Subgroups — C₃₁×#33393 #21490152

Order 7936.

1102 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2148	C₂	1	6	yes	maximal, complemented
5	C₂×C₃₁×#362	C₂	1	6	yes	maximal, complemented
6	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
7	C₃₁×#578	—	2	4	no	—
8	C₃₁×#578	C₄	1	8	yes	—
9	C₂×C₃₁×#137	—	2	3	no	—
10	C₂×C₃₁×#135	C₂²	1	3	yes	—
11	C₂²×Q₁₆×C₃₁	C₂²	1	3	yes	—
12	C₃₁×#568	—	2	1	no	—
13	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
14	C₃₁×#504	—	2	4	no	—
15	C₂×C₃₁×M₃₂	C₂²	1	6	yes	—
16	C₃₁×#430	—	2	3	no	—
17	C₂×C₃₁×#108	C₂²	1	6	yes	—
18	C₃₁×#362	C₂²	1	12	yes	—
19	C₃₁×#361	C₂²	1	4	yes	—
20	C₃₁×2₋¹⁺⁴	—	2	8	no	—
21	C₃₁×2₋¹⁺⁴	—	4	4	no	—
22	C₂×#41×C₃₁	—	2	3	no	—
23	C₂×#41×C₃₁	—	4	3	no	—
24	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×Q₈×C₃₁	—	2	3	no	—
26	C₂²×Q₈×C₃₁	—	4	1	no	—
27	C₃₁×#137	—	2	24	no	—
28	C₃₁×#135	—	2	6	no	—
29	C₃₁×#135	C₂×C₄	1	12	yes	—
30	C₂×Q₁₆×C₃₁	—	2	3	no	—
31	C₂×Q₁₆×C₃₁	—	2	6	no	—
32	C₂×Q₁₆×C₃₁	C₂×C₄	1	12	yes	—
33	C₂×Q₁₆×C₃₁	C₂³	1	3	yes	—
34	C₂×SD₁₆×C₃₁	—	2	3	no	—
35	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×#131	—	4	4	no	—
37	C₃₁×#131	C₂×C₄	1	4	yes	—
38	C₃₁×#131	D₄	1	4	yes	—
39	C₂×M₁₆×C₃₁	—	2	6	no	—
40	C₂×M₁₆×C₃₁	D₄	1	3	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
42	C₃₁×M₃₂	—	2	6	no	—
43	C₃₁×M₃₂	C₂³	1	4	yes	—
44	C₂×C₁₆×C₃₁	—	2	6	no	—
45	C₃₁×#108	—	2	6	no	—
46	C₃₁×#108	C₂³	1	4	yes	—
47	#41×C₃₁	—	2	12	no	—
48	#41×C₃₁	—	4	24	no	—
49	#41×C₃₁	—	4	6	no	—
50	#41×C₃₁	C₂²⋊C₄	1	4	yes	—
51	#41×C₃₁	D₈	1	4	yes	—
52	C₂×Q₈×C₃₁	—	2	12	no	—
53	C₂×Q₈×C₃₁	—	2	3	no	—
54	C₂×Q₈×C₃₁	—	4	8	no	—
55	C₂×Q₈×C₃₁	—	4	6	no	—
56	C₂×Q₈×C₃₁	—	4	3	no	—
57	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
58	C₂×D₄×C₃₁	—	2	1	no	—
59	C₂×D₄×C₃₁	—	4	3	no	—
60	C₂×D₄×C₃₁	SD₁₆	1	3	yes	—
61	C₂²×C₄×C₃₁	—	4	3	no	—
62	C₂²×C₄×C₃₁	—	4	3	no	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
64	Q₁₆×C₃₁	—	2	12	no	—
65	Q₁₆×C₃₁	—	2	3	no	—
66	Q₁₆×C₃₁	C₂²×C₄	1	6	yes	—
67	SD₁₆×C₃₁	—	2	12	no	—
68	D₈×C₃₁	—	2	1	no	—
69	D₈×C₃₁	C₂²×C₄	1	2	yes	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	4	6	no	—
72	M₁₆×C₃₁	C₂²⋊C₄	1	12	yes	—
73	C₂×C₈×C₃₁	—	4	6	no	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	6	yes	—
75	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
77	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₃₁	—	2	4	no	—
79	#33393	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	4	1	no	—
81	C₂³×C₃₁	C₂×SD₁₆	1	3	yes	—
82	Q₈×C₃₁	—	2	6	no	—
83	Q₈×C₃₁	—	4	3	no	—
84	Q₈×C₃₁	—	4	12	no	—
85	Q₈×C₃₁	#102	1	4	yes	—
86	D₄×C₃₁	—	2	2	no	—
87	D₄×C₃₁	—	4	12	no	—
88	D₄×C₃₁	—	4	1	no	—
89	D₄×C₃₁	#102	1	12	yes	—
90	C₂×C₄×C₃₁	—	4	3	no	—
91	C₂×C₄×C₃₁	—	4	12	no	—
92	C₂×C₄×C₃₁	—	4	12	no	—
93	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
94	C₂×C₄×C₃₁	C₂×D₈	1	6	yes	—
95	C₂×C₄×C₃₁	C₂×SD₁₆	1	3	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₃₁	—	4	4	no	—
98	C₈×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
99	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×#578	C₂×C₃₁	1	1	yes	characteristic
102	#3407	C₂×C₃₁	1	1	yes	characteristic, complemented
103	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
104	#2148	C₂×C₃₁	1	6	yes	complemented
105	C₂×#362	C₂×C₃₁	1	6	yes	complemented
106	C₂²×C₃₁	—	4	3	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	8	2	no	—
109	C₂²×C₃₁	C₂×#102	1	6	yes	—
110	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
111	C₄×C₃₁	—	4	6	no	—
112	C₄×C₃₁	C₂×#102	1	6	yes	—
113	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
115	C₂×2₋¹⁺⁴	—	2	1	no	—
116	#578	—	2	4	no	—
117	#578	C₄×C₃₁	1	8	yes	—
118	C₂×#137	—	2	3	no	—
119	C₂×#135	C₂²×C₃₁	1	3	yes	—
120	C₂²×Q₁₆	C₂²×C₃₁	1	3	yes	—
121	#568	—	2	1	no	—
122	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
