Subgroups — C₂×C₃₁×#2641 #21491379

Order 7936.

3326 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2641	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#394	C₂	1	8	yes	maximal, complemented
5	C₂×C₃₁×#390	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
8	C₂³×D₄×C₃₁	C₂²	1	4	yes	—
9	C₂×C₃₁×#127	—	2	8	no	—
10	C₂×C₃₁×#127	C₂²	1	8	yes	—
11	C₂×C₃₁×#125	—	2	8	no	—
12	C₂×C₃₁×#125	C₂²	1	8	yes	complemented
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	16	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#394	C₂²	1	64	yes	complemented
19	C₃₁×#390	C₂²	1	32	yes	complemented
20	C₃₁×#376	C₂²	1	8	yes	complemented
21	C₂×C₃₁×#95	C₂²	1	4	yes	—
22	C₂⁵×C₃₁	C₂³	1	4	yes	—
23	C₂²×D₄×C₃₁	—	2	16	no	—
24	C₂²×D₄×C₃₁	—	2	8	no	—
25	C₂²×D₄×C₃₁	—	2	16	no	—
26	C₂²×D₄×C₃₁	—	4	16	no	—
27	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
28	C₂²×D₄×C₃₁	C₂³	1	16	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#127	—	2	32	no	—
34	C₃₁×#125	—	2	32	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
39	C₂×C₂²⋊C₄×C₃₁	—	4	32	no	—
40	C₂×C₂²⋊C₄×C₃₁	C₂³	1	64	yes	—
41	C₂×C₄²×C₃₁	—	2	8	no	—
42	C₂×C₄²×C₃₁	D₄	1	8	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
45	C₃₁×#95	C₂³	1	16	yes	—
46	C₂⁴×C₃₁	—	2	8	no	—
47	C₂⁴×C₃₁	—	2	16	no	—
48	C₂⁴×C₃₁	—	2	32	no	—
49	C₂⁴×C₃₁	C₂⁴	1	8	yes	—
50	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×D₄×C₃₁	—	2	32	no	—
52	C₂×D₄×C₃₁	—	2	16	no	—
53	C₂×D₄×C₃₁	—	2	32	no	—
54	C₂×D₄×C₃₁	—	4	32	no	—
55	C₂×D₄×C₃₁	—	4	64	no	—
56	C₂×D₄×C₃₁	—	4	32	no	—
57	C₂²×C₄×C₃₁	—	2	8	no	—
58	C₂²×C₄×C₃₁	—	2	8	no	—
59	C₂²×C₄×C₃₁	—	2	8	no	—
60	C₂²×C₄×C₃₁	—	2	16	no	—
61	C₂²×C₄×C₃₁	—	2	16	no	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₃₁	#41	1	8	yes	—
65	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
69	C₄⋊C₄×C₃₁	—	2	32	no	—
70	C₄⋊C₄×C₃₁	—	2	16	no	—
71	C₂²⋊C₄×C₃₁	—	4	64	no	—
72	C₄²×C₃₁	—	2	16	no	—
73	C₄²×C₃₁	C₂×D₄	1	16	yes	—
74	C₂×#2641	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	—	2	8	no	—
76	C₂³×C₃₁	—	2	16	no	—
77	C₂³×C₃₁	—	2	32	no	—
78	C₂³×C₃₁	—	4	16	no	—
79	C₂³×C₃₁	—	4	32	no	—
80	C₂³×C₃₁	—	4	64	no	—
81	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
82	C₂³×C₃₁	C₂×#41	1	4	yes	—
83	C₂³×C₃₁	H(5,𝔽₂)	1	8	yes	—
84	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
85	D₄×C₃₁	—	4	64	no	—
86	D₄×C₃₁	—	4	32	no	—
87	C₂×C₄×C₃₁	—	2	16	no	—
88	C₂×C₄×C₃₁	—	2	8	no	—
89	C₂×C₄×C₃₁	—	2	8	no	—
90	C₂×C₄×C₃₁	—	2	8	no	—
91	C₂×C₄×C₃₁	—	2	16	no	—
92	C₂×C₄×C₃₁	—	2	16	no	—
93	C₂×C₄×C₃₁	—	4	16	no	—
94	C₂×C₄×C₃₁	—	4	16	no	—
95	C₂×C₄×C₃₁	#124	1	16	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
97	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
98	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
