Subgroups — C₃₁×#2693 #16842626

Order 3968.

568 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#123	C₂	1	2	yes	maximal, complemented
2	C₃₁×#400	C₂	1	2	yes	maximal, complemented
3	C₃₁×#397	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#396	C₂	1	2	yes	maximal, complemented
5	C₃₁×#392	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#386	C₂	1	2	yes	maximal, complemented
7	C₃₁×#386	C₂	1	2	yes	maximal, complemented
8	C₃₁×#385	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#380	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
11	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#123	—	2	8	no	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#95	C₂²	1	2	yes	—
27	C₃₁×#95	C₂²	1	2	yes	—
28	C₃₁×#95	C₂²	1	1	yes	characteristic
29	C₃₁×#95	C₂²	1	1	yes	characteristic
30	C₃₁×#95	C₂²	1	2	yes	—
31	C₃₁×#95	C₂²	1	2	yes	—
32	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
33	C₂⁴×C₃₁	C₂³	1	2	yes	—
34	C₂×D₄×C₃₁	—	2	4	no	—
35	C₂×D₄×C₃₁	—	2	4	no	—
36	C₂×D₄×C₃₁	—	2	4	no	—
37	C₂×D₄×C₃₁	—	4	1	no	—
38	C₂×D₄×C₃₁	—	4	2	no	—
39	C₂²×C₄×C₃₁	—	2	4	no	—
40	C₂²×C₄×C₃₁	—	2	4	no	—
41	C₂²×C₄×C₃₁	—	4	2	no	—
42	C₂²×C₄×C₃₁	—	4	1	no	—
43	C₂²×C₄×C₃₁	—	4	2	no	—
44	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	C₄⋊C₄×C₃₁	—	2	4	no	—
52	C₄⋊C₄×C₃₁	—	2	4	no	—
53	C₄⋊C₄×C₃₁	—	4	2	no	—
54	C₄⋊C₄×C₃₁	—	4	1	no	—
55	C₂²⋊C₄×C₃₁	—	2	4	no	—
56	C₂²⋊C₄×C₃₁	—	2	4	no	—
57	C₂²⋊C₄×C₃₁	—	4	2	no	—
58	C₂²⋊C₄×C₃₁	—	4	2	no	—
59	C₂²⋊C₄×C₃₁	—	4	1	no	—
60	C₂²⋊C₄×C₃₁	—	4	2	no	—
61	C₂²⋊C₄×C₃₁	D₄	1	4	yes	complemented
62	C₂³×C₃₁	—	2	2	no	—
63	C₂³×C₃₁	—	2	2	no	—
64	C₂³×C₃₁	—	2	2	no	—
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	2	1	no	—
67	C₂³×C₃₁	—	2	2	no	—
68	C₂³×C₃₁	—	2	2	no	—
69	C₂³×C₃₁	—	2	2	no	—
70	C₂³×C₃₁	—	2	1	no	—
71	C₂³×C₃₁	—	2	2	no	—
72	C₂³×C₃₁	C₂×D₄	1	2	yes	—
73	C₂³×C₃₁	#41	1	4	yes	—
74	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
75	D₄×C₃₁	—	4	8	no	—
76	D₄×C₃₁	—	4	4	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	4	8	no	—
96	C₂×C₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
98	#2693	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₃₁	—	2	2	no	—
100	C₂²×C₃₁	—	2	4	no	—
101	C₂²×C₃₁	—	2	4	no	—
102	C₂²×C₃₁	—	2	4	no	—
103	C₂²×C₃₁	—	2	4	no	—
104	C₂²×C₃₁	—	4	2	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	4	1	no	—
109	C₂²×C₃₁	—	4	1	no	—
110	C₂²×C₃₁	—	4	2	no	—
111	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₃₁	C₂×#41	1	2	yes	—
113	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
115	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
116	C₄×C₃₁	—	2	4	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₄×C₃₁	—	4	2	no	—
120	C₄×C₃₁	—	4	1	no	—
121	C₄×C₃₁	—	4	1	no	—
122	C₄×C₃₁	—	4	2	no	—
123	C₂×#123	C₂×C₃₁	1	2	yes	complemented
124	#400	C₂×C₃₁	1	2	yes	complemented
125	#397	C₂×C₃₁	1	1	yes	characteristic, complemented
126	#396	C₂×C₃₁	1	2	yes	complemented
127	#392	C₂×C₃₁	1	1	yes	characteristic
128	#386	C₂×C₃₁	1	2	yes	complemented
129	#386	C₂×C₃₁	1	2	yes	complemented
130	#385	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#380	C₂×C₃₁	1	2	yes	complemented
132	C₂×C₃₁	—	4	2	no	—
133	C₂×C₃₁	—	4	4	no	—
134	C₂×C₃₁	#537	1	1	yes	characteristic
135	C₂×C₃₁	#539	1	2	yes	—
136	C₂×C₃₁	#546	1	2	yes	—
137	C₂×C₃₁	#551	1	1	yes	characteristic
138	C₂×C₃₁	#560	1	1	yes	characteristic
139	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
140	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
142	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
143	#123	—	2	8	no	—
144	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
155	#95	C₂²×C₃₁	1	1	yes	characteristic
156	#95	C₂²×C₃₁	1	2	yes	—
157	#95	C₂²×C₃₁	1	2	yes	—
158	#95	C₂²×C₃₁	1	2	yes	—
159	#95	C₂²×C₃₁	1	2	yes	—
160	#95	C₂²×C₃₁	1	1	yes	characteristic
161	C₃₁	#2693	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
163	C₂⁴	C₂³×C₃₁	1	2	yes	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
176	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
178	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
179	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	4	1	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	D₄×C₃₁	1	4	yes	complemented
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	C₂×D₄×C₃₁	1	2	yes	—
203	C₂³	#41×C₃₁	1	4	yes	—
204	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
205	D₄	—	4	8	no	—
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
241	C₂²	C₂×#41×C₃₁	1	2	yes	—
242	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
244	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₃₁×#537	1	1	yes	characteristic
255	C₂	C₃₁×#539	1	2	yes	—
256	C₂	C₃₁×#546	1	2	yes	—
257	C₂	C₃₁×#551	1	1	yes	characteristic
258	C₂	C₃₁×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₃₁×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#123	C₂	semidirect
2. C₃₁×#400	C₂	semidirect
3. C₃₁×#397	C₂	semidirect
4. C₃₁×#396	C₂	semidirect
6. C₃₁×#386	C₂	semidirect
7. C₃₁×#386	C₂	semidirect
8. C₃₁×#385	C₂	semidirect
9. C₃₁×#380	C₂	semidirect
61. C₂²⋊C₄×C₃₁	D₄	semidirect
98. #2693	C₃₁	direct (N × H)
123. C₂×#123	C₂×C₃₁	semidirect
124. #400	C₂×C₃₁	semidirect
125. #397	C₂×C₃₁	semidirect
126. #396	C₂×C₃₁	semidirect
128. #386	C₂×C₃₁	semidirect
129. #386	C₂×C₃₁	semidirect
130. #385	C₂×C₃₁	semidirect
131. #380	C₂×C₃₁	semidirect
161. C₃₁	#2693	direct (N × H)
191. C₂²⋊C₄	D₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#123	2
C₃₁×#400	2
C₃₁×#397	2
C₃₁×#396	2
C₃₁×#392	2
C₃₁×#386	2
C₃₁×#385	2
C₃₁×#380	2
#2693	31

Subgroups — C31×#2693 #16842626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2693 #16842626