Subgroups — C₃₁×#3400 #16843333

Order 3968.

362 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#559	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#497	C₂	1	2	yes	maximal, complemented
4	C₃₁×#492	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#464	C₂	1	2	yes	maximal, complemented
6	C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#461	C₂	1	2	yes	maximal, complemented
8	C₃₁×#460	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#440	C₂	1	2	yes	maximal, complemented
11	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
17	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₃₁×#125	—	2	1	no	—
21	C₃₁×#125	—	2	2	no	—
22	C₃₁×#124	—	2	1	no	—
23	C₃₁×#124	C₂²	1	2	yes	—
24	C₃₁×#123	—	2	2	no	—
25	C₄×Q₈×C₃₁	—	2	1	no	—
26	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₃₁×#107	C₂²	1	1	yes	characteristic
33	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
35	C₃₁×#103	C₂²	1	1	yes	characteristic
36	C₃₁×#103	C₂²	1	1	yes	characteristic
37	C₃₁×#103	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	—	2	1	no	—
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	Q₁₆×C₃₁	—	2	2	no	—
50	Q₁₆×C₃₁	—	4	2	no	—
51	SD₁₆×C₃₁	—	2	4	no	—
52	SD₁₆×C₃₁	—	4	2	no	—
53	SD₁₆×C₃₁	—	4	2	no	—
54	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	2	1	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	—	2	1	no	—
74	C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₃₁	—	2	2	no	—
77	Q₈×C₃₁	—	2	2	no	—
78	Q₈×C₃₁	—	2	2	no	—
79	Q₈×C₃₁	—	2	1	no	—
80	Q₈×C₃₁	—	2	1	no	—
81	Q₈×C₃₁	—	2	1	no	—
82	Q₈×C₃₁	—	2	2	no	—
83	Q₈×C₃₁	—	2	1	no	—
84	D₄×C₃₁	—	2	2	no	—
85	D₄×C₃₁	—	2	2	no	—
86	D₄×C₃₁	—	4	2	no	—
87	D₄×C₃₁	—	8	1	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	8	2	no	—
94	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	—	2	2	no	—
101	C₈×C₃₁	#41	1	2	yes	—
102	#3400	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₃₁	—	4	2	no	—
104	C₂²×C₃₁	—	4	2	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	4	1	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	—	4	1	no	—
113	C₄×C₃₁	—	4	1	no	—
114	C₄×C₃₁	—	4	1	no	—
115	C₄×C₃₁	#137	1	2	yes	—
116	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
118	#559	C₂×C₃₁	1	1	yes	characteristic
119	#557	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#497	C₂×C₃₁	1	2	yes	complemented
121	#492	C₂×C₃₁	1	1	yes	characteristic
122	#464	C₂×C₃₁	1	2	yes	complemented
123	#462	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#461	C₂×C₃₁	1	2	yes	complemented
125	#460	C₂×C₃₁	1	1	yes	characteristic
126	#446	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#440	C₂×C₃₁	1	2	yes	complemented
128	C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₂×C₃₁	—	8	2	no	—
130	C₂×C₃₁	#550	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
132	C₂×C₃₁	#577	1	1	yes	characteristic
133	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
135	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
136	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
137	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
138	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
139	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
140	#127	C₂²×C₃₁	1	1	yes	characteristic
141	#125	—	2	1	no	—
142	#125	—	2	2	no	—
143	#124	—	2	1	no	—
144	#124	C₂²×C₃₁	1	2	yes	—
145	#123	—	2	2	no	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
149	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	C₂²×C₃₁	1	2	yes	—
153	#107	C₂²×C₃₁	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
156	#103	C₂²×C₃₁	1	1	yes	characteristic
157	#103	C₂²×C₃₁	1	1	yes	characteristic
158	#103	C₂²×C₃₁	1	2	yes	—
159	#102	C₂²×C₃₁	1	2	yes	—
160	#97	C₂²×C₃₁	1	2	yes	—
161	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₃₁	#3400	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
166	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	C₂³×C₃₁	1	2	yes	—
170	C₂²×C₄	—	2	2	no	—
171	Q₁₆	—	2	2	no	—
172	Q₁₆	—	4	2	no	—
173	SD₁₆	—	2	4	no	—
174	SD₁₆	—	4	2	no	—
175	SD₁₆	—	4	2	no	—
176	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₃₁	1	2	yes	—
178	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
187	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
188	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₃₁	1	1	yes	characteristic
197	C₄²	C₂³×C₃₁	1	2	yes	—
198	C₂³	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	2	1	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	#41×C₃₁	1	2	yes	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	C₃₁×#137	1	2	yes	—
237	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
238	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂	—	8	2	no	—
240	C₂	C₃₁×#550	1	1	yes	characteristic
241	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
242	C₂	C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#557	C₂	semidirect
3. C₃₁×#497	C₂	semidirect
5. C₃₁×#464	C₂	semidirect
6. C₃₁×#462	C₂	semidirect
7. C₃₁×#461	C₂	semidirect
9. C₃₁×#446	C₂	semidirect
10. C₃₁×#440	C₂	semidirect
11. C₄×Q₁₆×C₃₁	C₂	semidirect
102. #3400	C₃₁	direct (N × H)
119. #557	C₂×C₃₁	semidirect
120. #497	C₂×C₃₁	semidirect
122. #464	C₂×C₃₁	semidirect
123. #462	C₂×C₃₁	semidirect
124. #461	C₂×C₃₁	semidirect
126. #446	C₂×C₃₁	semidirect
127. #440	C₂×C₃₁	semidirect
128. C₄×Q₁₆	C₂×C₃₁	semidirect
162. C₃₁	#3400	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#559	2
C₃₁×#557	2
C₃₁×#497	2
C₃₁×#492	2
C₃₁×#464	2
C₃₁×#462	2
C₃₁×#461	2
C₃₁×#460	2
C₃₁×#446	2
C₃₁×#440	2
C₄×Q₁₆×C₃₁	2
#3400	31

Subgroups — C31×#3400 #16843333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3400 #16843333