Subgroups — C₃₁×#3285 #16843218

Order 3968.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#548	C₂	1	2	yes	maximal, complemented
4	C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#464	C₂	1	2	yes	maximal, complemented
8	C₃₁×#452	C₂	1	2	yes	maximal, complemented
9	C₃₁×#448	C₂	1	2	yes	maximal, complemented
10	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	2	yes	—
13	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#135	—	2	4	no	—
15	C₂×Q₁₆×C₃₁	—	2	4	no	—
16	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
21	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#124	—	2	2	no	—
25	C₃₁×#124	C₂²	1	2	yes	—
26	C₃₁×C₂²⋊Q₈	—	2	2	no	—
27	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₃₁×#121	—	2	2	no	—
29	C₃₁×#121	C₂²	1	2	yes	—
30	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#107	C₂²	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#103	C₂²	1	2	yes	—
37	C₃₁×#102	C₂²	1	2	yes	—
38	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	#41×C₃₁	—	2	4	no	—
42	#41×C₃₁	—	4	2	no	—
43	C₂×Q₈×C₃₁	—	2	2	no	—
44	C₂×Q₈×C₃₁	—	2	4	no	—
45	C₂×Q₈×C₃₁	—	4	2	no	—
46	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	Q₁₆×C₃₁	—	2	4	no	—
57	Q₁₆×C₃₁	—	2	2	no	—
58	Q₁₆×C₃₁	D₄	1	4	yes	complemented
59	SD₁₆×C₃₁	—	2	4	no	—
60	SD₁₆×C₃₁	—	4	2	no	—
61	D₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₄²×C₃₁	—	2	2	no	—
79	C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₃₁	—	2	2	no	—
81	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₃₁	—	2	2	no	—
84	Q₈×C₃₁	—	2	2	no	—
85	Q₈×C₃₁	—	2	2	no	—
86	Q₈×C₃₁	—	4	4	no	—
87	Q₈×C₃₁	C₂×D₄	1	4	yes	—
88	D₄×C₃₁	—	2	2	no	—
89	D₄×C₃₁	—	2	2	no	—
90	D₄×C₃₁	—	2	2	no	—
91	D₄×C₃₁	—	4	4	no	—
92	D₄×C₃₁	—	4	4	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	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	—	4	4	no	—
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	C₂×D₄	1	2	yes	—
111	#3285	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	2	1	no	—
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	4	2	no	—
118	C₂²×C₃₁	—	4	2	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	#135	1	2	yes	complemented
121	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₃₁	—	2	1	no	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	4	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	—	4	2	no	—
129	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
132	C₂²×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#548	C₂×C₃₁	1	2	yes	complemented
134	#495	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#467	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#466	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#464	C₂×C₃₁	1	2	yes	complemented
138	#452	C₂×C₃₁	1	2	yes	complemented
139	#448	C₂×C₃₁	1	2	yes	complemented
140	C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×C₃₁	—	2	2	no	—
143	C₂×C₃₁	—	4	1	no	—
144	C₂×C₃₁	—	8	2	no	—
145	C₂×C₃₁	D₄²	1	1	yes	characteristic
146	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
147	C₂×C₃₁	#578	1	1	yes	characteristic
148	C₂×#41	C₂²×C₃₁	1	2	yes	—
149	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
150	#135	—	2	4	no	—
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
153	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
156	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
157	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	#124	—	2	2	no	—
161	#124	C₂²×C₃₁	1	2	yes	—
162	C₂²⋊Q₈	—	2	2	no	—
163	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
164	#121	—	2	2	no	—
165	#121	C₂²×C₃₁	1	2	yes	—
166	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
169	#107	C₂²×C₃₁	1	1	yes	characteristic
170	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
171	#103	C₂²×C₃₁	1	2	yes	—
172	#103	C₂²×C₃₁	1	2	yes	—
173	#102	C₂²×C₃₁	1	2	yes	—
174	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
175	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
176	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
177	C₃₁	#3285	1	1	yes	characteristic, direct product factor, Sylow
178	#41	—	2	4	no	—
179	#41	—	4	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
184	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
188	C₂×D₄	C₂³×C₃₁	1	2	yes	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	2	2	no	—
195	Q₁₆	D₄×C₃₁	1	4	yes	complemented
196	SD₁₆	—	2	4	no	—
197	SD₁₆	—	4	2	no	—
198	D₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
202	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
208	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
214	C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
215	C₄²	—	2	2	no	—
216	C₄²	C₂³×C₃₁	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	2	2	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	4	4	no	—
224	Q₈	C₂×D₄×C₃₁	1	4	yes	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	C₂×D₄×C₃₁	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₃₁×#135	1	2	yes	complemented
257	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
266	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	8	2	no	—
270	C₂	D₄²×C₃₁	1	1	yes	characteristic
271	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
272	C₂	C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#135	C₂	semidirect
2. C₂²×Q₁₆×C₃₁	C₂	semidirect
3. C₃₁×#548	C₂	semidirect
4. C₃₁×#495	C₂	semidirect
5. C₃₁×#467	C₂	semidirect
6. C₃₁×#466	C₂	semidirect
7. C₃₁×#464	C₂	semidirect
8. C₃₁×#452	C₂	semidirect
9. C₃₁×#448	C₂	semidirect
10. C₄×Q₁₆×C₃₁	C₂	semidirect
11. C₈×D₄×C₃₁	C₂	semidirect
58. Q₁₆×C₃₁	D₄	semidirect
111. #3285	C₃₁	direct (N × H)
120. C₂²×C₃₁	#135	semidirect
131. C₂×#135	C₂×C₃₁	semidirect
132. C₂²×Q₁₆	C₂×C₃₁	semidirect
133. #548	C₂×C₃₁	semidirect
134. #495	C₂×C₃₁	semidirect
135. #467	C₂×C₃₁	semidirect
136. #466	C₂×C₃₁	semidirect
137. #464	C₂×C₃₁	semidirect
138. #452	C₂×C₃₁	semidirect
139. #448	C₂×C₃₁	semidirect
140. C₄×Q₁₆	C₂×C₃₁	semidirect
141. C₈×D₄	C₂×C₃₁	semidirect
177. C₃₁	#3285	direct (N × H)
195. Q₁₆	D₄×C₃₁	semidirect
256. C₂²	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#135	2
C₂²×Q₁₆×C₃₁	2
C₃₁×#548	2
C₃₁×#495	2
C₃₁×#467	2
C₃₁×#466	2
C₃₁×#464	2
C₃₁×#452	2
C₃₁×#448	2
C₄×Q₁₆×C₃₁	2
C₈×D₄×C₃₁	2
#3285	31

Subgroups — C31×#3285 #16843218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3285 #16843218