Subgroups — C₂×C₃₁×#1331 #21468348

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1331	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₃₁	—	2	1	no	—
7	C₂²×C₁₆×C₃₁	—	2	1	no	—
8	C₄×D₈×C₃₁	—	2	1	no	—
9	C₄×D₈×C₃₁	—	2	1	no	—
10	C₄×D₈×C₃₁	C₄	1	2	yes	—
11	C₄×D₈×C₃₁	C₂²	1	2	yes	—
12	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
14	C₂×C₃₁×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₃₁	—	2	1	no	—
17	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
18	C₃₁×#335	—	2	1	no	—
19	C₃₁×#335	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₃₁	—	2	1	no	—
21	C₂³×C₄×C₃₁	—	4	1	no	—
22	C₂×D₈×C₃₁	—	2	1	no	—
23	C₂×D₈×C₃₁	—	2	2	no	—
24	C₂×D₈×C₃₁	C₈	1	4	yes	complemented
25	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₃₁	—	4	1	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₄×D₄×C₃₁	—	2	2	no	—
30	C₄×D₄×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	4	1	no	—
32	C₄×D₄×C₃₁	—	4	1	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
35	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₃₁	—	2	4	no	—
37	C₂×C₁₆×C₃₁	—	2	2	no	—
38	C₃₁×#107	—	2	1	no	—
39	C₃₁×#107	C₂×C₄	1	2	yes	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	4	1	no	—
42	C₃₁×#102	—	2	2	no	—
43	C₃₁×#102	—	2	2	no	—
44	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂⁴×C₃₁	—	4	1	no	—
49	C₂×D₄×C₃₁	—	2	4	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	4	1	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	—	4	1	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	—	8	2	no	—
57	C₂²×C₄×C₃₁	—	8	2	no	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
61	D₈×C₃₁	—	2	4	no	—
62	D₈×C₃₁	—	2	2	no	—
63	D₈×C₃₁	C₂×C₈	1	4	yes	complemented
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	—	8	2	no	—
67	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
71	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₄⋊C₄×C₃₁	—	4	1	no	—
75	C₂²⋊C₄×C₃₁	—	4	2	no	—
76	C₂²⋊C₄×C₃₁	—	4	2	no	—
77	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₃₁	D₈	1	1	yes	characteristic
79	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₃₁	—	2	4	no	—
82	C₂×#1331	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	4	4	no	—
84	C₂³×C₃₁	—	4	2	no	—
85	C₂³×C₃₁	—	8	2	no	—
86	C₂³×C₃₁	—	8	2	no	—
87	C₂³×C₃₁	#102	1	1	yes	characteristic
88	D₄×C₃₁	—	2	4	no	—
89	D₄×C₃₁	—	4	4	no	—
90	D₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	8	2	no	—
94	C₂×C₄×C₃₁	—	8	4	no	—
95	C₂×C₄×C₃₁	—	8	2	no	—
96	C₂×C₄×C₃₁	—	8	2	no	—
97	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
99	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#104	1	2	yes	—
102	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
103	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	8	2	no	—
110	C₈×C₃₁	C₂²⋊C₈	1	2	yes	—
111	C₈×C₃₁	C₂²×C₈	1	1	yes	characteristic
112	C₈×C₃₁	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
114	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
115	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
116	#1331	C₂×C₃₁	1	4	yes	direct product factor
117	C₂²×C₃₁	—	4	4	no	—
118	C₂²×C₃₁	—	8	8	no	—
119	C₂²×C₃₁	—	8	2	no	—
120	C₂²×C₃₁	—	8	2	no	—
121	C₂²×C₃₁	D₄⋊C₈	1	2	yes	—
122	C₂²×C₃₁	#357	1	2	yes	—
123	C₂²×C₃₁	#359	1	2	yes	—
124	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	8	2	no	—
128	C₄×C₃₁	D₄⋊C₈	1	2	yes	—
129	C₄×C₃₁	#357	1	2	yes	—
130	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
132	C₄×C₃₁	C₂×D₁₆	1	1	yes	characteristic
133	C₄×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
134	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic
135	C₂×C₄×D₄	—	2	1	no	—
136	C₂²×C₁₆	—	2	1	no	—
137	C₄×D₈	—	2	1	no	—
138	C₄×D₈	—	2	1	no	—
139	C₄×D₈	C₄×C₃₁	1	2	yes	—
140	C₄×D₈	C₂²×C₃₁	1	2	yes	—
141	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×#102	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₄×C₁₆	—	2	1	no	—
