Subgroups — C₂×C₃₁×#2208 #21490333

Order 7936.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×SD₃₂	C₂	1	2	yes	maximal
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2208	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₄×D₄×C₃₁	—	2	1	no	—
12	C₂×C₃₁×SD₃₂	—	2	8	no	—
13	C₂×C₃₁×SD₃₂	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#494	—	2	2	no	—
16	C₃₁×#494	C₂²	1	4	yes	complemented
17	C₄×D₈×C₃₁	—	2	2	no	—
18	C₄×D₈×C₃₁	C₂²	1	4	yes	—
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#102	—	2	1	no	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₃₁×#358	C₂²	1	8	yes	complemented
24	C₂²×Q₈×C₃₁	—	2	2	no	—
25	C₂²×D₄×C₃₁	—	2	1	no	—
26	C₂³×C₄×C₃₁	—	4	1	no	—
27	C₂×Q₁₆×C₃₁	—	2	4	no	—
28	C₂×Q₁₆×C₃₁	—	2	2	no	—
29	C₂×Q₁₆×C₃₁	—	2	4	no	—
30	C₂×Q₁₆×C₃₁	—	2	4	no	—
31	C₂×Q₁₆×C₃₁	—	4	4	no	—
32	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
33	C₂×D₈×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	2	2	no	—
35	C₂×D₈×C₃₁	D₄	1	4	yes	—
36	C₂×D₈×C₃₁	C₂³	1	2	yes	—
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	C₄×D₄×C₃₁	—	2	4	no	—
42	C₄×D₄×C₃₁	—	4	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
46	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₃₁×SD₃₂	—	2	16	no	—
49	C₂×C₁₆×C₃₁	—	2	4	no	—
50	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
51	C₃₁×#107	—	2	1	no	—
52	C₃₁×#107	C₂³	1	2	yes	—
53	C₃₁×#102	—	2	4	no	—
54	C₄×C₈×C₃₁	D₄	1	2	yes	—
55	C₄×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂⁴×C₃₁	—	4	1	no	—
57	C₂×Q₈×C₃₁	—	2	4	no	—
58	C₂×Q₈×C₃₁	—	4	2	no	—
59	C₂×Q₈×C₃₁	—	4	4	no	—
60	C₂×Q₈×C₃₁	—	4	4	no	—
61	C₂×D₄×C₃₁	—	2	4	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	—	4	1	no	—
64	C₂×D₄×C₃₁	—	4	2	no	—
65	C₂²×C₄×C₃₁	—	4	1	no	—
66	C₂²×C₄×C₃₁	—	4	2	no	—
67	C₂²×C₄×C₃₁	—	4	4	no	—
68	C₂²×C₄×C₃₁	—	8	4	no	—
69	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₃₁	—	2	4	no	—
73	Q₁₆×C₃₁	—	2	4	no	—
74	Q₁₆×C₃₁	—	4	8	no	—
75	Q₁₆×C₃₁	—	4	4	no	—
76	D₈×C₃₁	—	2	4	no	—
77	D₈×C₃₁	—	2	2	no	—
78	D₈×C₃₁	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₃₁	#41	1	2	yes	—
84	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	4	1	no	—
87	C₄⋊C₄×C₃₁	—	4	4	no	—
88	C₄⋊C₄×C₃₁	—	4	4	no	—
89	C₂²⋊C₄×C₃₁	—	4	4	no	—
90	C₄²×C₃₁	D₈	1	2	yes	—
91	C₄²×C₃₁	C₂×D₄	1	2	yes	—
92	C₁₆×C₃₁	—	2	4	no	—
93	C₂×#2208	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	—	4	4	no	—
96	C₂³×C₃₁	—	8	2	no	—
97	C₂³×C₃₁	—	8	2	no	—
98	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₃₁	—	4	4	no	—
100	Q₈×C₃₁	—	4	4	no	—
101	Q₈×C₃₁	—	8	4	no	—
102	D₄×C₃₁	—	2	4	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	8	4	no	—
109	C₂×C₄×C₃₁	—	8	2	no	—
110	C₂×C₄×C₃₁	—	8	8	no	—
111	C₂×C₄×C₃₁	SD₃₂	1	4	yes	complemented
112	C₂×C₄×C₃₁	#121	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₃₁	—	2	2	no	—
119	C₈×C₃₁	#121	1	2	yes	—
120	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
122	C₂²×SD₃₂	C₂×C₃₁	1	2	yes	—
123	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
125	#2208	C₂×C₃₁	1	8	yes	direct product factor
126	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
127	C₂×#358	C₂×C₃₁	1	2	yes	complemented
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	—	8	2	no	—
130	C₂²×C₃₁	—	8	2	no	—
131	C₂²×C₃₁	—	8	8	no	—
132	C₂²×C₃₁	#459	1	2	yes	—
133	C₂²×C₃₁	C₂×SD₃₂	1	2	yes	—
134	C₂²×C₃₁	#510	1	2	yes	—
135	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₃₁	—	2	2	no	—
137	C₄×C₃₁	—	8	2	no	—
138	C₄×C₃₁	—	8	4	no	—
139	C₄×C₃₁	#459	1	2	yes	—
140	C₄×C₃₁	C₂×SD₃₂	1	4	yes	complemented
141	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
142	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂²×Q₁₆	—	2	1	no	—
144	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
145	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×C₄⋊Q₈	—	2	1	no	—
147	C₂×C₄×D₄	—	2	1	no	—
148	C₂×SD₃₂	—	2	8	no	—
149	C₂×SD₃₂	C₂²×C₃₁	1	8	yes	—