99	C₂²×#127	C₂×C₃₁	1	1	yes	characteristic
100	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
101	#2641	C₂×C₃₁	1	16	yes	direct product factor
102	C₂×#394	C₂×C₃₁	1	8	yes	complemented
103	C₂×#390	C₂×C₃₁	1	4	yes	complemented
104	C₂×#376	C₂×C₃₁	1	1	yes	characteristic, complemented
105	C₂²×C₃₁	—	4	8	no	—
106	C₂²×C₃₁	—	4	16	no	—
107	C₂²×C₃₁	—	4	32	no	—
108	C₂²×C₃₁	—	8	32	no	—
109	C₂²×C₃₁	C₂×#124	1	4	yes	—
110	C₂²×C₃₁	#535	1	8	yes	—
111	C₂²×C₃₁	#540	1	16	yes	—
112	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
113	C₂²×C₃₁	C₂²×#41	1	2	yes	—
114	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	4	yes	—
115	C₄×C₃₁	—	2	8	no	—
116	C₄×C₃₁	—	4	8	no	—
117	C₄×C₃₁	—	4	8	no	—
118	C₄×C₃₁	C₂×#124	1	8	yes	—
119	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
120	C₂³×D₄	C₂²×C₃₁	1	4	yes	—
121	C₂×#127	—	2	8	no	—
122	C₂×#127	C₂²×C₃₁	1	8	yes	—
123	C₂×#125	—	2	8	no	—
124	C₂×#125	C₂²×C₃₁	1	8	yes	complemented
125	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
127	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	16	yes	—
128	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
129	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
130	#394	C₂²×C₃₁	1	64	yes	complemented
131	#390	C₂²×C₃₁	1	32	yes	complemented
132	#376	C₂²×C₃₁	1	8	yes	complemented
133	C₂×#95	C₂²×C₃₁	1	4	yes	—
134	C₂×C₃₁	—	8	8	no	—
135	C₂×C₃₁	#2641	1	8	yes	direct product factor
136	C₂×C₃₁	C₂²×#124	1	1	yes	characteristic
137	C₂×C₃₁	C₂×#535	1	2	yes	—
138	C₂×C₃₁	C₂×#540	1	4	yes	—
139	C₂⁵	C₂³×C₃₁	1	4	yes	—
140	C₂²×D₄	—	2	16	no	—
141	C₂²×D₄	—	2	8	no	—
142	C₂²×D₄	—	2	16	no	—
143	C₂²×D₄	—	4	16	no	—
144	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
145	C₂²×D₄	C₂³×C₃₁	1	16	yes	—
146	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
147	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
148	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
149	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
150	#127	—	2	32	no	—
151	#125	—	2	32	no	—
152	C₂×C₄⋊C₄	—	2	16	no	—
153	C₂×C₄⋊C₄	—	2	8	no	—
154	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
155	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
156	C₂×C₂²⋊C₄	—	4	32	no	—
157	C₂×C₂²⋊C₄	C₂³×C₃₁	1	64	yes	—
158	C₂×C₄²	—	2	8	no	—
159	C₂×C₄²	D₄×C₃₁	1	8	yes	—
160	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
161	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
162	#95	C₂³×C₃₁	1	16	yes	—
163	C₃₁	C₂×#2641	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	—	2	8	no	—
165	C₂⁴	—	2	16	no	—
166	C₂⁴	—	2	32	no	—
167	C₂⁴	C₂⁴×C₃₁	1	8	yes	—
168	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
169	C₂×D₄	—	2	32	no	—
170	C₂×D₄	—	2	16	no	—
171	C₂×D₄	—	2	32	no	—
172	C₂×D₄	—	4	32	no	—
173	C₂×D₄	—	4	64	no	—
