Subgroups — D₃₁×#430 #16854315

Order 3968.

460 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₃₁	C₂	1	2	yes	maximal
2	C₂×M₁₆×D₃₁	C₂	1	1	yes	maximal, characteristic
3	#14521098	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521097	C₂	1	2	yes	maximal, complemented
5	#14521096	C₂	1	4	yes	maximal, complemented
6	#108×D₃₁	C₂	1	4	yes	maximal, complemented
7	C₃₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×#92474	C₂²	1	2	yes	—
12	C₂×#92474	C₂²	1	1	yes	characteristic
13	M₁₆×D₃₁	—	2	2	no	—
14	M₁₆×D₃₁	—	2	2	no	—
15	M₁₆×D₃₁	C₄	1	4	yes	—
16	M₁₆×D₃₁	C₄	1	4	yes	—
17	M₁₆×D₃₁	C₂²	1	4	yes	—
18	M₁₆×D₃₁	C₂²	1	4	yes	—
19	C₂×#92469	—	2	2	no	—
20	C₂×#92469	C₂²	1	2	yes	—
21	C₂×C₈×D₃₁	—	2	2	no	—
22	C₂×C₈×D₃₁	C₂²	1	2	yes	—
23	C₃₁×#108	C₂²	1	4	yes	complemented
24	#11758410	C₂²	1	8	yes	complemented
25	#11758407	C₂²	1	4	yes	complemented
26	C₂³×D₃₁	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃₁	Q₈	1	1	yes	characteristic
29	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₃₁	D₄	1	2	yes	—
32	C₂×C₄×D₃₁	D₄	1	2	yes	—
33	C₂×C₄×D₃₁	Q₈	1	1	yes	characteristic
34	C₂×C₄×D₃₁	Q₈	1	1	yes	characteristic
35	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₃₁	—	2	2	no	—
38	M₁₆×C₃₁	C₂×C₄	1	4	yes	—
39	M₁₆×C₃₁	C₂³	1	4	yes	—
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂×C₈×C₃₁	C₂³	1	2	yes	—
42	#92474	—	2	2	no	—
43	#92474	C₂×C₄	1	4	yes	—
44	#92474	C₂³	1	4	yes	—
45	C₂×C₃₁⋊C₈	—	2	2	no	—
46	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
47	#92469	—	2	4	no	—
48	#92469	—	2	4	no	—
49	#92469	C₂×C₄	1	8	yes	—
50	C₈×D₃₁	—	2	4	no	—
51	C₈×D₃₁	—	2	4	no	—
52	C₈×D₃₁	C₂×C₄	1	4	yes	—
53	C₈×D₃₁	C₂×C₄	1	4	yes	—
54	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×D₃₁	—	4	2	no	—
56	C₂²×D₃₁	C₄⋊C₄	1	2	yes	—
57	C₂²×D₃₁	C₄⋊C₄	1	2	yes	—
58	C₂²×D₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×D₃₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
65	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
66	C₂×Dic₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂×Dic₃₁	C₂×Q₈	1	1	yes	characteristic
68	C₄×D₃₁	C₄⋊C₄	1	2	yes	—
69	C₄×D₃₁	C₄⋊C₄	1	2	yes	—
70	C₄×D₃₁	C₄⋊C₄	1	2	yes	—
71	C₄×D₃₁	C₄⋊C₄	1	2	yes	—
72	C₄×D₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₄×D₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₄×D₃₁	C₂²×C₄	1	2	yes	—
75	C₄×D₃₁	C₂×D₄	1	1	yes	characteristic
76	C₄×D₃₁	C₂×D₄	1	2	yes	—
77	C₄×D₃₁	C₂×D₄	1	1	yes	characteristic
78	C₈×C₃₁	—	2	4	no	—
79	C₈×C₃₁	C₂²×C₄	1	4	yes	—
80	C₃₁⋊C₈	—	2	4	no	—
81	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
82	C₂×#430	—	31	1	no	maximal, Sylow
83	C₂²×C₃₁	—	4	1	no	—
84	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
86	C₂×D₃₁	—	2	4	no	—
87	C₂×D₃₁	—	2	2	no	—
88	C₂×D₃₁	—	4	2	no	—
89	C₂×D₃₁	—	4	1	no	—
90	C₂×D₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×D₃₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×D₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	Dic₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	Dic₃₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂²×M₁₆	—	31	1	no	—
100	C₂²×M₁₆	—	31	2	no	—
101	#430	—	31	2	no	—
102	#430	—	31	1	no	—
103	#430	—	31	4	no	—
104	#430	D₃₁	1	1	yes	characteristic, direct product factor
105	C₂×#108	—	31	4	no	—
106	C₂×C₃₁	—	2	2	no	—
107	C₂×C₃₁	—	2	1	no	—
108	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	D₃₁	—	2	2	no	—