146	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
147	#335	—	2	1	no	—
148	#335	C₂²×C₃₁	1	2	yes	complemented
149	C₂×C₃₁	—	8	4	no	—
150	C₂×C₃₁	#1331	1	4	yes	direct product factor
151	C₂×C₃₁	C₂×D₄⋊C₈	1	1	yes	characteristic
152	C₂×C₃₁	C₂×#357	1	1	yes	characteristic
153	C₂×C₃₁	C₂×#359	1	1	yes	characteristic
154	C₂²×D₄	—	2	1	no	—
155	C₂³×C₄	—	4	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	C₈×C₃₁	1	4	yes	complemented
159	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
160	C₂²×C₈	—	4	1	no	—
161	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	4	1	no	—
166	C₄×D₄	—	4	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₂²⋊C₄	—	4	1	no	—
169	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
170	C₂×C₁₆	—	2	4	no	—
171	C₂×C₁₆	—	2	2	no	—
172	#107	—	2	1	no	—
173	#107	C₂×C₄×C₃₁	1	2	yes	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	4	1	no	—
176	#102	—	2	2	no	—
177	#102	—	2	2	no	—
178	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
182	C₃₁	C₂×#1331	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	4	1	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	8	2	no	—
192	C₂²×C₄	—	8	2	no	—
193	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
196	D₈	—	2	4	no	—
197	D₈	—	2	2	no	—
198	D₈	C₂×C₈×C₃₁	1	4	yes	complemented
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	8	2	no	—
202	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
205	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₄²	D₈×C₃₁	1	1	yes	characteristic
214	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₁₆	—	2	4	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	8	2	no	—
221	C₂³	C₃₁×#102	1	1	yes	characteristic
222	D₄	—	2	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
232	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
234	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
235	C₂×C₄	C₃₁×#104	1	2	yes	—
236	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
237	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	8	2	no	—
244	C₈	C₃₁×C₂²⋊C₈	1	2	yes	—
245	C₈	C₂²×C₈×C₃₁	1	1	yes	characteristic
246	C₈	C₂×M₁₆×C₃₁	1	1	yes	characteristic
247	C₂²	—	4	4	no	—
248	C₂²	—	8	8	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	C₃₁×D₄⋊C₈	1	2	yes	—
252	C₂²	C₃₁×#357	1	2	yes	—
253	C₂²	C₃₁×#359	1	2	yes	—
254	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	C₃₁×D₄⋊C₈	1	2	yes	—
259	C₄	C₃₁×#357	1	2	yes	—
260	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
261	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
262	C₄	C₂×C₃₁×D₁₆	1	1	yes	characteristic
263	C₄	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
264	C₂	—	8	4	no	—
265	C₂	C₃₁×#1331	1	4	yes	direct product factor
266	C₂	C₂×C₃₁×D₄⋊C₈	1	1	yes	characteristic
267	C₂	C₂×C₃₁×#357	1	1	yes	characteristic
268	C₂	C₂×C₃₁×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	D₈
commutator	C₈	C₂²×C₈×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂³×C₃₁	#102
prime radical	C₂²×D₈×C₃₁	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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×C₄×C₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×#335	C₂	semidirect
4. C₃₁×#1331	C₂	direct (N × H)
17. C₄×C₁₆×C₃₁	C₂²	semidirect
19. C₃₁×#335	C₂²	semidirect
24. C₂×D₈×C₃₁	C₈	semidirect
63. D₈×C₃₁	C₂×C₈	semidirect
82. C₂×#1331	C₃₁	direct (N × H)
114. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
115. C₂×#335	C₂×C₃₁	semidirect
116. #1331	C₂×C₃₁	direct (N × H)
146. C₄×C₁₆	C₂²×C₃₁	semidirect
148. #335	C₂²×C₃₁	semidirect
150. C₂×C₃₁	#1331	direct (N × H)
158. C₂×D₈	C₈×C₃₁	semidirect
182. C₃₁	C₂×#1331	direct (N × H)
198. D₈	C₂×C₈×C₃₁	semidirect
265. C₂	C₃₁×#1331	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×C₄×D₈×C₃₁	2
C₂×C₄×C₁₆×C₃₁	2
C₂×C₃₁×#335	2
C₃₁×#1331	2
C₂×#1331	31

Subgroups — C2×C31×#1331 #21468348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1331 #21468348