Subgroups — C₃₁×#33849 #21490608

Order 7936.

676 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#508	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3283	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2216	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2208	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2189	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	4	yes	—
10	C₃₁×#547	—	2	2	no	—
11	C₃₁×#508	—	2	8	no	—
12	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#494	C₂²	1	2	yes	—
18	C₃₁×#467	C₂²	1	4	yes	—
19	C₃₁×#461	—	2	2	no	—
20	C₃₁×#452	—	2	4	no	—
21	C₄×D₈×C₃₁	C₂²	1	2	yes	—
22	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#366	C₂²	1	1	yes	characteristic
24	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₃₁×#358	C₂²	1	4	yes	—
26	C₃₁×#357	C₂²	1	2	yes	—
27	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×#41×C₃₁	—	2	4	no	—
30	C₃₁×#135	—	2	8	no	—
31	C₃₁×#135	—	4	4	no	—
32	C₂×Q₁₆×C₃₁	—	2	2	no	—
33	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
34	C₂×SD₁₆×C₃₁	—	2	4	no	—
35	C₂×D₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×#123	—	4	4	no	—
39	C₃₁×C₂²⋊Q₈	—	2	4	no	—
40	C₃₁×#121	—	4	2	no	—
41	C₄×D₄×C₃₁	—	2	2	no	—
42	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
44	C₃₁×Q₃₂	—	2	4	no	—
45	C₃₁×Q₃₂	—	4	2	no	—
46	C₃₁×SD₃₂	—	2	8	no	—
47	C₃₁×D₁₆	D₄	1	4	yes	complemented
48	C₂×C₁₆×C₃₁	—	2	4	no	—
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	C₂³	1	2	yes	—
53	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₃₁×#103	—	2	4	no	—
55	C₃₁×#102	—	2	2	no	—
56	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
57	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	#41×C₃₁	—	4	8	no	—
59	#41×C₃₁	—	8	4	no	—
60	C₂×Q₈×C₃₁	—	2	4	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	4	4	no	—
63	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	—	4	2	no	—
65	C₂²×C₄×C₃₁	—	4	4	no	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	Q₁₆×C₃₁	—	2	4	no	—
68	Q₁₆×C₃₁	—	2	4	no	—
69	Q₁₆×C₃₁	—	8	2	no	—
70	SD₁₆×C₃₁	—	4	8	no	—
71	D₈×C₃₁	—	2	2	no	—
72	D₈×C₃₁	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	4	4	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
83	C₂²⋊C₄×C₃₁	—	4	4	no	—
84	C₂²⋊C₄×C₃₁	—	4	2	no	—
85	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
86	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₃₁	—	2	2	no	—
88	C₁₆×C₃₁	—	2	1	no	—
89	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
90	#33849	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	4	2	no	—
92	C₂³×C₃₁	C₂×D₈	1	2	yes	—
93	Q₈×C₃₁	—	4	4	no	—
94	Q₈×C₃₁	—	4	4	no	—
95	D₄×C₃₁	—	2	4	no	—
96	D₄×C₃₁	—	2	2	no	—
97	D₄×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	8	8	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	8	8	no	—
104	C₂×C₄×C₃₁	—	8	4	no	—
105	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₃₁	—	2	2	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×#508	C₂×C₃₁	1	2	yes	complemented
114	#3283	C₂×C₃₁	1	2	yes	complemented
115	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
116	#2216	C₂×C₃₁	1	2	yes	complemented
117	#2208	C₂×C₃₁	1	2	yes	complemented
118	#2189	C₂×C₃₁	1	4	yes	complemented
119	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
120	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	2	2	no	—
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	4	4	no	—
125	C₂²×C₃₁	—	8	2	no	—
126	C₂²×C₃₁	—	8	2	no	—
127	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	8	2	no	—
131	C₄×C₃₁	—	8	4	no	—
132	C₄×C₃₁	#508	1	2	yes	complemented
133	C₄×C₃₁	D₄²	1	1	yes	characteristic
134	C₄×C₃₁	#576	1	1	yes	characteristic
135	C₂×#135	C₂²×C₃₁	1	4	yes	—
136	#547	—	2	2	no	—
137	#508	—	2	8	no	—
138	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
