Subgroups — C₃₁×#2395 #16842328

Order 3968.

560 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#398	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#397	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#397	C₂	1	2	yes	maximal, complemented
8	C₃₁×#394	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₃₁	—	2	4	no	—
16	C₃₁×#117	—	2	4	no	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
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	2	yes	—
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#95	C₂²	1	2	yes	—
32	C₃₁×#95	C₂²	1	2	yes	—
33	C₃₁×#95	C₂²	1	1	yes	characteristic
34	C₃₁×#95	C₂²	1	2	yes	—
35	C₃₁×#95	C₂²	1	1	yes	characteristic
36	C₃₁×#95	C₂²	1	2	yes	—
37	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
38	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₃₁	—	2	2	no	—
40	C₂×D₄×C₃₁	—	2	2	no	—
41	C₂×D₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	—	2	4	no	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	D₄	1	4	yes	—
46	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	2	4	no	—
60	C₂²⋊C₄×C₃₁	—	2	4	no	—
61	C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂²⋊C₄×C₃₁	—	2	4	no	—
63	C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂²⋊C₄×C₃₁	—	2	4	no	—
65	C₂²⋊C₄×C₃₁	—	2	4	no	—
66	C₄²×C₃₁	—	2	4	no	—
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	4	no	—
69	C₂³×C₃₁	—	2	1	no	—
70	C₂³×C₃₁	—	2	2	no	—
71	C₂³×C₃₁	—	2	2	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	2	2	no	—
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	—	2	1	no	—
76	C₂³×C₃₁	—	2	1	no	—
77	C₂³×C₃₁	C₂×D₄	1	2	yes	—
78	C₂³×C₃₁	#41	1	2	yes	—
79	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
80	D₄×C₃₁	—	2	8	no	—
81	C₂×C₄×C₃₁	—	2	1	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	1	no	—
86	C₂×C₄×C₃₁	—	2	8	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₃₁	#41	1	4	yes	—
104	C₂×C₄×C₃₁	#41	1	2	yes	—
105	C₂×C₄×C₃₁	#41	1	4	yes	—
106	#2395	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	4	no	—
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	4	1	no	—
113	C₂²×C₃₁	—	4	2	no	—
114	C₂²×C₃₁	—	4	2	no	—
115	C₂²×C₃₁	—	4	2	no	—
116	C₂²×C₃₁	—	4	1	no	—
117	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₃₁	C₂×#41	1	2	yes	—
119	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
120	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
121	C₂²×C₃₁	C₂×#41	1	2	yes	—
122	C₄×C₃₁	—	2	2	no	—
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	2	4	no	—
125	C₄×C₃₁	—	4	1	no	—
126	C₄×C₃₁	—	4	2	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	—	4	1	no	—
129	C₄×C₃₁	—	4	2	no	—
130	C₄×C₃₁	#123	1	4	yes	—
131	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
132	C₂×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#400	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#399	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#398	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#397	C₂×C₃₁	1	2	yes	complemented
137	#397	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#394	C₂×C₃₁	1	2	yes	—
139	C₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
140	C₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
141	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×C₃₁	—	4	2	no	—
143	C₂×C₃₁	—	4	2	no	—
144	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
145	C₂×C₃₁	#525	1	1	yes	characteristic
146	C₂×C₃₁	#529	1	2	yes	—
147	C₂×C₃₁	#529	1	2	yes	—
148	C₂×C₃₁	#532	1	1	yes	characteristic
149	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
151	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
152	C₄×D₄	—	2	4	no	—
153	#117	—	2	4	no	—
154	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
158	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
165	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
168	#95	C₂²×C₃₁	1	2	yes	—
169	#95	C₂²×C₃₁	1	1	yes	characteristic
170	#95	C₂²×C₃₁	1	2	yes	—
171	#95	C₂²×C₃₁	1	1	yes	characteristic
172	#95	C₂²×C₃₁	1	2	yes	—
173	#95	C₂²×C₃₁	1	2	yes	—
174	C₃₁	#2395	1	1	yes	characteristic, direct product factor, Sylow
175	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
176	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	D₄×C₃₁	1	4	yes	—
184	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
185	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
188	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
189	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
191	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	C₂×D₄×C₃₁	1	2	yes	—
216	C₂³	#41×C₃₁	1	2	yes	—
217	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
218	D₄	—	2	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
240	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
241	C₂×C₄	#41×C₃₁	1	4	yes	—
242	C₂×C₄	#41×C₃₁	1	2	yes	—
243	C₂×C₄	#41×C₃₁	1	4	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	1	no	—
254	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
255	C₂²	C₂×#41×C₃₁	1	2	yes	—
256	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
257	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
258	C₂²	C₂×#41×C₃₁	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	C₃₁×#123	1	4	yes	—
268	C₂	—	4	2	no	—
269	C₂	—	4	2	no	—
270	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
271	C₂	C₃₁×#525	1	1	yes	characteristic
272	C₂	C₃₁×#529	1	2	yes	—
273	C₂	C₃₁×#529	1	2	yes	—
274	C₂	C₃₁×#532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#117	C₂	semidirect
3. C₃₁×#400	C₂	semidirect
4. C₃₁×#399	C₂	semidirect
5. C₃₁×#398	C₂	semidirect
6. C₃₁×#397	C₂	semidirect
7. C₃₁×#397	C₂	semidirect
9. C₄×C₄⋊C₄×C₃₁	C₂	semidirect
10. C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
11. C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
106. #2395	C₃₁	direct (N × H)
132. C₂×#117	C₂×C₃₁	semidirect
133. #400	C₂×C₃₁	semidirect
134. #399	C₂×C₃₁	semidirect
135. #398	C₂×C₃₁	semidirect
136. #397	C₂×C₃₁	semidirect
137. #397	C₂×C₃₁	semidirect
139. C₄×C₄⋊C₄	C₂×C₃₁	semidirect
140. C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
141. C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
174. C₃₁	#2395	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×D₄×C₃₁	2
C₂×C₃₁×#117	2
C₃₁×#400	2
C₃₁×#399	2
C₃₁×#398	2
C₃₁×#397	2
C₃₁×#394	2
C₄×C₄⋊C₄×C₃₁	2
C₄×C₂²⋊C₄×C₃₁	2
#2395	31

Subgroups — C31×#2395 #16842328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2395 #16842328