Subgroups — C₂×C₃₁×#2230 #21490394

Order 7936.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2230	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#360	C₂	1	2	yes	maximal
7	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#131	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
11	C₃₁×#498	—	2	2	no	—
12	C₃₁×#498	C₂²	1	4	yes	complemented
13	C₃₁×#443	—	2	2	no	—
14	C₃₁×#443	C₂²	1	4	yes	—
15	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	—	2	1	no	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#104	—	2	1	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#365	C₂²	1	8	yes	complemented
21	C₃₁×#364	C₂²	1	4	yes	complemented
22	C₃₁×#360	C₂²	1	8	yes	—
23	C₂×#41×C₃₁	—	2	1	no	—
24	C₂²×Q₈×C₃₁	—	4	1	no	—
25	C₃₁×#135	—	2	1	no	—
26	C₃₁×#135	—	2	1	no	—
27	C₃₁×#135	Q₈	1	2	yes	complemented
28	C₃₁×#135	C₂³	1	2	yes	—
29	C₂×Q₁₆×C₃₁	Q₈	1	1	yes	characteristic, complemented
30	C₂×SD₁₆×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	Q₈	1	1	yes	characteristic, complemented
32	C₃₁×#131	—	2	4	no	—
33	C₃₁×#131	—	4	2	no	—
34	C₂×M₁₆×C₃₁	—	2	1	no	—
35	C₂×M₁₆×C₃₁	—	4	1	no	—
36	C₂²×C₈×C₃₁	—	4	1	no	—
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×C₄⋊Q₈	—	2	4	no	—
40	C₃₁×C₄⋊Q₈	—	4	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
43	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
44	C₃₁×M₃₂	—	2	2	no	—
45	C₃₁×M₃₂	C₂³	1	4	yes	—
46	C₂×C₁₆×C₃₁	—	2	2	no	—
47	C₃₁×#108	—	2	1	no	—
48	C₃₁×#108	C₂³	1	2	yes	—
49	C₃₁×#106	—	2	4	no	—
50	C₃₁×#106	—	2	2	no	—
51	C₃₁×#106	C₂³	1	4	yes	—
52	C₃₁×#104	—	2	4	no	—
53	C₄×C₈×C₃₁	D₄	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	—	4	1	no	—
58	#41×C₃₁	—	4	1	no	—
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	—	4	4	no	—
61	C₂×Q₈×C₃₁	—	8	4	no	—
62	C₂×D₄×C₃₁	—	2	1	no	—
63	C₂×D₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	—	2	1	no	—
65	C₂²×C₄×C₃₁	—	4	1	no	—
66	C₂²×C₄×C₃₁	—	4	2	no	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₃₁	—	2	1	no	—
69	Q₁₆×C₃₁	C₂×Q₈	1	2	yes	complemented
70	SD₁₆×C₃₁	—	2	2	no	—
71	SD₁₆×C₃₁	—	2	2	no	—
72	D₈×C₃₁	—	2	1	no	—
73	D₈×C₃₁	C₂×Q₈	1	2	yes	complemented
74	M₁₆×C₃₁	—	2	2	no	—
75	M₁₆×C₃₁	—	4	1	no	—
76	M₁₆×C₃₁	—	4	4	no	—
77	C₂×C₈×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	—	4	4	no	—
79	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
80	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
81	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
82	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
87	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	4	4	no	—
91	C₄⋊C₄×C₃₁	—	4	2	no	—
92	C₄²×C₃₁	—	2	1	no	—
93	C₄²×C₃₁	C₂×D₄	1	2	yes	—
94	C₁₆×C₃₁	—	2	4	no	—
95	C₂×#2230	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	4	1	no	—
97	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
98	Q₈×C₃₁	—	2	2	no	—
99	Q₈×C₃₁	—	4	1	no	—
100	Q₈×C₃₁	—	8	8	no	—
101	D₄×C₃₁	—	2	2	no	—
102	D₄×C₃₁	—	4	2	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	8	4	no	—
112	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₃₁	—	4	2	no	—
119	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
121	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
122	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
123	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
125	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
127	#2230	C₂×C₃₁	1	8	yes	direct product factor
128	C₂×#365	C₂×C₃₁	1	2	yes	complemented
129	C₂×#364	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂×#360	C₂×C₃₁	1	2	yes	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	4	2	no	—
133	C₂²×C₃₁	—	8	2	no	—
134	C₂²×C₃₁	#476	1	1	yes	characteristic
135	C₂²×C₃₁	#476	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	4	2	no	—
139	C₄×C₃₁	—	8	4	no	—
140	C₄×C₃₁	#476	1	1	yes	characteristic
141	C₄×C₃₁	#476	1	1	yes	characteristic
142	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×#131	—	2	1	no	—
146	C₂×C₄⋊Q₈	—	2	1	no	—
147	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
148	#498	—	2	2	no	—
149	#498	C₂²×C₃₁	1	4	yes	complemented
150	#443	—	2	2	no	—
151	#443	C₂²×C₃₁	1	4	yes	—