139	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
140	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
141	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
143	#494	C₂²×C₃₁	1	2	yes	—
144	#467	C₂²×C₃₁	1	4	yes	—
145	#461	—	2	2	no	—
146	#452	—	2	4	no	—
147	C₄×D₈	C₂²×C₃₁	1	2	yes	—
148	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
149	#366	C₂²×C₃₁	1	1	yes	characteristic
150	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
151	#358	C₂²×C₃₁	1	4	yes	—
152	#357	C₂²×C₃₁	1	2	yes	—
153	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
154	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₃₁	—	4	2	no	—
156	C₂×C₃₁	—	8	4	no	—
157	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
158	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
159	C₂×C₃₁	#3417	1	1	yes	characteristic
160	C₂×#41	—	2	4	no	—
161	#135	—	2	8	no	—
162	#135	—	4	4	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
165	C₂×SD₁₆	—	2	4	no	—
166	C₂×D₈	C₂³×C₃₁	1	2	yes	—
167	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
168	C₄⋊Q₈	—	2	2	no	—
169	#123	—	4	4	no	—
170	C₂²⋊Q₈	—	2	4	no	—
171	#121	—	4	2	no	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	4	2	no	—
175	Q₃₂	—	2	4	no	—
176	Q₃₂	—	4	2	no	—
177	SD₃₂	—	2	8	no	—
178	D₁₆	D₄×C₃₁	1	4	yes	complemented
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
181	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
182	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
183	#107	C₂³×C₃₁	1	2	yes	—
184	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
185	#103	—	2	4	no	—
186	#102	—	2	2	no	—
187	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
188	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
189	C₃₁	#33849	1	1	yes	characteristic, direct product factor, Sylow
190	#41	—	4	8	no	—
191	#41	—	8	4	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	8	2	no	—
202	SD₁₆	—	4	8	no	—
203	D₈	—	2	2	no	—
204	D₈	C₂×D₄×C₃₁	1	4	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
209	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
218	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₁₆	—	2	2	no	—
220	C₁₆	—	2	1	no	—
221	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂×D₈×C₃₁	1	2	yes	—
224	Q₈	—	4	4	no	—
225	Q₈	—	4	4	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	8	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	1	no	—
242	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
243	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	4	no	—
255	C₄	C₃₁×#508	1	2	yes	complemented
256	C₄	D₄²×C₃₁	1	1	yes	characteristic
257	C₄	C₃₁×#576	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	8	4	no	—
260	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
261	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
262	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#508	C₂	semidirect
2. C₃₁×#3283	C₂	semidirect
3. C₃₁×#2247	C₂	semidirect
4. C₃₁×#2216	C₂	semidirect
5. C₃₁×#2208	C₂	semidirect
6. C₃₁×#2189	C₂	semidirect
7. C₄×C₃₁×D₁₆	C₂	semidirect
8. D₄×C₁₆×C₃₁	C₂	semidirect
47. C₃₁×D₁₆	D₄	semidirect
90. #33849	C₃₁	direct (N × H)
113. C₂×#508	C₂×C₃₁	semidirect
114. #3283	C₂×C₃₁	semidirect
115. #2247	C₂×C₃₁	semidirect
116. #2216	C₂×C₃₁	semidirect
117. #2208	C₂×C₃₁	semidirect
118. #2189	C₂×C₃₁	semidirect
119. C₄×D₁₆	C₂×C₃₁	semidirect
120. D₄×C₁₆	C₂×C₃₁	semidirect
132. C₄×C₃₁	#508	semidirect
178. D₁₆	D₄×C₃₁	semidirect
189. C₃₁	#33849	direct (N × H)
255. C₄	C₃₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#508	2
C₃₁×#3283	2
C₃₁×#2247	2
C₃₁×#2216	2
C₃₁×#2208	2
C₃₁×#2189	2
C₄×C₃₁×D₁₆	2
D₄×C₁₆×C₃₁	2
#33849	31

Subgroups — C31×#33849 #21490608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33849 #21490608