150	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
151	#494	—	2	2	no	—
152	#494	C₂²×C₃₁	1	4	yes	complemented
153	C₄×D₈	—	2	2	no	—
154	C₄×D₈	C₂²×C₃₁	1	4	yes	—
155	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×#102	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	complemented
159	#358	C₂²×C₃₁	1	8	yes	complemented
160	C₂×C₃₁	—	8	4	no	—
161	C₂×C₃₁	#2208	1	4	yes	direct product factor
162	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
163	C₂×C₃₁	C₂²×SD₃₂	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×D₄	—	2	1	no	—
167	C₂³×C₄	—	4	1	no	—
168	C₂×Q₁₆	—	2	4	no	—
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	4	4	no	—
173	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	D₄×C₃₁	1	4	yes	—
177	C₂×D₈	C₂³×C₃₁	1	2	yes	—
178	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
180	C₄⋊Q₈	—	2	4	no	—
181	C₄⋊Q₈	—	4	2	no	—
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₂²⋊C₄	—	4	1	no	—
188	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
189	SD₃₂	—	2	16	no	—
190	C₂×C₁₆	—	2	4	no	—
191	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
192	#107	—	2	1	no	—
193	#107	C₂³×C₃₁	1	2	yes	—
194	#102	—	2	4	no	—
195	C₄×C₈	D₄×C₃₁	1	2	yes	—
196	C₄×C₈	C₂³×C₃₁	1	2	yes	—
197	C₃₁	C₂×#2208	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	—	4	1	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	8	4	no	—
211	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	2	4	no	—
216	Q₁₆	—	4	8	no	—
217	Q₁₆	—	4	4	no	—
218	D₈	—	2	4	no	—
219	D₈	—	2	2	no	—
220	D₈	C₂×D₄×C₃₁	1	4	yes	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
225	C₂×C₈	#41×C₃₁	1	2	yes	—
226	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	D₈×C₃₁	1	2	yes	—
233	C₄²	C₂×D₄×C₃₁	1	2	yes	—
234	C₁₆	—	2	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	8	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₃₁×SD₃₂	1	4	yes	complemented
253	C₂×C₄	C₃₁×#121	1	2	yes	—
254	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
255	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
257	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
258	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	C₃₁×#121	1	2	yes	—
261	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
262	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
263	C₂²	—	4	4	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	8	no	—
267	C₂²	C₃₁×#459	1	2	yes	—
268	C₂²	C₂×C₃₁×SD₃₂	1	2	yes	—
269	C₂²	C₃₁×#510	1	2	yes	—
270	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₃₁×#459	1	2	yes	—
275	C₄	C₂×C₃₁×SD₃₂	1	4	yes	complemented
276	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
277	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	C₃₁×#2208	1	4	yes	direct product factor
280	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
281	C₂	C₂²×C₃₁×SD₃₂	1	1	yes	characteristic
282	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#494	C₂	semidirect
4. C₃₁×#2208	C₂	direct (N × H)
5. C₂×C₃₁×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₃₁×#358	C₂	semidirect
16. C₃₁×#494	C₂²	semidirect
22. C₃₁×C₄⋊C₁₆	C₂²	semidirect
23. C₃₁×#358	C₂²	semidirect
93. C₂×#2208	C₃₁	direct (N × H)
111. C₂×C₄×C₃₁	SD₃₂	semidirect
123. C₂×#494	C₂×C₃₁	semidirect
125. #2208	C₂×C₃₁	direct (N × H)
126. C₂×C₄⋊C₁₆	C₂×C₃₁	semidirect
127. C₂×#358	C₂×C₃₁	semidirect
140. C₄×C₃₁	C₂×SD₃₂	semidirect
152. #494	C₂²×C₃₁	semidirect
158. C₄⋊C₁₆	C₂²×C₃₁	semidirect
159. #358	C₂²×C₃₁	semidirect
161. C₂×C₃₁	#2208	direct (N × H)
197. C₃₁	C₂×#2208	direct (N × H)
252. C₂×C₄	C₃₁×SD₃₂	semidirect
275. C₄	C₂×C₃₁×SD₃₂	semidirect
279. C₂	C₃₁×#2208	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×SD₃₂	2
C₂×C₃₁×#494	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2208	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#358	2
C₂×#2208	31

Subgroups — C2×C31×#2208 #21490333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2208 #21490333