Subgroups — C₂×C₃₁×#2209 #21490336

Order 7936.

702 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	C₂²×C₃₁×Q₃₂	C₂	1	2	yes	maximal
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2209	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#358	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
12	C₂×C₃₁×Q₃₂	—	2	8	no	—
13	C₂×C₃₁×Q₃₂	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	—
17	C₄×Q₁₆×C₃₁	—	2	2	no	—
18	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	—	2	1	no	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	—
23	C₃₁×#358	C₂²	1	8	yes	—
24	C₂²×Q₈×C₃₁	—	2	1	no	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂×Q₁₆×C₃₁	—	2	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	2	no	—
31	C₂×Q₁₆×C₃₁	—	2	4	no	—
32	C₂×Q₁₆×C₃₁	—	4	4	no	—
33	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
34	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
35	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×C₄⋊Q₈	—	2	4	no	—
39	C₃₁×C₄⋊Q₈	—	4	2	no	—
40	C₄×Q₈×C₃₁	—	2	4	no	—
41	C₄×Q₈×C₃₁	—	4	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
46	C₂×C₄²×C₃₁	—	4	1	no	—
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₃₁×Q₃₂	—	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×#103	—	2	4	no	—
54	C₄×C₈×C₃₁	D₄	1	2	yes	—
55	C₄×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂×Q₈×C₃₁	—	2	4	no	—
57	C₂×Q₈×C₃₁	—	2	2	no	—
58	C₂×Q₈×C₃₁	—	2	4	no	—
59	C₂×Q₈×C₃₁	—	4	1	no	—
60	C₂×Q₈×C₃₁	—	4	2	no	—
61	C₂×Q₈×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	4	4	no	—
63	C₂×Q₈×C₃₁	—	4	4	no	—
64	C₂²×C₄×C₃₁	—	4	1	no	—
65	C₂²×C₄×C₃₁	—	4	1	no	—
66	C₂²×C₄×C₃₁	—	4	2	no	—
67	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₃₁	—	2	4	no	—
71	Q₁₆×C₃₁	—	2	4	no	—
72	Q₁₆×C₃₁	—	2	2	no	—
73	Q₁₆×C₃₁	—	2	4	no	—
74	Q₁₆×C₃₁	—	4	8	no	—
75	Q₁₆×C₃₁	—	4	4	no	—
76	Q₁₆×C₃₁	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₃₁	#41	1	2	yes	—
82	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₄⋊C₄×C₃₁	—	4	4	no	—
86	C₄⋊C₄×C₃₁	—	4	4	no	—
87	C₄⋊C₄×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	—	4	4	no	—
89	C₄²×C₃₁	D₈	1	2	yes	—
90	C₄²×C₃₁	C₂×D₄	1	2	yes	—
91	C₁₆×C₃₁	—	2	4	no	—
92	C₂×#2209	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
94	Q₈×C₃₁	—	2	4	no	—
95	Q₈×C₃₁	—	4	2	no	—
96	Q₈×C₃₁	—	4	4	no	—
97	Q₈×C₃₁	—	4	4	no	—
98	Q₈×C₃₁	—	4	4	no	—
99	Q₈×C₃₁	—	8	4	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	8	4	no	—
106	C₂×C₄×C₃₁	—	8	2	no	—
107	C₂×C₄×C₃₁	Q₃₂	1	4	yes	complemented
108	C₂×C₄×C₃₁	#121	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₃₁	—	2	2	no	—
115	C₈×C₃₁	#121	1	2	yes	—
116	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
118	C₂²×Q₃₂	C₂×C₃₁	1	2	yes	—
119	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
120	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
121	#2209	C₂×C₃₁	1	8	yes	direct product factor
122	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
123	C₂×#358	C₂×C₃₁	1	2	yes	—
124	C₂²×C₃₁	#459	1	2	yes	—
125	C₂²×C₃₁	C₂×Q₃₂	1	2	yes	—
126	C₂²×C₃₁	#510	1	2	yes	—
127	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	4	4	no	—
130	C₄×C₃₁	—	8	2	no	—
131	C₄×C₃₁	—	8	4	no	—
132	C₄×C₃₁	#459	1	2	yes	—
133	C₄×C₃₁	C₂×Q₃₂	1	4	yes	complemented
134	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
135	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂²×Q₁₆	—	2	1	no	—
137	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
138	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
139	C₂×C₄⋊Q₈	—	2	1	no	—
