Subgroups — C₃₁×#33442 #21490201

Order 7936.

642 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×#3407	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2955	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2171	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2170	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2153	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#350	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#575	C₄	1	1	yes	characteristic
9	C₂×C₃₁×#131	C₂²	1	2	yes	—
10	C₃₁×#504	—	2	4	no	—
11	C₃₁×#504	C₂²	1	8	yes	complemented
12	C₂×C₃₁×M₃₂	—	2	4	no	—
13	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#444	C₄	1	4	yes	—
17	C₃₁×#443	C₂²	1	4	yes	—
18	C₃₁×#430	C₄	1	1	yes	characteristic
19	C₃₁×#421	C₄	1	2	yes	—
20	C₃₁×#405	C₂²	1	1	yes	characteristic
21	C₃₁×#364	C₂²	1	4	yes	complemented
22	C₃₁×#350	C₂²	1	4	yes	complemented
23	C₃₁×#350	C₂²	1	4	yes	complemented
24	C₃₁×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
26	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
27	C₃₁×#137	C₈	1	4	yes	—
28	C₃₁×AGL(1,ℤ/8)	C₈	1	4	yes	—
29	C₃₁×#135	C₂×C₄	1	4	yes	—
30	C₃₁×#131	—	2	2	no	—
31	C₃₁×#131	—	2	2	no	—
32	C₃₁×#131	C₂×C₄	1	4	yes	—
33	C₃₁×#131	C₂³	1	4	yes	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
40	C₃₁×M₃₂	—	2	2	no	—
41	C₃₁×M₃₂	—	2	8	no	—
42	C₃₁×M₃₂	D₄	1	4	yes	—
43	C₃₁×M₃₂	C₂³	1	4	yes	—
44	C₂×C₁₆×C₃₁	—	2	2	no	—
45	C₂×C₁₆×C₃₁	—	2	2	no	—
46	C₂×C₁₆×C₃₁	—	2	8	no	—
47	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#108	C₂×C₄	1	4	yes	—
51	C₃₁×#104	C₂×C₄	1	8	yes	—
52	C₃₁×#97	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	2	yes	—
54	#41×C₃₁	—	2	2	no	—
55	#41×C₃₁	—	2	2	no	—
56	#41×C₃₁	C₂×C₈	1	4	yes	—
57	#41×C₃₁	C₂²×C₄	1	4	yes	—
58	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	Q₁₆×C₃₁	C₂×C₈	1	4	yes	—
64	SD₁₆×C₃₁	C₂×C₈	1	8	yes	—
65	D₈×C₃₁	C₂×C₈	1	4	yes	—
66	M₁₆×C₃₁	—	2	2	no	—
67	M₁₆×C₃₁	—	2	4	no	—
68	M₁₆×C₃₁	—	2	4	no	—
69	M₁₆×C₃₁	C₂×C₈	1	4	yes	—
70	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₃₁	#41	1	2	yes	—
78	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
82	C₁₆×C₃₁	—	2	4	no	—
83	C₁₆×C₃₁	—	2	1	no	—
84	C₁₆×C₃₁	—	2	1	no	—
85	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
86	#33442	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	#131	1	1	yes	characteristic
89	Q₈×C₃₁	—	2	2	no	—
90	Q₈×C₃₁	C₂²×C₈	1	4	yes	—
91	D₄×C₃₁	—	2	4	no	—
92	D₄×C₃₁	—	2	4	no	—
93	D₄×C₃₁	—	2	2	no	—
94	D₄×C₃₁	C₂²×C₈	1	4	yes	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₃₁	—	2	4	no	—
104	C₈×C₃₁	C₄×D₄	1	2	yes	—
105	C₈×C₃₁	C₂²×C₈	1	4	yes	—
106	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
108	#3407	C₂×C₃₁	1	2	yes	complemented
109	#2955	C₂×C₃₁	1	1	yes	characteristic
110	#2171	C₂×C₃₁	1	4	yes	complemented
111	#2170	C₂×C₃₁	1	4	yes	complemented
112	#2153	C₂×C₃₁	1	1	yes	characteristic, complemented
113	C₂×#350	C₂×C₃₁	1	2	yes	complemented
114	#2108	C₂×C₃₁	1	1	yes	characteristic, complemented
115	C₂²×C₃₁	—	2	4	no	—
116	C₂²×C₃₁	—	4	1	no	—
117	C₂²×C₃₁	—	8	2	no	—
118	C₂²×C₃₁	C₈×D₄	1	2	yes	—
119	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
120	C₄×C₃₁	—	2	4	no	—
121	C₄×C₃₁	—	4	2	no	—
122	C₄×C₃₁	C₈×D₄	1	2	yes	—
123	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
124	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
125	#575	C₄×C₃₁	1	1	yes	characteristic
126	C₂×#131	C₂²×C₃₁	1	2	yes	—
127	#504	—	2	4	no	—
128	#504	C₂²×C₃₁	1	8	yes	complemented
129	C₂×M₃₂	—	2	4	no	—
130	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
131	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
132	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
133	#444	C₄×C₃₁	1	4	yes	—
134	#443	C₂²×C₃₁	1	4	yes	—
135	#430	C₄×C₃₁	1	1	yes	characteristic
136	#421	C₄×C₃₁	1	2	yes	—
137	#405	C₂²×C₃₁	1	1	yes	characteristic
138	#364	C₂²×C₃₁	1	4	yes	complemented
