Subgroups — C₃₁×#33658 #21490417

Order 7936.

534 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3059	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2239	C₂	1	8	yes	maximal, complemented
4	C₃₁×#2155	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#362	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2115	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#568	—	2	1	no	—
9	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
10	C₃₁×#536	—	2	1	no	—
11	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
12	C₃₁×#470	C₂²	1	4	yes	—
13	C₃₁×#467	—	2	2	no	—
14	C₃₁×#430	—	2	1	no	—
15	C₃₁×#430	C₂²	1	4	yes	—
16	C₂×C₃₁×#108	—	2	2	no	—
17	C₂×C₃₁×#108	C₂²	1	2	yes	—
18	C₃₁×#427	C₂²	1	1	yes	characteristic
19	C₃₁×#419	—	2	1	no	—
20	C₃₁×#365	C₂²	1	8	yes	—
21	C₃₁×#362	C₂²	1	8	yes	—
22	C₃₁×#349	C₂²	1	4	yes	—
23	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×Q₈×C₃₁	—	2	1	no	—
25	C₂×Q₁₆×C₃₁	—	2	1	no	—
26	C₂×Q₁₆×C₃₁	—	2	2	no	—
27	C₂×Q₁₆×C₃₁	—	2	2	no	—
28	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
29	C₃₁×#131	—	2	2	no	—
30	C₃₁×#131	—	2	2	no	—
31	C₃₁×#131	—	4	4	no	—
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂×M₁₆×C₃₁	—	2	2	no	—
34	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×C₄⋊Q₈	—	4	1	no	—
39	C₃₁×#124	—	4	1	no	—
40	C₃₁×#123	—	4	2	no	—
41	C₃₁×C₂²⋊Q₈	—	2	4	no	—
42	C₃₁×#117	—	2	1	no	—
43	C₃₁×M₃₂	—	2	2	no	—
44	C₃₁×M₃₂	C₂³	1	4	yes	—
45	C₂×C₁₆×C₃₁	—	2	2	no	—
46	C₃₁×#108	—	2	2	no	—
47	C₃₁×#108	—	2	4	no	—
48	C₃₁×#108	C₂³	1	4	yes	—
49	C₃₁×#107	—	2	1	no	—
50	C₃₁×#106	C₂³	1	2	yes	—
51	C₃₁×#103	—	2	4	no	—
52	#41×C₃₁	—	2	2	no	—
53	#41×C₃₁	—	2	2	no	—
54	C₂×Q₈×C₃₁	—	2	2	no	—
55	C₂×Q₈×C₃₁	—	4	1	no	—
56	C₂×Q₈×C₃₁	—	4	2	no	—
57	C₂×Q₈×C₃₁	—	4	2	no	—
58	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
59	C₂×D₄×C₃₁	SD₁₆	1	2	yes	—
60	C₂×D₄×C₃₁	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	Q₁₆×C₃₁	—	2	4	no	—
65	Q₁₆×C₃₁	—	4	1	no	—
66	Q₁₆×C₃₁	—	4	1	no	—
67	M₁₆×C₃₁	—	2	2	no	—
68	M₁₆×C₃₁	—	2	4	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	4	4	no	—
71	M₁₆×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	4	4	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	#41	1	2	yes	—
79	C₂×C₈×C₃₁	#41	1	2	yes	—
80	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	4	4	no	—
83	C₂²⋊C₄×C₃₁	—	4	1	no	—
84	C₂²⋊C₄×C₃₁	—	4	4	no	—
85	C₄²×C₃₁	—	4	1	no	—
86	C₁₆×C₃₁	—	2	4	no	—
87	#33658	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
89	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
90	Q₈×C₃₁	—	2	2	no	—
91	Q₈×C₃₁	—	4	2	no	—
92	Q₈×C₃₁	—	4	2	no	—
93	Q₈×C₃₁	—	8	2	no	—
94	D₄×C₃₁	—	2	4	no	—
95	D₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	8	2	no	—
101	C₂×C₄×C₃₁	#123	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	—	4	4	no	—
108	C₈×C₃₁	#123	1	1	yes	characteristic
109	C₈×C₃₁	#123	1	1	yes	characteristic
110	C₈×C₃₁	C₂×#41	1	2	yes	—
111	#3059	C₂×C₃₁	1	1	yes	characteristic
112	#2916	C₂×C₃₁	1	1	yes	characteristic
113	#2239	C₂×C₃₁	1	8	yes	complemented
114	#2155	C₂×C₃₁	1	2	yes	complemented
115	C₂×#362	C₂×C₃₁	1	2	yes	complemented
116	#2115	C₂×C₃₁	1	1	yes	characteristic
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	4	1	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	#481	1	2	yes	—
121	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