174	C₂×D₄	—	4	32	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	16	no	—
179	C₂²×C₄	—	2	16	no	—
180	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
181	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
182	C₂²×C₄	#41×C₃₁	1	8	yes	—
183	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
184	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
185	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
186	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	16	no	—
189	C₂²⋊C₄	—	4	64	no	—
190	C₄²	—	2	16	no	—
191	C₄²	C₂×D₄×C₃₁	1	16	yes	—
192	C₂³	—	2	8	no	—
193	C₂³	—	2	16	no	—
194	C₂³	—	2	32	no	—
195	C₂³	—	4	16	no	—
196	C₂³	—	4	32	no	—
197	C₂³	—	4	64	no	—
198	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
199	C₂³	C₂×#41×C₃₁	1	4	yes	—
200	C₂³	C₃₁×H(5,𝔽₂)	1	8	yes	—
201	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
202	D₄	—	4	64	no	—
203	D₄	—	4	32	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	4	16	no	—
211	C₂×C₄	—	4	16	no	—
212	C₂×C₄	C₃₁×#124	1	16	yes	—
213	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
214	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
215	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
216	C₂²	—	4	8	no	—
217	C₂²	—	4	16	no	—
218	C₂²	—	4	32	no	—
219	C₂²	—	8	32	no	—
220	C₂²	C₂×C₃₁×#124	1	4	yes	—
221	C₂²	C₃₁×#535	1	8	yes	—
222	C₂²	C₃₁×#540	1	16	yes	—
223	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
224	C₂²	C₂²×#41×C₃₁	1	2	yes	—
225	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	4	yes	—
226	C₄	—	2	8	no	—
227	C₄	—	4	8	no	—
228	C₄	—	4	8	no	—
229	C₄	C₂×C₃₁×#124	1	8	yes	—
230	C₂	—	8	8	no	—
231	C₂	C₃₁×#2641	1	8	yes	direct product factor
232	C₂	C₂²×C₃₁×#124	1	1	yes	characteristic
233	C₂	C₂×C₃₁×#535	1	2	yes	—
234	C₂	C₂×C₃₁×#540	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#125	C₂	semidirect
3. C₃₁×#2641	C₂	direct (N × H)
4. C₂×C₃₁×#394	C₂	semidirect
5. C₂×C₃₁×#390	C₂	semidirect
6. C₂×C₃₁×#376	C₂	semidirect
12. C₂×C₃₁×#125	C₂²	semidirect
18. C₃₁×#394	C₂²	semidirect
19. C₃₁×#390	C₂²	semidirect
20. C₃₁×#376	C₂²	semidirect
74. C₂×#2641	C₃₁	direct (N × H)
100. C₂²×#125	C₂×C₃₁	semidirect
101. #2641	C₂×C₃₁	direct (N × H)
102. C₂×#394	C₂×C₃₁	semidirect
103. C₂×#390	C₂×C₃₁	semidirect
104. C₂×#376	C₂×C₃₁	semidirect
124. C₂×#125	C₂²×C₃₁	semidirect
130. #394	C₂²×C₃₁	semidirect
131. #390	C₂²×C₃₁	semidirect
132. #376	C₂²×C₃₁	semidirect
135. C₂×C₃₁	#2641	direct (N × H)
163. C₃₁	C₂×#2641	direct (N × H)
231. C₂	C₃₁×#2641	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×#127	2
C₂²×C₃₁×#125	2
C₃₁×#2641	2
C₂×C₃₁×#394	2
C₂×C₃₁×#390	2
C₂×C₃₁×#376	2
C₂×#2641	31

Subgroups — C2×C31×#2641 #21491379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2641 #21491379