139	#350	C₂²×C₃₁	1	4	yes	complemented
140	#350	C₂²×C₃₁	1	4	yes	complemented
141	#346	C₂²×C₃₁	1	2	yes	complemented
142	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
143	C₂×C₃₁	—	2	1	no	—
144	C₂×C₃₁	—	2	2	no	—
145	C₂×C₃₁	—	4	4	no	—
146	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
147	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
148	#137	C₈×C₃₁	1	4	yes	—
149	AGL(1,ℤ/8)	C₈×C₃₁	1	4	yes	—
150	#135	C₂×C₄×C₃₁	1	4	yes	—
151	#131	—	2	2	no	—
152	#131	—	2	2	no	—
153	#131	C₂×C₄×C₃₁	1	4	yes	—
154	#131	C₂³×C₃₁	1	4	yes	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
157	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
160	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
161	M₃₂	—	2	2	no	—
162	M₃₂	—	2	8	no	—
163	M₃₂	D₄×C₃₁	1	4	yes	—
164	M₃₂	C₂³×C₃₁	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	8	no	—
168	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
169	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
171	#108	C₂×C₄×C₃₁	1	4	yes	—
172	#104	C₂×C₄×C₃₁	1	8	yes	—
173	#97	C₂×C₄×C₃₁	1	2	yes	—
174	C₄×C₈	C₂³×C₃₁	1	2	yes	—
175	C₃₁	#33442	1	1	yes	characteristic, direct product factor, Sylow
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	C₂×C₈×C₃₁	1	4	yes	—
179	#41	C₂²×C₄×C₃₁	1	4	yes	—
180	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
185	Q₁₆	C₂×C₈×C₃₁	1	4	yes	—
186	SD₁₆	C₂×C₈×C₃₁	1	8	yes	—
187	D₈	C₂×C₈×C₃₁	1	4	yes	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	C₂×C₈×C₃₁	1	4	yes	—
192	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
197	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
199	C₂×C₈	#41×C₃₁	1	2	yes	—
200	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
204	C₁₆	—	2	4	no	—
205	C₁₆	—	2	1	no	—
206	C₁₆	—	2	1	no	—
207	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	C₃₁×#131	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	C₂²×C₈×C₃₁	1	4	yes	—
212	D₄	—	2	4	no	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	D₄	C₂²×C₈×C₃₁	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
221	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
222	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
223	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₄×D₄×C₃₁	1	2	yes	—
226	C₈	C₂²×C₈×C₃₁	1	4	yes	—
227	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
228	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	C₈×D₄×C₃₁	1	2	yes	—
233	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	C₈×D₄×C₃₁	1	2	yes	—
237	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
238	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	4	no	—
242	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×C₈×D₄
prime radical	C₃₁×#575	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₁ ◁ C₂×C₃₁×M₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3407	C₂	semidirect
3. C₃₁×#2171	C₂	semidirect
4. C₃₁×#2170	C₂	semidirect
5. C₃₁×#2153	C₂	semidirect
6. C₂×C₃₁×#350	C₂	semidirect
7. C₃₁×#2108	C₂	semidirect
11. C₃₁×#504	C₂²	semidirect
21. C₃₁×#364	C₂²	semidirect
22. C₃₁×#350	C₂²	semidirect
23. C₃₁×#350	C₂²	semidirect
24. C₃₁×#346	C₂²	semidirect
25. C₄×C₁₆×C₃₁	C₂²	semidirect
86. #33442	C₃₁	direct (N × H)
108. #3407	C₂×C₃₁	semidirect
110. #2171	C₂×C₃₁	semidirect
111. #2170	C₂×C₃₁	semidirect
112. #2153	C₂×C₃₁	semidirect
113. C₂×#350	C₂×C₃₁	semidirect
114. #2108	C₂×C₃₁	semidirect
128. #504	C₂²×C₃₁	semidirect
138. #364	C₂²×C₃₁	semidirect
139. #350	C₂²×C₃₁	semidirect
140. #350	C₂²×C₃₁	semidirect
141. #346	C₂²×C₃₁	semidirect
142. C₄×C₁₆	C₂²×C₃₁	semidirect
175. C₃₁	#33442	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×#3407	2
C₃₁×#2955	2
C₃₁×#2171	2
C₃₁×#2170	2
C₃₁×#2153	2
C₂×C₃₁×#350	2
C₃₁×#2108	2
#33442	31

Subgroups — C31×#33442 #21490201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33442 #21490201