122	C₄×C₃₁	—	2	2	no	—
123	C₄×C₃₁	—	8	2	no	—
124	C₄×C₃₁	—	8	2	no	—
125	C₄×C₃₁	#481	1	2	yes	—
126	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
127	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
129	#568	—	2	1	no	—
130	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
131	#536	—	2	1	no	—
132	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
133	#470	C₂²×C₃₁	1	4	yes	—
134	#467	—	2	2	no	—
135	#430	—	2	1	no	—
136	#430	C₂²×C₃₁	1	4	yes	—
137	C₂×#108	—	2	2	no	—
138	C₂×#108	C₂²×C₃₁	1	2	yes	—
139	#427	C₂²×C₃₁	1	1	yes	characteristic
140	#419	—	2	1	no	—
141	#365	C₂²×C₃₁	1	8	yes	—
142	#362	C₂²×C₃₁	1	8	yes	—
143	#349	C₂²×C₃₁	1	4	yes	—
144	C₂×C₃₁	—	2	1	no	—
145	C₂×C₃₁	—	2	2	no	—
146	C₂×C₃₁	—	4	2	no	—
147	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
148	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
149	C₂²×Q₈	—	2	1	no	—
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
154	#131	—	2	2	no	—
155	#131	—	2	2	no	—
156	#131	—	4	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
160	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
161	C₂²×C₈	D₄×C₃₁	1	2	yes	—
162	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
163	C₄⋊Q₈	—	4	1	no	—
164	#124	—	4	1	no	—
165	#123	—	4	2	no	—
166	C₂²⋊Q₈	—	2	4	no	—
167	#117	—	2	1	no	—
168	M₃₂	—	2	2	no	—
169	M₃₂	C₂³×C₃₁	1	4	yes	—
170	C₂×C₁₆	—	2	2	no	—
171	#108	—	2	2	no	—
172	#108	—	2	4	no	—
173	#108	C₂³×C₃₁	1	4	yes	—
174	#107	—	2	1	no	—
175	#106	C₂³×C₃₁	1	2	yes	—
176	#103	—	2	4	no	—
177	C₃₁	#33658	1	1	yes	characteristic, direct product factor, Sylow
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
185	C₂×D₄	SD₁₆×C₃₁	1	2	yes	—
186	C₂×D₄	SD₁₆×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
190	Q₁₆	—	2	4	no	—
191	Q₁₆	—	4	1	no	—
192	Q₁₆	—	4	1	no	—
193	M₁₆	—	2	2	no	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	#41×C₃₁	1	2	yes	—
205	C₂×C₈	#41×C₃₁	1	2	yes	—
206	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	—	4	1	no	—
212	C₁₆	—	2	4	no	—
213	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
214	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	8	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₃₁×#123	1	2	yes	—
227	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
228	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
230	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	4	4	no	—
233	C₈	C₃₁×#123	1	1	yes	characteristic
234	C₈	C₃₁×#123	1	1	yes	characteristic
235	C₈	C₂×#41×C₃₁	1	2	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	C₃₁×#481	1	2	yes	—
240	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	C₃₁×#481	1	2	yes	—
245	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
246	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×C₃₁×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2239	C₂	semidirect
4. C₃₁×#2155	C₂	semidirect
5. C₂×C₃₁×#362	C₂	semidirect
87. #33658	C₃₁	direct (N × H)
113. #2239	C₂×C₃₁	semidirect
114. #2155	C₂×C₃₁	semidirect
115. C₂×#362	C₂×C₃₁	semidirect
177. C₃₁	#33658	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×#3059	2
C₃₁×#2916	2
C₃₁×#2239	2
C₃₁×#2155	2
C₂×C₃₁×#362	2
C₃₁×#2115	2
#33658	31

Subgroups — C31×#33658 #21490417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33658 #21490417