110	D₃₁	—	2	1	no	—
111	D₃₁	#430	1	2	yes	direct product factor
112	C₂³×C₄	—	31	1	no	—
113	C₂×M₁₆	—	31	1	no	—
114	C₂×M₁₆	—	31	4	no	—
115	C₂×M₁₆	—	31	4	no	—
116	C₂×M₁₆	—	31	2	no	—
117	C₂×M₁₆	—	31	2	no	—
118	C₂×M₁₆	—	31	4	no	—
119	C₂×M₁₆	—	31	4	no	—
120	C₂×M₁₆	—	62	2	no	—
121	C₂×M₁₆	—	62	2	no	—
122	C₂×M₁₆	—	62	2	no	—
123	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
124	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
125	C₂²×C₈	—	31	2	no	—
126	C₂²×C₈	—	62	2	no	—
127	#108	—	31	8	no	—
128	#108	—	31	4	no	—
129	#108	C₂×D₃₁	1	4	yes	complemented
130	C₃₁	C₂×#430	1	1	yes	characteristic, complemented, Sylow
131	C₂⁴	—	31	1	no	—
132	C₂²×C₄	—	31	1	no	—
133	C₂²×C₄	—	31	2	no	—
134	C₂²×C₄	—	31	2	no	—
135	C₂²×C₄	—	31	2	no	—
136	C₂²×C₄	—	31	2	no	—
137	C₂²×C₄	—	31	1	no	—
138	C₂²×C₄	—	31	1	no	—
139	C₂²×C₄	—	31	1	no	—
140	C₂²×C₄	—	31	1	no	—
141	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	31	4	no	—
144	M₁₆	—	31	8	no	—
145	M₁₆	—	31	4	no	—
146	M₁₆	—	62	4	no	—
147	M₁₆	—	62	4	no	—
148	M₁₆	—	62	2	no	—
149	M₁₆	C₄×D₃₁	1	4	yes	—
150	M₁₆	C₂²×D₃₁	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	31	2	no	—
153	C₂×C₈	—	31	4	no	—
154	C₂×C₈	—	31	4	no	—
155	C₂×C₈	—	62	4	no	—
156	C₂×C₈	—	62	4	no	—
157	C₂×C₈	—	62	2	no	—
158	C₂×C₈	C₂²×D₃₁	1	2	yes	—
159	C₂³	—	31	2	no	—
160	C₂³	—	31	1	no	—
161	C₂³	—	31	2	no	—
162	C₂³	—	31	1	no	—
163	C₂³	—	124	2	no	—
164	C₂³	Q₈×D₃₁	1	1	yes	characteristic
165	C₂×C₄	—	31	2	no	—
166	C₂×C₄	—	31	2	no	—
167	C₂×C₄	—	31	1	no	—
168	C₂×C₄	—	31	1	no	—
169	C₂×C₄	—	31	2	no	—
170	C₂×C₄	—	31	2	no	—
171	C₂×C₄	—	31	2	no	—
172	C₂×C₄	—	31	2	no	—
173	C₂×C₄	—	31	1	no	—
174	C₂×C₄	—	31	1	no	—
175	C₂×C₄	—	31	2	no	—
176	C₂×C₄	—	31	1	no	—
177	C₂×C₄	—	31	2	no	—
178	C₂×C₄	—	31	1	no	—
179	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
180	C₂×C₄	D₄×D₃₁	1	2	yes	—
181	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
182	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	31	4	no	—
185	C₈	—	62	4	no	—
186	C₈	C₂×C₄×D₃₁	1	4	yes	—
187	C₂²	—	4	1	no	—
188	C₂²	—	31	1	no	—
189	C₂²	—	31	1	no	—
190	C₂²	—	31	2	no	—
191	C₂²	—	62	4	no	—
192	C₂²	—	62	2	no	—
193	C₂²	—	124	2	no	—
194	C₂²	—	124	1	no	—
195	C₂²	C₄⋊C₄×D₃₁	1	2	yes	—
196	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
197	C₄	—	31	1	no	—
198	C₄	—	31	1	no	—
199	C₄	—	31	2	no	—
200	C₄	C₄⋊C₄×D₃₁	1	2	yes	—
201	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
202	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	2	1	no	—
205	C₂	—	31	2	no	—
206	C₂	—	62	1	no	—
207	C₂	—	62	2	no	—
208	C₂	C₂×C₄⋊C₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#430	C₂
socle	C₂×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂³×D₃₁	Q₈
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#430	D₃₁
hypocenter	C₃₁	C₂×#430
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
3. #14521098	C₂	semidirect
4. #14521097	C₂	semidirect
5. #14521096	C₂	semidirect
6. #108×D₃₁	C₂	semidirect
7. C₃₁×#430	C₂	semidirect
23. C₃₁×#108	C₂²	semidirect
24. #11758410	C₂²	semidirect
25. #11758407	C₂²	semidirect
104. #430	D₃₁	direct (N × H)
111. D₃₁	#430	direct (N × H)
129. #108	C₂×D₃₁	semidirect
130. C₃₁	C₂×#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₃₁	2
#14521098	2
#14521097	2
#14521096	2
#108×D₃₁	2
C₃₁×#430	2
C₂×#430	31

Subgroups — D31×#430 #16854315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#430 #16854315