140	C₂×C₄×Q₈	—	2	1	no	—
141	C₂×Q₃₂	—	2	8	no	—
142	C₂×Q₃₂	C₂²×C₃₁	1	8	yes	—
143	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
144	#494	—	2	2	no	—
145	#494	C₂²×C₃₁	1	4	yes	—
146	C₄×Q₁₆	—	2	2	no	—
147	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
148	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×#103	—	2	1	no	—
150	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	—
152	#358	C₂²×C₃₁	1	8	yes	—
153	C₂×C₃₁	#2209	1	4	yes	direct product factor
154	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
155	C₂×C₃₁	C₂²×Q₃₂	1	1	yes	characteristic
156	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	—	4	4	no	—
166	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
167	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
168	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
169	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	4	no	—
172	C₄⋊Q₈	—	4	2	no	—
173	C₄×Q₈	—	2	4	no	—
174	C₄×Q₈	—	4	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	4	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	4	1	no	—
179	C₂×C₄²	—	4	1	no	—
180	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
181	Q₃₂	—	2	16	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
184	#107	—	2	1	no	—
185	#107	C₂³×C₃₁	1	2	yes	—
186	#103	—	2	4	no	—
187	C₄×C₈	D₄×C₃₁	1	2	yes	—
188	C₄×C₈	C₂³×C₃₁	1	2	yes	—
189	C₃₁	C₂×#2209	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	2	2	no	—
207	Q₁₆	—	2	4	no	—
208	Q₁₆	—	4	8	no	—
209	Q₁₆	—	4	4	no	—
210	Q₁₆	C₂×D₄×C₃₁	1	4	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
215	C₂×C₈	#41×C₃₁	1	2	yes	—
216	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	D₈×C₃₁	1	2	yes	—
224	C₄²	C₂×D₄×C₃₁	1	2	yes	—
225	C₁₆	—	2	4	no	—
226	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	4	4	no	—
232	Q₈	—	8	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₃₁×Q₃₂	1	4	yes	complemented
241	C₂×C₄	C₃₁×#121	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
243	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
245	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
246	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	C₃₁×#121	1	2	yes	—
249	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
250	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
251	C₂²	C₃₁×#459	1	2	yes	—
252	C₂²	C₂×C₃₁×Q₃₂	1	2	yes	—
253	C₂²	C₃₁×#510	1	2	yes	—
254	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	4	no	—
259	C₄	C₃₁×#459	1	2	yes	—
260	C₄	C₂×C₃₁×Q₃₂	1	4	yes	complemented
261	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
262	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂	C₃₁×#2209	1	4	yes	direct product factor
264	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
265	C₂	C₂²×C₃₁×Q₃₂	1	1	yes	characteristic
266	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
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
4. C₃₁×#2209	C₂	direct (N × H)
92. C₂×#2209	C₃₁	direct (N × H)
107. C₂×C₄×C₃₁	Q₃₂	semidirect
121. #2209	C₂×C₃₁	direct (N × H)
133. C₄×C₃₁	C₂×Q₃₂	semidirect
153. C₂×C₃₁	#2209	direct (N × H)
189. C₃₁	C₂×#2209	direct (N × H)
240. C₂×C₄	C₃₁×Q₃₂	semidirect
260. C₄	C₂×C₃₁×Q₃₂	semidirect
263. C₂	C₃₁×#2209	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×Q₃₂	2
C₂×C₃₁×#494	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2209	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#358	2
C₂×#2209	31

Subgroups — C2×C31×#2209 #21490336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2209 #21490336