Subgroups — C₃₁×#3221 #16843154

Order 3968.

388 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#463	C₂	1	2	yes	maximal, complemented
8	C₃₁×#461	C₂	1	2	yes	maximal, complemented
9	C₃₁×#449	C₂	1	2	yes	maximal, complemented
10	C₃₁×#448	C₂	1	2	yes	maximal, complemented
11	C₃₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
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	C₂×D₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃₁×#127	C₂²	1	1	yes	characteristic
21	C₃₁×#125	C₂²	1	1	yes	characteristic
22	C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₃₁×#124	C₂²	1	1	yes	characteristic
24	C₃₁×#123	—	2	2	no	—
25	C₃₁×#123	—	2	2	no	—
26	C₃₁×C₂²⋊Q₈	—	2	2	no	—
27	C₃₁×#121	—	2	2	no	—
28	C₃₁×#121	C₂²	1	2	yes	—
29	C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
32	C₃₁×#103	C₂²	1	2	yes	—
33	C₃₁×#103	C₂²	1	2	yes	—
34	C₃₁×#103	C₂²	1	2	yes	—
35	C₃₁×#102	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₃₁×#97	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	#41×C₃₁	—	4	2	no	—
40	C₂×Q₈×C₃₁	—	2	1	no	—
41	C₂×Q₈×C₃₁	—	4	2	no	—
42	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	—	2	1	no	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	SD₁₆×C₃₁	—	4	2	no	—
55	SD₁₆×C₃₁	—	4	2	no	—
56	SD₁₆×C₃₁	—	4	2	no	—
57	D₈×C₃₁	—	4	2	no	—
58	C₂×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂×C₈×C₃₁	C₂³	1	2	yes	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₃₁	—	2	1	no	—
73	C₂³×C₃₁	—	2	2	no	—
74	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₃₁	—	2	2	no	—
76	Q₈×C₃₁	—	2	2	no	—
77	Q₈×C₃₁	—	2	2	no	—
78	Q₈×C₃₁	—	4	2	no	—
79	D₄×C₃₁	—	2	2	no	—
80	D₄×C₃₁	—	2	2	no	—
81	D₄×C₃₁	—	2	2	no	—
82	D₄×C₃₁	—	4	2	no	—
83	D₄×C₃₁	—	8	2	no	—
84	D₄×C₃₁	—	8	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	8	2	no	—
93	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	2	2	no	—
99	#3221	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	4	1	no	—
103	C₂²×C₃₁	—	4	2	no	—
104	C₂²×C₃₁	—	4	2	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	—	4	1	no	—
112	C₄×C₃₁	—	4	1	no	—
113	C₄×C₃₁	—	4	1	no	—
114	C₄×C₃₁	—	4	2	no	—
115	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
116	#540	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#536	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#496	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#492	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#491	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#488	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#463	C₂×C₃₁	1	2	yes	complemented
123	#461	C₂×C₃₁	1	2	yes	complemented
124	#449	C₂×C₃₁	1	2	yes	complemented
125	#448	C₂×C₃₁	1	2	yes	complemented
126	#433	C₂×C₃₁	1	1	yes	characteristic, complemented
127	C₂×C₃₁	—	4	1	no	—
128	C₂×C₃₁	—	8	1	no	—
129	C₂×C₃₁	—	8	2	no	—
130	C₂×C₃₁	#535	1	1	yes	characteristic
131	C₂×C₃₁	#577	1	2	yes	—
132	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
133	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
135	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
136	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
137	C₂×D₈	C₂²×C₃₁	1	2	yes	—
138	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
139	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
140	#127	C₂²×C₃₁	1	1	yes	characteristic
141	#125	C₂²×C₃₁	1	1	yes	characteristic
142	#124	C₂²×C₃₁	1	1	yes	characteristic
143	#124	C₂²×C₃₁	1	1	yes	characteristic
144	#123	—	2	2	no	—
145	#123	—	2	2	no	—
146	C₂²⋊Q₈	—	2	2	no	—
147	#121	—	2	2	no	—
148	#121	C₂²×C₃₁	1	2	yes	—
149	C₄×D₄	C₂²×C₃₁	1	2	yes	—
150	#117	C₂²×C₃₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
152	#103	C₂²×C₃₁	1	2	yes	—
153	#103	C₂²×C₃₁	1	2	yes	—
154	#103	C₂²×C₃₁	1	2	yes	—
155	#102	C₂²×C₃₁	1	2	yes	—
156	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
157	#97	C₂²×C₃₁	1	1	yes	characteristic
158	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₃₁	#3221	1	1	yes	characteristic, direct product factor, Sylow
160	#41	—	4	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₃₁	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
175	SD₁₆	—	4	2	no	—
176	SD₁₆	—	4	2	no	—
177	SD₁₆	—	4	2	no	—
178	D₈	—	4	2	no	—
179	C₂×C₈	C₂³×C₃₁	1	2	yes	—
180	C₂×C₈	C₂³×C₃₁	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
185	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
191	C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	C₄²	C₂³×C₃₁	1	1	yes	characteristic
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	4	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	2	no	—
205	D₄	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	2	no	—
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	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
236	C₂	—	4	1	no	—
237	C₂	—	8	1	no	—
238	C₂	—	8	2	no	—
239	C₂	C₃₁×#535	1	1	yes	characteristic
240	C₂	C₃₁×#577	1	2	yes	—

	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 ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#540	C₂	semidirect
2. C₃₁×#536	C₂	semidirect
3. C₃₁×#496	C₂	semidirect
4. C₃₁×#492	C₂	semidirect
5. C₃₁×#491	C₂	semidirect
6. C₃₁×#488	C₂	semidirect
7. C₃₁×#463	C₂	semidirect
8. C₃₁×#461	C₂	semidirect
9. C₃₁×#449	C₂	semidirect
10. C₃₁×#448	C₂	semidirect
11. C₃₁×#433	C₂	semidirect
99. #3221	C₃₁	direct (N × H)
116. #540	C₂×C₃₁	semidirect
117. #536	C₂×C₃₁	semidirect
118. #496	C₂×C₃₁	semidirect
119. #492	C₂×C₃₁	semidirect
120. #491	C₂×C₃₁	semidirect
121. #488	C₂×C₃₁	semidirect
122. #463	C₂×C₃₁	semidirect
123. #461	C₂×C₃₁	semidirect
124. #449	C₂×C₃₁	semidirect
125. #448	C₂×C₃₁	semidirect
126. #433	C₂×C₃₁	semidirect
159. C₃₁	#3221	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×#540	2
C₃₁×#536	2
C₃₁×#496	2
C₃₁×#492	2
C₃₁×#491	2
C₃₁×#488	2
C₃₁×#463	2
C₃₁×#461	2
C₃₁×#449	2
C₃₁×#448	2
C₃₁×#433	2
#3221	31

Subgroups — C31×#3221 #16843154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3221 #16843154