123	#504	—	2	4	no	—
124	C₂×M₃₂	C₂²×C₃₁	1	6	yes	—
125	#430	—	2	3	no	—
126	C₂×#108	C₂²×C₃₁	1	6	yes	—
127	#362	C₂²×C₃₁	1	12	yes	—
128	#361	C₂²×C₃₁	1	4	yes	—
129	C₂×C₃₁	—	2	1	no	—
130	C₂×C₃₁	—	2	6	no	—
131	C₂×C₃₁	—	8	2	no	—
132	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
133	2₋¹⁺⁴	—	2	8	no	—
134	2₋¹⁺⁴	—	4	4	no	—
135	C₂×#41	—	2	3	no	—
136	C₂×#41	—	4	3	no	—
137	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
138	C₂²×Q₈	—	2	3	no	—
139	C₂²×Q₈	—	4	1	no	—
140	#137	—	2	24	no	—
141	#135	—	2	6	no	—
142	#135	C₂×C₄×C₃₁	1	12	yes	—
143	C₂×Q₁₆	—	2	3	no	—
144	C₂×Q₁₆	—	2	6	no	—
145	C₂×Q₁₆	C₂×C₄×C₃₁	1	12	yes	—
146	C₂×Q₁₆	C₂³×C₃₁	1	3	yes	—
147	C₂×SD₁₆	—	2	3	no	—
148	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
149	#131	—	4	4	no	—
150	#131	C₂×C₄×C₃₁	1	4	yes	—
151	#131	D₄×C₃₁	1	4	yes	—
152	C₂×M₁₆	—	2	6	no	—
153	C₂×M₁₆	D₄×C₃₁	1	3	yes	—
154	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
155	M₃₂	—	2	6	no	—
156	M₃₂	C₂³×C₃₁	1	4	yes	—
157	C₂×C₁₆	—	2	6	no	—
158	#108	—	2	6	no	—
159	#108	C₂³×C₃₁	1	4	yes	—
160	C₃₁	#33393	1	1	yes	characteristic, direct product factor, Sylow
161	#41	—	2	12	no	—
162	#41	—	4	24	no	—
163	#41	—	4	6	no	—
164	#41	C₂²⋊C₄×C₃₁	1	4	yes	—
165	#41	D₈×C₃₁	1	4	yes	—
166	C₂×Q₈	—	2	12	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	4	8	no	—
169	C₂×Q₈	—	4	6	no	—
170	C₂×Q₈	—	4	3	no	—
171	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	—	4	3	no	—
174	C₂×D₄	SD₁₆×C₃₁	1	3	yes	—
175	C₂²×C₄	—	4	3	no	—
176	C₂²×C₄	—	4	3	no	—
177	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
178	Q₁₆	—	2	12	no	—
179	Q₁₆	—	2	3	no	—
180	Q₁₆	C₂²×C₄×C₃₁	1	6	yes	—
181	SD₁₆	—	2	12	no	—
182	D₈	—	2	1	no	—
183	D₈	C₂²×C₄×C₃₁	1	2	yes	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	4	6	no	—
186	M₁₆	C₂²⋊C₄×C₃₁	1	12	yes	—
187	C₂×C₈	—	4	6	no	—
188	C₂×C₈	C₂²×C₄×C₃₁	1	6	yes	—
189	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
190	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
191	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
192	C₁₆	—	2	4	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×SD₁₆×C₃₁	1	3	yes	—
195	Q₈	—	2	6	no	—
196	Q₈	—	4	12	no	—
197	Q₈	—	4	3	no	—
198	Q₈	C₃₁×#102	1	4	yes	—
199	D₄	—	2	2	no	—
200	D₄	—	4	12	no	—
201	D₄	—	4	1	no	—
202	D₄	C₃₁×#102	1	12	yes	—
203	C₂×C₄	—	4	12	no	—
204	C₂×C₄	—	4	3	no	—
205	C₂×C₄	—	4	12	no	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
207	C₂×C₄	C₂×D₈×C₃₁	1	6	yes	—
208	C₂×C₄	C₂×SD₁₆×C₃₁	1	3	yes	—
209	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
210	C₈	—	4	4	no	—
211	C₈	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
212	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
213	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
214	C₂²	—	4	3	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	8	2	no	—
217	C₂²	C₂×C₃₁×#102	1	6	yes	—
218	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
219	C₄	—	4	6	no	—
220	C₄	C₂×C₃₁×#102	1	6	yes	—
221	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
222	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	6	no	—
225	C₂	—	8	2	no	—
226	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂²×#102
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#578	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3407	C₂	semidirect
3. C₃₁×#2916	C₂	semidirect
4. C₃₁×#2148	C₂	semidirect
5. C₂×C₃₁×#362	C₂	semidirect
79. #33393	C₃₁	direct (N × H)
102. #3407	C₂×C₃₁	semidirect
103. #2916	C₂×C₃₁	semidirect
104. #2148	C₂×C₃₁	semidirect
105. C₂×#362	C₂×C₃₁	semidirect
160. C₃₁	#33393	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#578	2
C₃₁×#3407	2
C₃₁×#2916	2
C₃₁×#2148	2
C₂×C₃₁×#362	2
#33393	31

Subgroups — C31×#33393 #21490152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33393 #21490152