152	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×#106	—	2	1	no	—
154	C₂×#106	C₂²×C₃₁	1	2	yes	—
155	C₂×#104	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	#365	C₂²×C₃₁	1	8	yes	complemented
158	#364	C₂²×C₃₁	1	4	yes	complemented
159	#360	C₂²×C₃₁	1	8	yes	—
160	C₂×C₃₁	—	2	1	no	—
161	C₂×C₃₁	—	2	1	no	—
162	C₂×C₃₁	—	8	2	no	—
163	C₂×C₃₁	#2230	1	2	yes	direct product factor
164	C₂×C₃₁	C₂×#476	1	1	yes	characteristic
165	C₂×#41	—	2	1	no	—
166	C₂²×Q₈	—	4	1	no	—
167	#135	—	2	1	no	—
168	#135	—	2	1	no	—
169	#135	Q₈×C₃₁	1	2	yes	complemented
170	#135	C₂³×C₃₁	1	2	yes	—
171	C₂×Q₁₆	Q₈×C₃₁	1	1	yes	characteristic, complemented
172	C₂×SD₁₆	—	2	1	no	—
173	C₂×D₈	Q₈×C₃₁	1	1	yes	characteristic, complemented
174	#131	—	2	4	no	—
175	#131	—	4	2	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	4	1	no	—
178	C₂²×C₈	—	4	1	no	—
179	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	4	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	4	1	no	—
185	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
186	M₃₂	—	2	2	no	—
187	M₃₂	C₂³×C₃₁	1	4	yes	—
188	C₂×C₁₆	—	2	2	no	—
189	#108	—	2	1	no	—
190	#108	C₂³×C₃₁	1	2	yes	—
191	#106	—	2	4	no	—
192	#106	—	2	2	no	—
193	#106	C₂³×C₃₁	1	4	yes	—
194	#104	—	2	4	no	—
195	C₄×C₈	D₄×C₃₁	1	2	yes	—
196	C₄×C₈	C₂³×C₃₁	1	2	yes	—
197	C₃₁	C₂×#2230	1	1	yes	characteristic, direct product factor, Sylow
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	4	1	no	—
201	#41	—	4	1	no	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	8	4	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
211	Q₁₆	—	2	1	no	—
212	Q₁₆	C₂×Q₈×C₃₁	1	2	yes	complemented
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	2	2	no	—
215	D₈	—	2	1	no	—
216	D₈	C₂×Q₈×C₃₁	1	2	yes	complemented
217	M₁₆	—	2	2	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	4	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
223	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
224	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
227	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
229	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
230	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	C₂×D₄×C₃₁	1	2	yes	—
237	C₁₆	—	2	4	no	—
238	C₂³	—	4	1	no	—
239	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	Q₈	—	8	8	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
262	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
263	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
264	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
265	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
266	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	C₃₁×#476	1	1	yes	characteristic
271	C₂²	C₃₁×#476	1	1	yes	characteristic
272	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	8	4	no	—
276	C₄	C₃₁×#476	1	1	yes	characteristic
277	C₄	C₃₁×#476	1	1	yes	characteristic
278	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
279	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	C₃₁×#2230	1	2	yes	direct product factor
284	C₂	C₂×C₃₁×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#498	C₂	semidirect
3. C₃₁×#2230	C₂	direct (N × H)
4. C₂×C₃₁×#365	C₂	semidirect
5. C₂×C₃₁×#364	C₂	semidirect
12. C₃₁×#498	C₂²	semidirect
20. C₃₁×#365	C₂²	semidirect
21. C₃₁×#364	C₂²	semidirect
27. C₃₁×#135	Q₈	semidirect
29. C₂×Q₁₆×C₃₁	Q₈	semidirect
31. C₂×D₈×C₃₁	Q₈	semidirect
69. Q₁₆×C₃₁	C₂×Q₈	semidirect
73. D₈×C₃₁	C₂×Q₈	semidirect
95. C₂×#2230	C₃₁	direct (N × H)
125. C₂×#498	C₂×C₃₁	semidirect
127. #2230	C₂×C₃₁	direct (N × H)
128. C₂×#365	C₂×C₃₁	semidirect
129. C₂×#364	C₂×C₃₁	semidirect
149. #498	C₂²×C₃₁	semidirect
157. #365	C₂²×C₃₁	semidirect
158. #364	C₂²×C₃₁	semidirect
163. C₂×C₃₁	#2230	direct (N × H)
169. #135	Q₈×C₃₁	semidirect
171. C₂×Q₁₆	Q₈×C₃₁	semidirect
173. C₂×D₈	Q₈×C₃₁	semidirect
197. C₃₁	C₂×#2230	direct (N × H)
212. Q₁₆	C₂×Q₈×C₃₁	semidirect
216. D₈	C₂×Q₈×C₃₁	semidirect
283. C₂	C₃₁×#2230	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×#498	2
C₂×C₃₁×#443	2
C₃₁×#2230	2
C₂×C₃₁×#365	2
C₂×C₃₁×#364	2
C₂×C₃₁×#360	2
C₂×#2230	31

Subgroups — C2×C31×#2230 #21490394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2230 #21490394