characteristic, direct product factor, Sylow
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	C₂×C₈×C₃₁	1	4	yes	—
192	#41	C₂²×C₄×C₃₁	1	4	yes	—
193	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	Q₁₆	C₂×C₈×C₃₁	1	4	yes	—
200	SD₁₆	C₂×C₈×C₃₁	1	8	yes	—
201	D₈	C₂×C₈×C₃₁	1	4	yes	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
217	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
218	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
221	C₁₆	—	2	4	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	C₃₁×#131	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	C₂²×C₈×C₃₁	1	4	yes	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	D₄	C₂²×C₈×C₃₁	1	4	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
242	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
245	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
246	C₈	C₂²×C₈×C₃₁	1	2	yes	—
247	C₈	C₂²×C₈×C₃₁	1	2	yes	—
248	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
249	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
254	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
255	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	C₈×D₄×C₃₁	1	1	yes	characteristic
259	C₄	C₈×D₄×C₃₁	1	1	yes	characteristic
260	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
261	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	4	4	no	—
265	C₂	C₃₁×#2171	1	2	yes	direct product factor
266	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₈×D₄
prime radical	C₂×C₃₁×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂²×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#504	C₂	semidirect
3. C₃₁×#2171	C₂	direct (N × H)
4. C₂×C₃₁×#364	C₂	semidirect
5. C₂×C₃₁×#350	C₂	semidirect
6. C₂×C₃₁×#346	C₂	semidirect
10. C₃₁×#504	C₂²	semidirect
21. C₃₁×#364	C₂²	semidirect
22. C₃₁×#350	C₂²	semidirect
23. C₃₁×#346	C₂²	semidirect
90. C₂×#2171	C₃₁	direct (N × H)
117. C₂×#504	C₂×C₃₁	semidirect
119. #2171	C₂×C₃₁	direct (N × H)
120. C₂×#364	C₂×C₃₁	semidirect
121. C₂×#350	C₂×C₃₁	semidirect
122. C₂×#346	C₂×C₃₁	semidirect
138. #504	C₂²×C₃₁	semidirect
149. #364	C₂²×C₃₁	semidirect
150. #350	C₂²×C₃₁	semidirect
151. #346	C₂²×C₃₁	semidirect
155. C₂×C₃₁	#2171	direct (N × H)
188. C₃₁	C₂×#2171	direct (N × H)
265. C₂	C₃₁×#2171	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×#504	2
C₂×C₃₁×#443	2
C₃₁×#2171	2
C₂×C₃₁×#364	2
C₂×C₃₁×#350	2
C₂×C₃₁×#346	2
C₂×#2171	31

Subgroups — C2×C31×#2171 #21490199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2171 #21490199